CFRG R. L. Barnes
InternetDraft Cisco
Intended status: Informational D. Cook
Expires: 9 January 2025 ISRG
C. Patton
Cloudflare
P. Schoppmann
Google
8 July 2024
Verifiable Distributed Aggregation Functions
draftirtfcfrgvdaf10
Abstract
This document describes Verifiable Distributed Aggregation Functions
(VDAFs), a family of multiparty protocols for computing aggregate
statistics over user measurements. These protocols are designed to
ensure that, as long as at least one aggregation server executes the
protocol honestly, individual measurements are never seen by any
server in the clear. At the same time, VDAFs allow the servers to
detect if a malicious or misconfigured client submitted an
measurement that would result in an invalid aggregate result.
Discussion Venues
This note is to be removed before publishing as an RFC.
Discussion of this document takes place on the Crypto Forum Research
Group mailing list (cfrg@ietf.org), which is archived at
https://mailarchive.ietf.org/arch/search/?email_list=cfrg.
Source for this draft and an issue tracker can be found at
https://github.com/cjpatton/vdaf.
Status of This Memo
This InternetDraft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
InternetDrafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
working documents as InternetDrafts. The list of current Internet
Drafts is at https://datatracker.ietf.org/drafts/current/.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1. Change Log . . . . . . . . . . . . . . . . . . . . . . . 8
2. Conventions and Definitions . . . . . . . . . . . . . . . . . 13
3. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4. Definition of DAFs . . . . . . . . . . . . . . . . . . . . . 17
4.1. Sharding . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2. Preparation . . . . . . . . . . . . . . . . . . . . . . . 21
4.3. Validity of Aggregation Parameters . . . . . . . . . . . 21
4.4. Aggregation . . . . . . . . . . . . . . . . . . . . . . . 21
4.5. Unsharding . . . . . . . . . . . . . . . . . . . . . . . 22
4.6. Execution of a DAF . . . . . . . . . . . . . . . . . . . 23
5. Definition of VDAFs . . . . . . . . . . . . . . . . . . . . . 25
5.1. Sharding . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2. Preparation . . . . . . . . . . . . . . . . . . . . . . . 28
5.3. Validity of Aggregation Parameters . . . . . . . . . . . 30
5.4. Aggregation . . . . . . . . . . . . . . . . . . . . . . . 30
5.5. Unsharding . . . . . . . . . . . . . . . . . . . . . . . 31
5.6. Execution of a VDAF . . . . . . . . . . . . . . . . . . . 31
5.7. Communication Patterns for Preparation . . . . . . . . . 33
5.8. PingPong Topology (Only Two Aggregators) . . . . . . . . 35
5.9. Star Topology (Any Number of Aggregators) . . . . . . . . 41
6. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 41
6.1. Finite Fields . . . . . . . . . . . . . . . . . . . . . . 41
6.1.1. Auxiliary Functions . . . . . . . . . . . . . . . . . 44
6.1.2. FFTFriendly Fields . . . . . . . . . . . . . . . . . 45
6.1.3. Parameters . . . . . . . . . . . . . . . . . . . . . 45
6.2. Extendable Output Functions . . . . . . . . . . . . . . . 46
6.2.1. XofTurboShake128 . . . . . . . . . . . . . . . . . . 48
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6.2.2. XofFixedKeyAes128 . . . . . . . . . . . . . . . . . . 48
6.2.3. The Domain Separation Tag and Binder String . . . . . 50
7. Prio3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.1. Fully Linear Proof (FLP) Systems . . . . . . . . . . . . 52
7.1.1. Encoding the Input . . . . . . . . . . . . . . . . . 56
7.1.2. Multiple Proofs . . . . . . . . . . . . . . . . . . . 57
7.2. Construction . . . . . . . . . . . . . . . . . . . . . . 58
7.2.1. Sharding . . . . . . . . . . . . . . . . . . . . . . 60
7.2.2. Preparation . . . . . . . . . . . . . . . . . . . . . 66
7.2.3. Validity of Aggregation Parameters . . . . . . . . . 68
7.2.4. Aggregation . . . . . . . . . . . . . . . . . . . . . 69
7.2.5. Unsharding . . . . . . . . . . . . . . . . . . . . . 69
7.2.6. Auxiliary Functions . . . . . . . . . . . . . . . . . 69
7.2.7. Message Serialization . . . . . . . . . . . . . . . . 71
7.3. The FLP of BBCGGI19 . . . . . . . . . . . . . . . . . . . 74
7.3.1. Overview . . . . . . . . . . . . . . . . . . . . . . 74
7.3.2. Validity Circuits . . . . . . . . . . . . . . . . . . 77
7.3.3. Construction . . . . . . . . . . . . . . . . . . . . 79
7.4. Instantiations . . . . . . . . . . . . . . . . . . . . . 83
7.4.1. Prio3Count . . . . . . . . . . . . . . . . . . . . . 83
7.4.2. Prio3Sum . . . . . . . . . . . . . . . . . . . . . . 84
7.4.3. Prio3SumVec . . . . . . . . . . . . . . . . . . . . . 86
7.4.4. Prio3Histogram . . . . . . . . . . . . . . . . . . . 91
7.4.5. Prio3MultihotCountVec . . . . . . . . . . . . . . . . 94
8. Poplar1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8.1. Incremental Distributed Point Functions (IDPFs) . . . . . 100
8.1.1. Encoding inputs as indices . . . . . . . . . . . . . 103
8.2. Construction . . . . . . . . . . . . . . . . . . . . . . 103
8.2.1. Sharding . . . . . . . . . . . . . . . . . . . . . . 105
8.2.2. Preparation . . . . . . . . . . . . . . . . . . . . . 108
8.2.3. Validity of Aggregation Parameters . . . . . . . . . 111
8.2.4. Aggregation . . . . . . . . . . . . . . . . . . . . . 112
8.2.5. Unsharding . . . . . . . . . . . . . . . . . . . . . 113
8.2.6. Message Serialization . . . . . . . . . . . . . . . . 113
8.3. The IDPF scheme of BBCGGI21 . . . . . . . . . . . . . . . 117
8.3.1. Key Generation . . . . . . . . . . . . . . . . . . . 117
8.3.2. Key Evaluation . . . . . . . . . . . . . . . . . . . 119
8.3.3. Auxiliary Functions . . . . . . . . . . . . . . . . . 121
8.4. Instantiation . . . . . . . . . . . . . . . . . . . . . . 123
9. Security Considerations . . . . . . . . . . . . . . . . . . . 123
9.1. Requirements for the Verification Key . . . . . . . . . . 125
9.2. Requirements for the Nonce . . . . . . . . . . . . . . . 126
9.3. Requirements for the Public Share . . . . . . . . . . . . 126
9.4. Requirements for Aggregation Parameters . . . . . . . . . 126
9.4.1. Additional Privacy Considerations . . . . . . . . . . 126
9.4.2. Safe Usage of IDPF Outputs . . . . . . . . . . . . . 127
9.5. Requirements for XOFs . . . . . . . . . . . . . . . . . . 128
9.6. Choosing the Field Size . . . . . . . . . . . . . . . . . 129
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10. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 129
11. References . . . . . . . . . . . . . . . . . . . . . . . . . 130
11.1. Normative References . . . . . . . . . . . . . . . . . . 130
11.2. Informative References . . . . . . . . . . . . . . . . . 131
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 133
Test Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Prio3Count . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Prio3Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Prio3SumVec . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Prio3Histogram . . . . . . . . . . . . . . . . . . . . . . . . 134
Poplar1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 134
1. Introduction
(TO BE REMOVED BY RFC EDITOR: The source for this draft and and the
reference implementation can be found at https://github.com/cfrg/
draftirtfcfrgvdaf.)
The ubiquity of the Internet makes it an ideal platform for
measurement of largescale phenomena, whether public health trends or
the behavior of computer systems at scale. There is substantial
overlap, however, between information that is valuable to measure and
information that users consider private.
For example, consider an application that provides health information
to users. The operator of an application might want to know which
parts of their application are used most often, as a way to guide
future development of the application. Specific users' patterns of
usage, though, could reveal sensitive things about them, such as
which users are researching a given health condition.
In many situations, the measurement collector is only interested in
aggregate statistics, e.g., which portions of an application are most
used or what fraction of people have experienced a given disease.
Thus systems that provide aggregate statistics while protecting
individual measurements can deliver the value of the measurements
while protecting users' privacy.
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This problem is often formulated in terms of differential privacy
(DP) [Dwo06]. Roughly speaking, a data aggregation system that is
differentially private ensures that the degree to which any
individual measurement influences the value of the aggregate result
can be precisely controlled. For example, in systems like RAPPOR
[EPK14], each user samples noise from a wellknown distribution and
adds it to their measurement before submitting to the aggregation
server. The aggregation server then adds up the noisy measurements,
and because it knows the distribution from which the noise was
sampled, it can estimate the true sum with reasonable accuracy.
However, even when noise is added to the measurements, collecting
them in the clear still reveals a significant amount of information
to the collector. On the one hand, depending on the "amount" of
noise a client adds to its measurement, it may be possible for a
curious collector to make a reasonable guess of the measurement's
true value. On the other hand, the more noise the clients add, the
less reliable will be the server's estimate of the output. Thus
systems relying solely on a DP mechanism must strike a delicate
balance between privacy and utility.
The ideal goal for a privacypreserving measurement system is that of
secure multiparty computation (MPC): No participant in the protocol
should learn anything about an individual measurement beyond what it
can deduce from the differentially private aggregate [MPRV09]. In
this document, we describe Verifiable Distributed Aggregation
Functions (VDAFs) as a general class of delegated MPC protocols that
can be used to achieve this goal.
VDAF schemes achieve their privacy goal by distributing the
computation of the aggregate among a number of noncolluding
aggregation servers. As long as a subset of the servers executes the
protocol honestly, VDAFs guarantee that no measurement is ever
accessible to any party besides the client that submitted it. VDAFs
can also be composed with various DP mechanisms, thereby ensuring the
aggregate result does not leak too much information about any one
measurmment. At the same time, VDAFs are "verifiable" in the sense
that malformed measurements that would otherwise garble the result of
the computation can be detected and removed from the set of
measurements. We refer to this property as "robustness".
The cost of achieving these security properties is the need for
multiple servers to participate in the protocol, and the need to
ensure they do not collude to undermine the VDAF's privacy
guarantees. Recent implementation experience has shown that
practical challenges of coordinating multiple servers can be
overcome. The Prio system [CGB17] (essentially a VDAF) has been
deployed in systems supporting hundreds of millions of users: The
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Mozilla Origin Telemetry project [OriginTelemetry] and the Exposure
Notification Private Analytics collaboration among the Internet
Security Research Group (ISRG), Google, Apple, and others [ENPA].
The VDAF abstraction laid out in Section 5 represents a class of
multiparty protocols for privacypreserving measurement proposed in
the literature. These protocols vary in their operational and
security requirements, sometimes in subtle but consequential ways.
This document therefore has two important goals:
1. Providing higherlevel protocols like [DAP] with a simple,
uniform interface for accessing privacypreserving measurement
schemes, documenting relevant operational and security
requirements, and specifying constraints for safe usage:
1. General patterns of communications among the various actors
involved in the system (clients, aggregation servers, and the
collector of the aggregate result);
2. Capabilities of a malicious coalition of servers attempting
to divulge information about client measurements; and
3. Conditions that are necessary to ensure that malicious
clients cannot corrupt the computation.
2. Providing cryptographers with design criteria that provide a
clear deployment roadmap for new constructions.
This document also specifies two concrete VDAF schemes, each based on
a protocol from the literature.
* The aforementioned Prio system [CGB17] allows for the privacy
preserving computation of a variety aggregate statistics. The
basic idea underlying Prio is fairly simple:
1. Each client shards its measurement into a sequence of additive
shares and distributes the shares among the aggregation
servers.
2. Next, each server adds up its shares locally, resulting in an
additive share of the aggregate.
3. Finally, the aggregation servers send their aggregate shares
to the data collector, who combines them to obtain the
aggregate result.
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The difficult part of this system is ensuring that the servers
hold shares of a valid, aggregatable value, e.g., the measurement
is an integer in a specific range. Thus Prio specifies a multi
party protocol for accomplishing this task.
In Section 7 we describe Prio3, a VDAF that follows the same
overall framework as the original Prio protocol, but incorporates
techniques introduced in [BBCGGI19] that result in significant
performance gains.
* More recently, Boneh et al. [BBCGGI21] described a protocol
called Poplar for solving the theavyhitters problem in a
privacypreserving manner. Here each client holds a bitstring of
length n, and the goal of the aggregation servers is to compute
the set of strings that occur at least t times. The core
primitive used in their protocol is a specialized Distributed
Point Function (DPF) [GI14] that allows the servers to "query"
their DPF shares on any bitstring of length shorter than or equal
to n. As a result of this query, each of the servers has an
additive share of a bit indicating whether the string is a prefix
of the client's string. The protocol also specifies a multiparty
computation for verifying that at most one string among a set of
candidates is a prefix of the client's string.
In Section 8 we describe a VDAF called Poplar1 that implements
this functionality.
Finally, perhaps the most complex aspect of schemes like Prio3 and
Poplar1 is the process by which the clientgenerated measurements are
prepared for aggregation. Because these constructions are based on
secret sharing, the servers will be required to exchange some amount
of information in order to verify the measurement is valid and can be
aggregated. Depending on the construction, this process may require
multiple round trips over the network.
There are applications in which this verification step may not be
necessary, e.g., when the client's software is run a trusted
execution environment. To support these applications, this document
also defines Distributed Aggregation Functions (DAFs) as a simpler
class of protocols that aim to provide the same privacy guarantee as
VDAFs but fall short of being verifiable.
OPEN ISSUE Decide if we should give one or two example DAFs.
There are natural variants of Prio3 and Poplar1 that might be
worth describing.
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The remainder of this document is organized as follows: Section 3
gives a brief overview of DAFs and VDAFs; Section 4 defines the
syntax for DAFs; Section 5 defines the syntax for VDAFs; Section 6
defines various functionalities that are common to our constructions;
Section 7 describes the Prio3 construction; Section 8 describes the
Poplar1 construction; and Section 9 enumerates the security
considerations for VDAFs.
1.1. Change Log
(*) Indicates a change that breaks wire compatibility with the
previous draft.
10:
* Define Prio3MultihotCountVec, a variant of Prio3 for aggregating
bit vectors with bounded weight.
* FLP: Allow the output of the circuit to be a vector. This makes
it possible to skip joint randomness derivation in more cases.
* Poplar1: On the first round of preparation, handle None as an
error. Previously this message was interpreted as a length3
vector of zeros.
* Prio3: Move specification of the field from the FLP validity
circuit to the VDAF itself.
* Clarify the extent to which the attacker controls the network in
our threat models for privacy and robustness.
* Clean up various aspects of the code, including: follow existing
objectoriented programming patterns for Python more closely; make
the type hints enforceable; and avoid shadowing variables.
* Poplar1: Align terminology with [BBCGGI23].
* IDPF: Add guidance for encoding byte strings as indices.
09:
* Poplar1: Make prefix tree traversal stricter by requiring each
node to be a child of a node that was already visited. This
change is intended to make it harder for a malicious Aggregator to
steer traversal towards nonheavyhitting measurements.
* Prio3: Add more explicit guidance for choosing the field size.
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* IDPF: Define extractability and clarify (un)safe usage of
intermediate prefix counts. Accordingly, add text ensuring public
share consistency to security considerations.
08:
* Poplar1: Bind the report nonce to the authenticator vector
programmed into the IDPF. (*)
* IdpfPoplar: Modify extend() by stealing each control bit from its
corresponding seed. This improves performance by reducing the
number of AES calls per level from 3 to 2. The cost is a slight
reduction in the concrete privacy bound. (*)
* Prio3: Add support for generating and verifying mutliple proofs
per measurement. This enables a tradeoff between communication
cost and runtime: if more proofs are used, then a smaller field
can be used without impacting robustness. (*)
* Replace SHAKE128 with TurboSHAKE128. (*)
07:
* Rename PRG to XOF ("eXtendable Output Function"). Accordingly,
rename PrgSha3 to XofShake128 and PrgFixedKeyAes128 to
XofFixedKeyAes128. "PRG" is a misnomer since we don't actually
treat this object as a pseudorandom generator in existing security
analysis.
* Replace cSHAKE128 with SHAKE128, reimplementing domain separation
for the customization string using a simpler scheme. This change
addresses the reality that implementations of cSHAKE128 are less
common. (*)
* Define a new VDAF, called Prio3SumVec, that generalizes Prio3Sum
to a vector of summands.
* Prio3Histogram: Update the codepoint and use the parallel sum
optimization introduced by Prio3SumVec to reduce the proof size.
(*)
* Daf, Vdaf: Rename interface methods to match verbiage in the
draft.
* Daf: Align with Vdaf by adding a nonce to shard() and prep().
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* Vdaf: Have prep_init() compute the first prep share. This change
is intended to simplify the interface by making the input to
prep_next() not optional.
* Prio3: Split sharding into two auxiliary functions, one for
sharding with joint randomness and another without. This change
is intended to improve readability.
* Fix bugs in the pingpong interface discovered after implementing
it.
06:
* Vdaf: Define a wrapper interface for preparation that is suitable
for the "pingpong" topology in which two Aggregators exchange
messages over a request/response protocol, like HTTP, and take
turns executing the computation until input from the peer is
required.
* Prio3Histogram: Generalize the measurement type so that the
histogram can be used more easily with discrete domains. (*)
* Daf, Vdaf: Change the aggregation parameter validation algorithm
to take the set of previous parameters rather than a list. (The
order of the parameters is irrelevant.)
* Daf, Vdaf, Idpf: Add parameter RAND_SIZE that specifies the number
of random bytes consumed by the randomized algorithm (shard() for
Daf and Vdaf and gen() for Idpf).
05:
* IdpfPoplar: Replace PrgSha3 with PrgFixedKeyAes128, a fixedkey
mode for AES128 based on a construction from [GKWWY20]. This
change is intended to improve performance of IDPF evaluation.
Note that the new PRG is not suitable for all applications. (*)
* Idpf: Add a binder string to the keygeneration and evaluation
algorithms. This is used to plumb the nonce generated by the
Client to the PRG.
* Plumb random coins through the interface of randomized algorithms.
Specifically, add a random input to (V)DAF sharding algorithm and
IDPF keygeneration algorithm and require implementations to
specify the length of the random input. Accordingly, update
Prio3, Poplar1, and IdpfPoplar to match the new interface. This
change is intended to improve coverage of test vectors.
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* Use littleendian byteorder for field element encoding. (*)
* Poplar1: Move the last step of sketch evaluation from prep_next()
to prep_shares_to_prep().
04:
* Align security considerations with the security analysis of
[DPRS23].
* Vdaf: Pass the nonce to the sharding algorithm.
* Vdaf: Rather than allow the application to choose the nonce
length, have each implementation of the Vdaf interface specify the
expected nonce length. (*)
* Prg: Split "info string" into two components: the "customization
string", intended for domain separation; and the "binder string",
used to bind the output to ephemeral values, like the nonce,
associated with execution of a (V)DAF.
* Replace PrgAes128 with PrgSha3, an implementation of the Prg
interface based on SHA3, and use the new scheme as the default.
Accordingly, replace Prio3Aes128Count with Prio3Count,
Poplar1Aes128 with Poplar1, and so on. SHA3 is a safer choice
for instantiating a random oracle, which is used in the analysis
of Prio3 of [DPRS23]. (*)
* Prio3, Poplar1: Ensure each invocation of the Prg uses a distinct
customization string, as suggested by [DPRS23]. This is intended
to make domain separation clearer, thereby simplifying security
analysis. (*)
* Prio3: Replace "joint randomness hints" sent in each input share
with "joint randomness parts" sent in the public share. This
reduces communication overhead when the number of shares exceeds
two. (*)
* Prio3: Bind nonce to joint randomness parts. This is intended to
address birthday attacks on robustness pointed out by [DPRS23].
(*)
* Poplar1: Use different Prg invocations for producing the
correlated randomness for inner and leaf nodes of the IDPF tree.
This is intended to simplify implementations. (*)
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* Poplar1: Don't bind the candidate prefixes to the verifier
randomness. This is intended to improve performance, while not
impacting security. According to the analysis of [DPRS23], it is
necessary to restrict Poplar1 usage such that no report is
aggregated more than once at a given level of the IDPF tree;
otherwise, attacks on privacy may be possible. In light of this
restriction, there is no added benefit of binding to the prefixes
themselves. (*)
* Poplar1: During preparation, assert that all candidate prefixes
are unique and appear in order. Uniqueness is required to avoid
erroneously rejecting a valid report; the ordering constraint
ensures the uniqueness check can be performed efficiently. (*)
* Poplar1: Increase the maximum candidate prefix count in the
encoding of the aggregation parameter. (*)
* Poplar1: Bind the nonce to the correlated randomness derivation.
This is intended to provide defenseindepth by ensuring the
Aggregators reject the report if the nonce does not match what the
Client used for sharding. (*)
* Poplar1: Clarify that the aggregation parameter encoding is
OPTIONAL. Accordingly, update implementation considerations
around crossaggregation state.
* IdpfPoplar: Add implementation considerations around branching on
the values of control bits.
* IdpfPoplar: When decoding the the control bits in the public
share, assert that the trailing bits of the final byte are all
zero. (*)
03:
* Define codepoints for (V)DAFs and use them for domain separation
in Prio3 and Poplar1. (*)
* Prio3: Align joint randomness computation with revised paper
[BBCGGI19]. This change mitigates an attack on robustness. (*)
* Prio3: Remove an intermediate PRG evaluation from query randomness
generation. (*)
* Add additional guidance for choosing FFTfriendly fields.
02:
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* Complete the initial specification of Poplar1.
* Extend (V)DAF syntax to include a "public share" output by the
Client and distributed to all of the Aggregators. This is to
accommodate "extractable" IDPFs as required for Poplar1. (See
[BBCGGI21], Section 4.3 for details.)
* Extend (V)DAF syntax to allow the unsharding step to take into
account the number of measurements aggregated.
* Extend FLP syntax by adding a method for decoding the aggregate
result from a vector of field elements. The new method takes into
account the number of measurements.
* Prio3: Align aggregate result computation with updated FLP syntax.
* Prg: Add a method for statefully generating a vector of field
elements.
* Field: Require that field elements are fully reduced before
decoding. (*)
* Define new field Field255.
01:
* Require that VDAFs specify serialization of aggregate shares.
* Define Distributed Aggregation Functions (DAFs).
* Prio3: Move proof verifier check from prep_next() to
prep_shares_to_prep(). (*)
* Remove public parameter and replace verification parameter with a
"verification key" and "Aggregator ID".
2. Conventions and Definitions
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in
BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
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Algorithms in this document are written in Python 3. Type hints are
used to define input and output types. The type variable F is used
in signatures to signify any type that is a subclass of Field. A
fatal error in a program (e.g., failure to parse one of the function
parameters) is usually handled by raising an exception.
A variable with type bytes is a byte string. This document defines
several bytestring constants. When comprised of printable ASCII
characters, they are written as Python 3 bytestring literals (e.g.,
b'some constant string').
A global constant VERSION of type int is defined, which algorithms
are free to use as desired. Its value SHALL be 8.
This document describes algorithms for multiparty computations in
which the parties typically communicate over a network. Wherever a
quantity is defined that must be be transmitted from one party to
another, this document prescribes a particular encoding of that
quantity as a byte string.
OPEN ISSUE It might be better to not be prescriptive about how
quantities are encoded on the wire. See issue #58.
Some common functionalities:
* zeros(len: int) > bytes returns an array of zero bytes. The
length of output MUST be len.
* gen_rand(len: int) > bytes returns an array of random bytes
generated by a cryptographically secure pseudorandom number
generator (CSPRNG). The length of output MUST be len.
* byte(int: int) > bytes returns the representation of int as a
byte string. The value of int MUST be in [0,256).
* concat(parts: list[bytes]) > bytes returns the concatenation of
the input byte strings, i.e., parts[0]  ...  parts[len(parts)
1].
* front(length: int, vec: list[Any]) > (list[Any], list[Any])
splits vec into two vectors, where the first vector is made up of
the first length elements of the input. I.e., (vec[:length],
vec[length:]).
* xor(left: bytes, right: bytes) > bytes returns the bitwise XOR of
left and right. An exception is raised if the inputs are not the
same length.
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* to_be_bytes(val: int, length: int) > bytes converts val to big
endian bytes; its value MUST be in range [0, 2^(8*length)).
Function from_be_bytes(encoded: bytes) > int computes the
inverse.
* to_le_bytes(val: int, length: int) > bytes converts val to
littleendian bytes; its value MUST be in range [0, 2^(8*length)).
Function from_le_bytes(encoded: bytes) > int computes the
inverse.
* next_power_of_2(n: int) > int returns the smallest integer
greater than or equal to n that is also a power of two.
* additive_secret_share(vec: list[Field], num_shares: int, field:
type) > list[list[Field]] takes a vector of field elements and
returns multiple vectors of the same length, such that they all
add up to the input vector, and each proper subset of the vectors
are indistinguishable from random.
* cast(typ: type, val: object) > object returns the input value
unchanged. This is only present to assist with static analysis of
the Python code. Type checkers will ignore the inferred type of
the input value, and assume the output value has the given type.
* range(stop) or range(start, stop[, step]) is the range function
from the Python standard library. The oneargument form returns
the integers from zero (inclusive) to stop, exclusive. The two
and threeargument forms allow overriding the start of the range
and overriding the step between successive output values.
3. Overview
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++
+> Aggregator 0 +
 ++ 
 ^ 
  
 V 
 ++ 
 +> Aggregator 1 + 
  ++  
+++  ^  +>++
 Client +  +> Collector > Aggregate
+++ +>++
 ... 
 
  
 V 
 ++ 
+> Aggregator N1 +
++
Input shares Aggregate shares
Figure 1: Overall data flow of a (V)DAF
In a DAF or VDAFbased private measurement system, we distinguish
three types of actors: Clients, Aggregators, and Collectors. The
overall flow of the measurement process is as follows:
* To submit an individual measurement, the Client shards the
measurement into "input shares" and sends one input share to each
Aggregator. We sometimes refer to this sequence of input shares
collectively as the Client's "report".
* The Aggregators refine their input shares into "output shares".
 Output shares are in onetoone correspondence with the input
shares.
 Just as each Aggregator receives one input share of each
measurement, if this process succeeds, then each aggregator
holds one output share.
 In VDAFs, Aggregators will need to exchange information among
themselves as part of the validation process.
* Each Aggregator combines the output shares in the batch to compute
the "aggregate share" for that batch, i.e., its share of the
desired aggregate result.
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* The Aggregators submit their aggregate shares to the Collector,
who combines them to obtain the aggregate result over the batch.
Aggregators are a new class of actor relative to traditional
measurement systems where Clients submit measurements to a single
server. They are critical for both the privacy properties of the
system and, in the case of VDAFs, the correctness of the measurements
obtained. The privacy properties of the system are assured by non
collusion among Aggregators, and Aggregators are the entities that
perform validation of Client measurements. Thus Clients trust
Aggregators not to collude (typically it is required that at least
one Aggregator is honest), and Collectors trust Aggregators to
correctly run the protocol.
Within the bounds of the noncollusion requirements of a given (V)DAF
instance, it is possible for the same entity to play more than one
role. For example, the Collector could also act as an Aggregator,
effectively using the other Aggregator(s) to augment a basic client
server protocol.
In this document, we describe the computations performed by the
actors in this system. It is up to the higherlevel protocol making
use of the (V)DAF to arrange for the required information to be
delivered to the proper actors in the proper sequence. In general,
we assume that all communications are confidential and mutually
authenticated, with the exception that Clients submitting
measurements may be anonymous.
4. Definition of DAFs
By way of a gentle introduction to VDAFs, this section describes a
simpler class of schemes called Distributed Aggregation Functions
(DAFs). Unlike VDAFs, DAFs do not provide verifiability of the
computation. Clients must therefore be trusted to compute their
input shares correctly. Because of this fact, the use of a DAF is
NOT RECOMMENDED for most applications. See Section 9 for additional
discussion.
A DAF scheme is used to compute a particular "aggregation function"
over a set of measurements generated by Clients. Depending on the
aggregation function, the Collector might select an "aggregation
parameter" and disseminates it to the Aggregators. The semantics of
this parameter is specific to the aggregation function, but in
general it is used to represent the set of "queries" that can be made
on the measurement set. For example, the aggregation parameter is
used to represent the candidate prefixes in Poplar1 Section 8.
Execution of a DAF has four distinct stages:
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* Sharding  Each Client generates input shares from its measurement
and distributes them among the Aggregators.
* Preparation  Each Aggregator converts each input share into an
output share compatible with the aggregation function. This
computation involves the aggregation parameter. In general, each
aggregation parameter may result in a different an output share.
* Aggregation  Each Aggregator combines a sequence of output shares
into its aggregate share and sends the aggregate share to the
Collector.
* Unsharding  The Collector combines the aggregate shares into the
aggregate result.
Sharding and Preparation are done once per measurement. Aggregation
and Unsharding are done over a batch of measurements (more precisely,
over the recovered output shares).
A concrete DAF specifies an algorithm for the computation needed in
each of these stages. The interface of each algorithm is defined in
the remainder of this section. In addition, a concrete DAF defines
the associated constants and types enumerated in the following table.
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+=============+=====================================================+
 Parameter  Description 
+=============+=====================================================+
 ID: int  Algorithm identifier for this 
  DAF, in range(2**32). 
+++
 SHARES: int  Number of input shares into which 
  each measurement is sharded. 
+++
 NONCE_SIZE:  Size of the nonce passed by the 
 int  application. 
+++
 RAND_SIZE:  Size of the random byte string 
 int  passed to sharding algorithm. 
+++
 Measurement  Type of each measurement. 
+++
 PublicShare  Type of each public share. 
+++
 InputShare  Type of each input share. 
+++
 AggParam  Type of aggregation parameter. 
+++
 OutShare  Type of each output share. 
+++
 AggShare  Type of the aggregate share. 
+++
 AggResult  Type of the aggregate result. 
+++
Table 1: Constants and types defined by each concrete DAF.
These types define the inputs and outputs of DAF methods at various
stages of the computation. Some of these values need to be written
to the network in order to carry out the computation. In particular,
it is RECOMMENDED that concrete instantiations of the Daf interface
specify a method of encoding the PublicShare, InputShare, and
AggShare.
Each DAF is identified by a unique, 32bit integer ID. Identifiers
for each (V)DAF specified in this document are defined in Table 23.
4.1. Sharding
In order to protect the privacy of its measurements, a DAF Client
shards its measurements into a sequence of input shares. The shard
method is used for this purpose.
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* daf.shard(measurement: Measurement, nonce: bytes, rand: bytes) >
tuple[PublicShare, list[InputShare]] is the randomized sharding
algorithm run by each Client that consumes a measurement and nonce
and produces a "public share" distributed to each of the Aggregate
and a corresponding sequence of input shares, one for each
Aggregator.
Preconditions:
 nonce MUST have length equal to NONCE_SIZE and MUST be
generated using a CSPRNG.
 rand consists of the random bytes consumed by the algorithm.
It MUST have length equal to RAND_SIZE and MUST be generated
using a CSPRNG.
Postconditions:
 The number of input shares MUST equal SHARES.
Client
======
measurement

V
++
 shard 
++
   
  ...  public_share
   
 +++
     
V  V  V 
input_share_0 input_share_1 input_share_[SHARES1]
    ...  
V V V V V V
Aggregator 0 Aggregator 1 Aggregator SHARES1
Figure 2: The Client divides its measurement into input shares
and distributes them to the Aggregators. The public share is
broadcast to all Aggregators.
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4.2. Preparation
Once an Aggregator has received the public share and one of the input
shares, the next step is to prepare the input share for aggregation.
This is accomplished using the following algorithm:
* daf.prep(agg_id: int, agg_param: AggParam, nonce: bytes,
public_share: PublicShare, input_share: InputShare) > OutShare is
the deterministic preparation algorithm. It takes as input the
public share and one of the input shares generated by a Client,
the Aggregator's unique identifier, the aggregation parameter
selected by the Collector, and a nonce and returns an output
share.
Preconditions:
 agg_id MUST be in range range(SHARES) and match the index of
input_share in the sequence of input shares produced by the
Client.
 nonce MUST have length NONCE_SIZE.
4.3. Validity of Aggregation Parameters
Concrete DAFs implementations MAY impose certain restrictions for
input shares and aggregation parameters. Protocols using a DAF MUST
ensure that for each input share and aggregation parameter agg_param,
daf.prep is only called if daf.is_valid(agg_param,
previous_agg_params) returns True, where previous_agg_params contains
all aggregation parameters that have previously been used with the
same input share.
DAFs MUST implement the following function:
* daf.is_valid(agg_param: AggParam, previous_agg_params:
set[AggParam]) > bool: Checks if the agg_param is compatible with
all elements of previous_agg_params.
4.4. Aggregation
Once an Aggregator holds output shares for a batch of measurements
(where batches are defined by the application), it combines them into
a share of the desired aggregate result:
* daf.aggregate(agg_param: AggParam, out_shares: list[OutShare]) >
AggShare is the deterministic aggregation algorithm. It is run by
each Aggregator a set of recovered output shares.
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Aggregator 0 Aggregator 1 Aggregator SHARES1
============ ============ ===================
out_share_0_0 out_share_1_0 out_share_[SHARES1]_0
out_share_0_1 out_share_1_1 out_share_[SHARES1]_1
out_share_0_2 out_share_1_2 out_share_[SHARES1]_2
... ... ...
out_share_0_B out_share_1_B out_share_[SHARES1]_B
  
V V V
++ ++ ++
 aggregate   aggregate  ...  aggregate 
++ ++ ++
  
V V V
agg_share_0 agg_share_1 agg_share_[SHARES1]
Figure 3: Aggregation of output shares. `B` indicates the number of
measurements in the batch.
For simplicity, we have written this algorithm in a "oneshot" form,
where all output shares for a batch are provided at the same time.
Many DAFs may also support a "streaming" form, where shares are
processed one at a time.
Implementation note: For most natural DAFs (and VDAFs) it is not
necessary for an Aggregator to store all output shares individually
before aggregating. Typically it is possible to merge output shares
into aggregate shares as they arrive, merge these into other
aggregate shares, and so on. In particular, this is the case when
the output shares are vectors over some finite field and aggregating
them involves merely adding up the vectors elementwise. Such is the
case for Prio3 Section 7 and Poplar1 Section 8.
4.5. Unsharding
After the Aggregators have aggregated a sufficient number of output
shares, each sends its aggregate share to the Collector, who runs the
following algorithm to recover the following output:
* daf.unshard(agg_param: AggParam, agg_shares: list[AggShare],
num_measurements: int) > AggResult is run by the Collector in
order to compute the aggregate result from the Aggregators'
shares. The length of agg_shares MUST be SHARES. num_measurements
is the number of measurements that contributed to each of the
aggregate shares. This algorithm is deterministic.
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Aggregator 0 Aggregator 1 Aggregator SHARES1
============ ============ ===================
agg_share_0 agg_share_1 agg_share_[SHARES1]
  
V V V
++
 unshard 
++

V
agg_result
Collector
=========
Figure 4: Computation of the final aggregate result from
aggregate shares.
4.6. Execution of a DAF
Securely executing a DAF involves emulating the following procedure.
def run_daf(
daf: Daf[
Measurement,
AggParam,
PublicShare,
InputShare,
OutShare,
AggShare,
AggResult,
],
agg_param: AggParam,
measurements: list[Measurement],
nonces: list[bytes]) > AggResult:
"""
Run a DAF on a list of measurements.
Preconditions:
 `type(agg_param) == daf.AggParam`
 `type(measurement) == daf.Measurement` for each
`measurement` in `measurements`
 `len(nonce) == daf.NONCE_SIZE` for each `nonce` in `nonces`
 `len(nonces) == len(measurements)`
"""
if any(len(nonce) != daf.NONCE_SIZE for nonce in nonces):
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raise ValueError("incorrect nonce size")
if len(nonces) != len(measurements):
raise ValueError(
"measurements and nonces lists have different lengths"
)
out_shares: list[list[OutShare]]
out_shares = [[] for j in range(daf.SHARES)]
for (measurement, nonce) in zip(measurements, nonces):
# Each Client shards its measurement into input shares and
# distributes them among the Aggregators.
rand = gen_rand(daf.RAND_SIZE)
(public_share, input_shares) = \
daf.shard(measurement, nonce, rand)
# Each Aggregator prepares its input share for aggregation.
for j in range(daf.SHARES):
out_shares[j].append(
daf.prep(j, agg_param, nonce,
public_share, input_shares[j]))
# Each Aggregator aggregates its output shares into an aggregate
# share and sends it to the Collector.
agg_shares = []
for j in range(daf.SHARES):
agg_share_j = daf.aggregate(agg_param,
out_shares[j])
agg_shares.append(agg_share_j)
# Collector unshards the aggregate result.
num_measurements = len(measurements)
agg_result = daf.unshard(agg_param, agg_shares,
num_measurements)
return agg_result
Figure 5: Execution of a DAF.
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The inputs to this procedure are the same as the aggregation function
computed by the DAF: An aggregation parameter and a sequence of
measurements. The procedure prescribes how a DAF is executed in a
"benign" environment in which there is no adversary and the messages
are passed among the protocol participants over secure pointtopoint
channels. In reality, these channels need to be instantiated by some
"wrapper protocol", such as [DAP], that realizes these channels using
suitable cryptographic mechanisms. Moreover, some fraction of the
Aggregators (or Clients) may be malicious and diverge from their
prescribed behaviors. Section 9 describes the execution of the DAF
in various adversarial environments and what properties the wrapper
protocol needs to provide in each.
5. Definition of VDAFs
Like DAFs described in the previous section, a VDAF scheme is used to
compute a particular aggregation function over a set of Client
generated measurements. Evaluation of a VDAF involves the same four
stages as for DAFs: Sharding, Preparation, Aggregation, and
Unsharding. However, the Preparation stage will require interaction
among the Aggregators in order to facilitate verifiability of the
computation's correctness. Accommodating this interaction will
require syntactic changes.
Overall execution of a VDAF comprises the following stages:
* Sharding  Computing input shares from an individual measurement
* Preparation  Conversion and verification of input shares to
output shares compatible with the aggregation function being
computed
* Aggregation  Combining a sequence of output shares into an
aggregate share
* Unsharding  Combining a sequence of aggregate shares into an
aggregate result
In contrast to DAFs, the Preparation stage for VDAFs now performs an
additional task: Verification of the validity of the recovered output
shares. This process ensures that aggregating the output shares will
not lead to a garbled aggregate result.
The remainder of this section defines the VDAF interface. The
attributes are listed in Table 2 are defined by each concrete VDAF.
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+=================+==========================================+
 Parameter  Description 
+=================+==========================================+
 ID  Algorithm identifier for this VDAF. 
+++
 VERIFY_KEY_SIZE  Size (in bytes) of the verification key 
  (Section 5.2). 
+++
 RAND_SIZE  Size of the random byte string passed to 
  sharding algorithm. 
+++
 NONCE_SIZE  Size (in bytes) of the nonce. 
+++
 ROUNDS  Number of rounds of communication during 
  the Preparation stage (Section 5.2). 
+++
 SHARES  Number of input shares into which each 
  measurement is sharded (Section 5.1). 
+++
 Measurement  Type of each measurement. 
+++
 PublicShare  Type of each public share. 
+++
 InputShare  Type of each input share. 
+++
 AggParam  Type of aggregation parameter. 
+++
 OutShare  Type of each output share. 
+++
 AggShare  Type of the aggregate share. 
+++
 AggResult  Type of the aggregate result. 
+++
 PrepState  Aggregator's state during preparation. 
+++
 PrepShare  Type of each prep share. 
+++
 PrepMessage  Type of each prep message. 
+++
Table 2: Constants and types defined by each concrete VDAF.
Some of these values need to be written to the network in order to
carry out the computation. In particular, it is RECOMMENDED that
concrete instantiations of the Vdaf interface specify a method of
encoding the PublicShare, InputShare, AggShare, PrepShare, and
PrepMessage.
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Each VDAF is identified by a unique, 32bit integer ID. Identifiers
for each (V)DAF specified in this document are defined in Table 23.
The following method is defined for every VDAF:
def domain_separation_tag(self, usage: int) > bytes:
"""
Format domain separation tag for this VDAF with the given usage.
Preconditions:
 `usage` in `range(2**16)`
"""
return format_dst(0, self.ID, usage)
It is used to construct a domain separation tag for an instance of
Xof used by the VDAF. (See Section 6.2.)
5.1. Sharding
Sharding transforms a measurement into input shares as it does in
DAFs (cf. Section 4.1); in addition, it takes a nonce as input and
produces a public share:
* vdaf.shard(measurement: Measurement, nonce: bytes, rand: bytes) >
tuple[PublicShare, list[InputShare]] is the randomized sharding
algorithm run by each Client that consumes a measurement and nonce
and produces a public share distributed to each of the Aggregate
and a corresponding sequence of input shares, one for each
Aggregator. Depending on the VDAF, the input shares may encode
additional information used to verify the recovered output shares
(e.g., the "proof shares" in Prio3 Section 7)
Preconditions:
 nonce MUST have length equal to NONCE_SIZE and MUST be
generated using a CSPRNG. (See Section 9 for details.)
 rand consists of the random bytes consumed by the algorithm.
It MUST have length equal to RAND_SIZE and MUST be generated
using a CSPRNG.
Postconditions:
 The number of input shares MUST equal SHARES.
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5.2. Preparation
To recover and verify output shares, the Aggregators interact with
one another over ROUNDS rounds. Prior to each round, each Aggregator
constructs an outbound message. Next, the sequence of outbound
messages is combined into a single message, called a "preparation
message", or "prep message" for short. (Each of the outbound
messages are called "preparationmessage shares", or "prep shares"
for short.) Finally, the preparation message is distributed to the
Aggregators to begin the next round.
An Aggregator begins the first round with its input share and it
begins each subsequent round with the previous prep message. Its
output in the last round is its output share and its output in each
of the preceding rounds is a prep share.
This process involves a value called the "aggregation parameter" used
to map the input shares to output shares. The Aggregators need to
agree on this parameter before they can begin preparing the
measurement shares for aggregation.
Aggregator 0 Aggregator 1 Aggregator SHARES1
============ ============ ===================
input_share_0 input_share_1 input_share_[SHARES1]
  ... 
V V V
++ ++ ++
 prep_init   prep_init   prep_init 
++ ++ ++
  ... 
V V V
++ \
 prep_shares_to_prep  
++ 
  ...  
V V V  x ROUNDS
++ ++ ++ 
 prep_next   prep_next   prep_next  
++ ++ ++ 
  ...  
V V V /
... ... ...
  ... 
V V V
out_share_0 out_share_1 out_share_[SHARES1]
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Figure 6: VDAF preparation process on the input shares for a single
measurement. At the end of the computation, each Aggregator holds an
output share or an error.
To facilitate the preparation process, a concrete VDAF implements the
following methods:
* vdaf.prep_init(verify_key: bytes, agg_id: int, agg_param:
AggParam, nonce: bytes, public_share: PublicShare, input_share:
InputShare) > tuple[PrepState, PrepShare] is the deterministic
preparationstate initialization algorithm run by each Aggregator
to begin processing its input share into an output share. Its
inputs are the shared verification key (verify_key), the
Aggregator's unique identifier (agg_id), the aggregation parameter
(agg_param), the nonce provided by the environment (nonce, see
Figure 7), the public share (public_share), and one of the input
shares generated by the Client (input_share). Its output is the
Aggregator's initial preparation state and initial prep share.
It is up to the high level protocol in which the VDAF is used to
arrange for the distribution of the verification key prior to
generating and processing reports. (See Section 9 for details.)
Protocols MUST ensure that public share consumed by each of the
Aggregators is identical. This is security critical for VDAFs
such as Poplar1.
Preconditions:
 verify_key MUST have length vdaf.VERIFY_KEY_SIZE.
 agg_id MUST be the integer in range(0, vdaf.SHARES) that
matches the index of input_share in the sequence of input
shares output by the Client.
 nonce MUST have length vdaf.NONCE_SIZE.
* vdaf.prep_next(prep_state: PrepState, prep_msg: PrepMessage) >
Union[tuple[PrepState, PrepShare], OutShare] is the deterministic
preparationstate update algorithm run by each Aggregator. It
updates the Aggregator's preparation state (prep_state) and
returns either its next preparation state and its message share
for the current round or, if this is the last round, its output
share. An exception is raised if a valid output share could not
be recovered. The input of this algorithm is the inbound
preparation message.
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* vdaf.prep_shares_to_prep(agg_param: AggParam, prep_shares:
list[PrepShare]) > PrepMessage is the deterministic preparation
message preprocessing algorithm. It combines the prep shares
generated by the Aggregators in the previous round into the prep
message consumed by each in the next round.
In effect, each Aggregator moves through a linear state machine with
ROUNDS states. The Aggregator enters the first state on using the
initialization algorithm, and the update algorithm advances the
Aggregator to the next state. Thus, in addition to defining the
number of rounds (ROUNDS), a VDAF instance defines the state of the
Aggregator after each round.
The preparationstate update accomplishes two tasks: recovery of
output shares from the input shares and ensuring that the recovered
output shares are valid. The abstraction boundary is drawn so that
an Aggregator only recovers an output share if it is deemed valid (at
least, based on the Aggregator's view of the protocol). Another way
to draw this boundary would be to have the Aggregators recover output
shares first, then verify that they are valid. However, this would
allow the possibility of misusing the API by, say, aggregating an
invalid output share. Moreover, in protocols like Prio+ [AGJOP21]
based on oblivious transfer, it is necessary for the Aggregators to
interact in order to recover aggregatable output shares at all.
5.3. Validity of Aggregation Parameters
Similar to DAFs (see Section 4.3), VDAFs MAY impose restrictions for
input shares and aggregation parameters. Protocols using a VDAF MUST
ensure that for each input share and aggregation parameter agg_param,
the preparation phase (including vdaf.prep_init, vdaf.prep_next, and
vdaf.prep_shares_to_prep; see Section 5.2) is only called if
vdaf.is_valid(agg_param, previous_agg_params) returns True, where
previous_agg_params contains all aggregation parameters that have
previously been used with the same input share.
VDAFs MUST implement the following function:
* vdaf.is_valid(agg_param: AggParam, previous_agg_params:
list[AggParam]) > bool: Checks if the agg_param is compatible
with all elements of previous_agg_params.
5.4. Aggregation
VDAF Aggregation is identical to DAF Aggregation (cf. Section 4.4):
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* vdaf.aggregate(agg_param: AggParam, out_shares: list[OutShare]) >
AggShare is the deterministic aggregation algorithm. It is run by
each Aggregator over the output shares it has computed for a batch
of measurements.
The data flow for this stage is illustrated in Figure 3. Here again,
we have the aggregation algorithm in a "oneshot" form, where all
shares for a batch are provided at the same time. VDAFs typically
also support a "streaming" form, where shares are processed one at a
time.
5.5. Unsharding
VDAF Unsharding is identical to DAF Unsharding (cf. Section 4.5):
* vdaf.unshard(agg_param: AggParam, agg_shares: list[AggShare],
num_measurements: int) > AggResult is run by the Collector in
order to compute the aggregate result from the Aggregators'
shares. The length of agg_shares MUST be SHARES. num_measurements
is the number of measurements that contributed to each of the
aggregate shares. This algorithm is deterministic.
The data flow for this stage is illustrated in Figure 4.
5.6. Execution of a VDAF
Secure execution of a VDAF involves simulating the following
procedure.
def run_vdaf(
vdaf: Vdaf[
Measurement,
AggParam,
PublicShare,
InputShare,
list[F], # OutShare
AggShare,
AggResult,
PrepState,
PrepShare,
PrepMessage,
],
verify_key: bytes,
agg_param: AggParam,
nonces: list[bytes],
measurements: list[Measurement]) > AggResult:
"""
Run the VDAF on a list of measurements.
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Preconditions:
 `len(verify_key) == vdaf.VERIFY_KEY_SIZE`
 `len(nonces) == len(measurements)`
 `all(len(nonce) == vdaf.NONCE_SIZE for nonce in nonces)`
"""
if len(verify_key) != vdaf.VERIFY_KEY_SIZE:
raise ValueError("incorrect verify_key size")
if any(len(nonce) != vdaf.NONCE_SIZE for nonce in nonces):
raise ValueError("incorrect nonce size")
if len(nonces) != len(measurements):
raise ValueError(
"measurements and nonces lists have different lengths"
)
out_shares = []
for (nonce, measurement) in zip(nonces, measurements):
assert len(nonce) == vdaf.NONCE_SIZE
# Each Client shards its measurement into input shares.
rand = gen_rand(vdaf.RAND_SIZE)
(public_share, input_shares) = \
vdaf.shard(measurement, nonce, rand)
# Each Aggregator initializes its preparation state.
prep_states = []
outbound_prep_shares = []
for j in range(vdaf.SHARES):
(state, share) = vdaf.prep_init(verify_key, j,
agg_param,
nonce,
public_share,
input_shares[j])
prep_states.append(state)
outbound_prep_shares.append(share)
# Aggregators recover their output shares.
for i in range(vdaf.ROUNDS  1):
prep_msg = vdaf.prep_shares_to_prep(agg_param,
outbound_prep_shares)
outbound_prep_shares = []
for j in range(vdaf.SHARES):
out = vdaf.prep_next(prep_states[j], prep_msg)
assert isinstance(out, tuple)
(prep_states[j], prep_share) = out
outbound_prep_shares.append(prep_share)
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# The final outputs of the prepare phase are the output
# shares.
prep_msg = vdaf.prep_shares_to_prep(agg_param,
outbound_prep_shares)
outbound_out_shares = []
for j in range(vdaf.SHARES):
out_share = vdaf.prep_next(prep_states[j], prep_msg)
assert not isinstance(out_share, tuple)
outbound_out_shares.append(out_share)
out_shares.append(outbound_out_shares)
# Each Aggregator aggregates its output shares into an
# aggregate share. In a distributed VDAF computation, the
# aggregate shares are sent over the network.
agg_shares = []
for j in range(vdaf.SHARES):
out_shares_j = [out[j] for out in out_shares]
agg_share_j = vdaf.aggregate(agg_param, out_shares_j)
agg_shares.append(agg_share_j)
# Collector unshards the aggregate.
num_measurements = len(measurements)
agg_result = vdaf.unshard(agg_param, agg_shares,
num_measurements)
return agg_result
Figure 7: Execution of a VDAF.
The inputs to this algorithm are the aggregation parameter, a list of
measurements, and a nonce for each measurement. This document does
not specify how the nonces are chosen, but security requires that the
nonces be unique. See Section 9 for details. As explained in
Section 4.6, the secure execution of a VDAF requires the application
to instantiate secure channels between each of the protocol
participants.
5.7. Communication Patterns for Preparation
In each round of preparation, each Aggregator writes a prep share to
some broadcast channel, which is then processed into the prep message
using the public prep_shares_to_prep() algorithm and broadcast to the
Aggregators to start the next round. In this section we describe
some approaches to realizing this broadcast channel functionality in
protocols that use VDAFs.
The state machine of each Aggregator is shown in Figure 8.
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++
 
v 
Start > Continued(prep_state) > Finished(out_share)
 
 
+> Rejected <+
Figure 8: State machine for VDAF preparation.
State transitions are made when the state is acted upon by the host's
local inputs and/or messages sent by the peers. The initial state is
Start. The terminal states are Rejected, which indicates that the
report cannot be processed any further, and Finished(out_share),
which indicates that the Aggregator has recovered an output share
out_share.
class State:
pass
class Start(State):
pass
class Continued(State):
def __init__(self, prep_state):
self.prep_state = prep_state
class Finished(State):
def __init__(self, output_share):
self.output_share = output_share
class Rejected(State):
def __init__(self):
pass
Note that there is no representation of the Start state as it is
never instantiated in the pingpong topology.
For convenience, the methods described in this section are defined in
terms of opaque byte strings. A compatible Vdaf MUST specify methods
for encoding public shares, input shares, prep shares, prep messages,
and aggregation parameters. Minimally:
* vdaf.decode_public_share(encoded: bytes) > PublicShare decodes a
public share.
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* vdaf.decode_input_share(agg_id: int, encoded: bytes) > InputShare
decodes an input share, using the aggregator ID as optional
context.
* vdaf.encode_prep_share(prep_share: PrepShare) > bytes encodes a
prep share.
* vdaf.decode_prep_share(prep_state: PrepState, encoded: bytes) >
PrepShare decodes a prep share, using the prep state as optional
context.
* vdaf.encode_prep_msg(prep_msg: PrepMessage) > bytes encodes a
prep message.
* vdaf.decode_prep_msg(prep_state: PrepState, encoded: bytes) >
PrepMessage decodes a prep message, using the prep state as
optional decoding context.
* vdaf.decode_agg_param(encoded: bytes) > AggParam decodes an
aggregation parameter.
* vdaf.encode_agg_param(agg_param: AggParam) > bytes encodes an
aggregation parameter.
Implementations of Prio3 and Poplar1 MUST use the encoding scheme
specified in Section 7.2.7 and Section 8.2.6 respectively.
5.8. PingPong Topology (Only Two Aggregators)
For VDAFs with precisely two Aggregators (i.e., SHARES == 2), the
following "ping pong" communication pattern can be used. It is
compatible with any request/response transport protocol, such as
HTTP.
Let us call the initiating party the "Leader" and the responding
party the "Helper". The highlevel idea is that the Leader and
Helper will take turns running the computation locally until input
from their peer is required:
* For a 1round VDAF (e.g., Prio3 (Section 7)), the Leader sends its
prep share to the Helper, who computes the prep message locally,
computes its output share, then sends the prep message to the
Leader. Preparation requires just one round trip between the
Leader and the Helper.
* For a 2round VDAF (e.g., Poplar1 (Section 8)), the Leader sends
its firstround prep share to the Helper, who replies with the
firstround prep message and its secondround prep share. In the
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next request, the Leader computes its secondround prep share
locally, computes its output share, and sends the secondround
prep message to the Helper. Finally, the Helper computes its own
output share.
* In general, each request includes the Leader's prep share for the
previous round and/or the prep message for the current round;
correspondingly, each response consists of the prep message for
the current round and the Helper's prep share for the next round.
The Aggregators proceed in this pingponging fashion until a step of
the computation fails (indicating the report is invalid and should be
rejected) or preparation is completed. All told there there are
ceil((ROUNDS+1)/2) requests sent.
Each message in the pingpong protocol is structured as follows
(expressed in TLS syntax as defined in Section 3 of [RFC8446]):
enum {
initialize(0),
continue(1),
finish(2),
(255)
} MessageType;
struct {
MessageType type;
select (Message.type) {
case initialize:
opaque prep_share<0..2^321>;
case continue:
opaque prep_msg<0..2^321>;
opaque prep_share<0..2^321>;
case finish:
opaque prep_msg<0..2^321>;
};
} Message;
These messages are used to transition between the states described in
Section 5.7. They are encoded and decoded to or from byte buffers as
described Section 3 of [RFC8446]) using the following routines:
* encode_ping_pong_message(message: Message) > bytes encodes a
Message into an opaque byte buffer.
* decode_pong_pong_message(encoded: bytes) > Message decodes an
opaque byte buffer into a Message, raising an error if the bytes
are not a valid encoding.
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The Leader's initial transition is computed with the following
procedure:
def ping_pong_leader_init(
vdaf,
vdaf_verify_key: bytes,
agg_param: bytes,
nonce: bytes,
public_share: bytes,
input_share: bytes,
) > tuple[State, bytes]:
try:
(prep_state, prep_share) = vdaf.prep_init(
vdaf_verify_key,
0,
vdaf.decode_agg_param(agg_param),
nonce,
vdaf.decode_public_share(public_share),
vdaf.decode_input_share(0, input_share),
)
outbound = Message.initialize(
vdaf.encode_prep_share(prep_share))
return (
Continued(prep_state),
encode_ping_pong_message(outbound),
)
except:
return (Rejected(), None)
The output is the State to which the Leader has transitioned and an
encoded Message. If the Leader's state is Rejected, then processing
halts. Otherwise, if the state is Continued, then processing
continues.
The Leader sends the outbound message to the Helper. The Helper's
initial transition is computed using the following procedure:
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def ping_pong_helper_init(
vdaf,
vdaf_verify_key: bytes,
agg_param: bytes,
nonce: bytes,
public_share: bytes,
input_share: bytes,
inbound_encoded: bytes,
) > tuple[State, bytes]:
try:
(prep_state, prep_share) = vdaf.prep_init(
vdaf_verify_key,
1,
vdaf.decode_agg_param(agg_param),
nonce,
vdaf.decode_public_share(public_share),
vdaf.decode_input_share(1, input_share),
)
inbound = decode_ping_pong_message(inbound_encoded)
if inbound.type != 0: # initialize
return (Rejected(), None)
prep_shares = [
vdaf.decode_prep_share(prep_state, inbound.prep_share),
prep_share,
]
return vdaf.ping_pong_transition(
agg_param,
prep_shares,
prep_state,
)
except:
return (Rejected(), None)
Procedure ping_pong_transition() takes in the prep shares, combines
them into the prep message, and computes the next prep state of the
caller:
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def ping_pong_transition(
vdaf,
agg_param: AggParam,
prep_shares: list[PrepShare],
prep_state: PrepState,
) > (State, bytes):
prep_msg = vdaf.prep_shares_to_prep(agg_param,
prep_shares)
out = vdaf.prep_next(prep_state, prep_msg)
if type(out) == OutShare:
outbound = Message.finish(vdaf.encode_prep_msg(prep_msg))
return (Finished(out), encode_ping_pong_message(outbound))
(prep_state, prep_share) = out
outbound = Message.continue(
vdaf.encode_prep_msg(prep_msg),
vdaf.encode_prep_share(prep_share),
)
return (
Continued(prep_state),
encode_ping_pong_message(outbound),
)
The output is the State to which the Helper has transitioned and an
encoded Message. If the Helper's state is Finished or Rejected, then
processing halts. Otherwise, if the state is Continued, then
processing continues.
Next, the Helper sends the outbound message to the Leader. The
Leader computes its next state transition using the function
ping_pong_leader_continued:
def ping_pong_leader_continued(
vdaf,
agg_param: bytes,
state: State,
inbound_encoded: bytes,
) > (State, Optional[bytes]):
return vdaf.ping_pong_continued(
True,
agg_param,
state,
inbound_encoded,
)
def ping_pong_continued(
vdaf,
is_leader: bool,
agg_param: bytes,
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state: State,
inbound_encoded: bytes,
) > (State, Optional[bytes]):
try:
inbound = decode_ping_pong_message(inbound_encoded)
if inbound.type == 0: # initialize
return (Rejected(), None)
if !isinstance(state, Continued):
return (Rejected(), None)
prep_msg = vdaf.decode_prep_msg(
state.prep_state,
inbound.prep_msg,
)
out = vdaf.prep_next(state.prep_state, prep_msg)
if type(out) == tuple[PrepState, PrepShare] \
and inbound.type == 1:
# continue
(prep_state, prep_share) = out
prep_shares = [
vdaf.decode_prep_share(
prep_state,
inbound.prep_share,
),
prep_share,
]
if is_leader:
prep_shares.reverse()
return vdaf.ping_pong_transition(
vdaf.decode_agg_param(agg_param),
prep_shares,
prep_state,
)
elif type(out) == OutShare and inbound.type == 2:
# finish
return (Finished(out), None)
else:
return (Rejected(), None)
except:
return (Rejected(), None)
If the Leader's state is Finished or Rejected, then processing halts.
Otherwise, the Leader sends the outbound message to the Helper. The
Helper computes its next state transition using the function
ping_pong_helper_continued:
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def ping_pong_helper_continued(
vdaf,
agg_param: bytes,
state: State,
inbound_encoded: bytes,
) > (State, Optional[bytes]):
return vdaf.ping_pong_continued(
False,
agg_param,
state,
inbound_encoded,
)
They continue in this way until processing halts. Note that,
depending on the number of rounds of preparation that are required,
there may be one more message to send before the peer can also finish
processing (i.e., outbound != None).
5.9. Star Topology (Any Number of Aggregators)
The pingpong topology of the previous section is only suitable for
VDAFs involving exactly two Aggregators. If more Aggregators are
required, the star topology described in this section can be used
instead.
TODO Describe the Leaderemulated broadcast channel architecture
that was originally envisioned for DAP. (As of DAP05 we are
going with the ping pong architecture described in the previous
section.)
6. Preliminaries
This section describes the primitives that are common to the VDAFs
specified in this document.
6.1. Finite Fields
Both Prio3 and Poplar1 use finite fields of prime order. Finite
field elements are represented by a class Field with the following
associated parameters:
* MODULUS: int is the prime modulus that defines the field.
* ENCODED_SIZE: int is the number of bytes used to encode a field
element as a byte string.
A concrete Field also implements the following class methods:
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* Field.zeros(length: int) > list[Self] returns a vector of zeros.
Preconditions:
 length MUST be greater than or equal 0.
Postconditions:
 The length of the output MUST be length.
* Field.rand_vec(length: int) > list[Self] returns a vector of
random field elements. Same pre and postconditions as for
Field.zeros().
A field element is an instance of a concrete Field. The concrete
class defines the usual arithmetic operations on field elements. In
addition, it defines the following instance method for converting a
field element to an unsigned integer:
* elem.as_unsigned() > int returns the integer representation of
field element elem.
Likewise, each concrete Field implements a constructor for converting
an unsigned integer into a field element:
* Field(integer: int) returns integer represented as a field
element. The value of integer MUST be nonnegative and less than
Field.MODULUS.
Each concrete Field has two derived class methods, one for encoding a
vector of field elements as a byte string and another for decoding a
vector of field elements.
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def encode_vec(cls, vec: list[Self]) > bytes:
"""
Encode a vector of field elements `vec` as a byte string.
"""
encoded = bytes()
for x in vec:
encoded += to_le_bytes(x.as_unsigned(), cls.ENCODED_SIZE)
return encoded
def decode_vec(cls, encoded: bytes) > list[Self]:
"""
Parse a vector of field elements from `encoded`.
"""
L = cls.ENCODED_SIZE
if len(encoded) % L != 0:
raise ValueError(
'input length must be a multiple of the size of an '
'encoded field element')
vec = []
for i in range(0, len(encoded), L):
encoded_x = encoded[i:i+L]
x = from_le_bytes(encoded_x)
if x >= cls.MODULUS:
raise ValueError('modulus overflow')
vec.append(cls(x))
return vec
Figure 9: Derived class methods for finite fields.
Finally, Field implements the following methods for representing a
value as a sequence of field elements, each of which represents a bit
of the input.
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def encode_into_bit_vector(
cls,
val: int,
bits: int) > list[Self]:
"""
Encode the bit representation of `val` with at most `bits` number
of bits, as a vector of field elements.
Preconditions:
 `val >= 0`
 `bits >= 0`
"""
if val >= 2 ** bits:
# Sanity check we are able to represent `val` with `bits`
# number of bits.
raise ValueError("Number of bits is not enough to represent "
"the input integer.")
encoded = []
for l in range(bits):
encoded.append(cls((val >> l) & 1))
return encoded
def decode_from_bit_vector(cls, vec: list[Self]) > Self:
"""
Decode the field element from the bit representation, expressed
as a vector of field elements `vec`.
"""
bits = len(vec)
if cls.MODULUS >> bits == 0:
raise ValueError("Number of bits is too large to be "
"represented by field modulus.")
decoded = cls(0)
for (l, bit) in enumerate(vec):
decoded += cls(1 << l) * bit
return decoded
Figure 10: Derived class methods to encode integers into bit vector
representation.
6.1.1. Auxiliary Functions
The following auxiliary functions on vectors of field elements are
used in the remainder of this document. Note that an exception is
raised by each function if the operands are not the same length.
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def vec_sub(left: list[F], right: list[F]) > list[F]:
"""
Subtract the right operand from the left and return the result.
"""
if len(left) != len(right):
raise ValueError("mismatched vector sizes")
return list(map(lambda x: x[0]  x[1], zip(left, right)))
def vec_add(left: list[F], right: list[F]) > list[F]:
"""Add the right operand to the left and return the result."""
if len(left) != len(right):
raise ValueError("mismatched vector sizes")
return list(map(lambda x: x[0] + x[1], zip(left, right)))
def vec_neg(vec: list[F]) > list[F]:
"""Negate the input vector."""
return list(map(lambda x: x, vec))
Figure 11: Common functions for finite fields.
6.1.2. FFTFriendly Fields
Some VDAFs require fields that are suitable for efficient computation
of the discrete Fourier transform, as this allows for fast polynomial
interpolation. (One example is Prio3 (Section 7) when instantiated
with the FLP of Section 7.3.3.) Specifically, a field is said to be
"FFTfriendly" if, in addition to satisfying the interface described
in Section 6.1, it implements the following method:
* Field.gen() > Field returns the generator of a large subgroup of
the multiplicative group. To be FFTfriendly, the order of this
subgroup MUST be a power of 2. In addition, the size of the
subgroup dictates how large interpolated polynomials can be. It
is RECOMMENDED that a generator is chosen with order at least
2^20.
FFTfriendly fields also define the following parameter:
* GEN_ORDER: int is the order of a multiplicative subgroup generated
by Field.gen().
6.1.3. Parameters
The tables below define finite fields used in the remainder of this
document.
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+==============+================+=======================+==========+
 Parameter  Field64  Field128  Field255 
+==============+================+=======================+==========+
 MODULUS  2^32 *  2^66 *  2^255  
  4294967295 + 1  4611686018427387897 +  19 
   1  
+++++
 ENCODED_SIZE  8  16  32 
+++++
 Generator  7^4294967295  7^4611686018427387897  n/a 
+++++
 GEN_ORDER  2^32  2^66  n/a 
+++++
Table 3: Parameters for the finite fields used in this document.
6.2. Extendable Output Functions
VDAFs in this specification use extendable output functions (XOFs) to
extract short, fixedlength strings we call "seeds" from long input
strings and expand seeds into long output strings. We specify a
single interface that is suitable for both purposes.
XOFs are defined by a class Xof with the following associated
parameter and methods:
* SEED_SIZE: int is the size (in bytes) of a seed.
* Xof(seed: bytes, dst: bytes, binder: bytes) constructs an instance
of Xof from the given seed, domain separation tag, and binder
string. (See below for definitions of these.) The seed MUST be
of length SEED_SIZE and MUST be generated securely (i.e., it is
either the output of a CSPRNG or a previous invocation of the
XOF).
* xof.next(length: int) returns the next length bytes of output of
xof.
Each Xof has three derived methods. The first is used to derive a
fresh seed from an existing one. The second is used to compute a
sequence of field elements. The third is a convenience method to
construct an Xof from a seed, domain separation tag, and binder
string, and then use it to compute a sequence of field elements.
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def derive_seed(cls,
seed: bytes,
dst: bytes,
binder: bytes) > bytes:
"""
Derive a new seed.
Preconditions:
 `len(seed) == Xof.SEED_SIZE`
"""
xof = cls(seed, dst, binder)
return xof.next(cls.SEED_SIZE)
def next_vec(self, field: type[F], length: int) > list[F]:
"""
Output the next `length` field elements.
Preconditions:
 `field` is subclass of `Field`
 `length > 0`
"""
m = next_power_of_2(field.MODULUS)  1
vec: list[F] = []
while len(vec) < length:
x = from_le_bytes(self.next(field.ENCODED_SIZE))
x &= m
if x < field.MODULUS:
vec.append(field(x))
return vec
def expand_into_vec(cls,
field: type[F],
seed: bytes,
dst: bytes,
binder: bytes,
length: int) > list[F]:
"""
Expand the input `seed` into vector of `length` field elements.
Preconditions:
 `field` is subclass of `Field`
 `len(seed) == Xof.SEED_SIZE`
 `length > 0`
"""
xof = cls(seed, dst, binder)
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return xof.next_vec(field, length)
Figure 12: Derived methods for XOFs.
6.2.1. XofTurboShake128
This section describes XofTurboShake128, an XOF based on the
TurboSHAKE128 [TurboSHAKE]. This XOF is RECOMMENDED for all use
cases within VDAFs. The length of the domain separation string dst
passed to XofTurboShake128 MUST NOT exceed 255 bytes.
class XofTurboShake128(Xof):
"""XOF wrapper for TurboSHAKE128."""
# Associated parameters
SEED_SIZE = 16
def __init__(self, seed: bytes, dst: bytes, binder: bytes):
self.l = 0
self.m = to_le_bytes(len(dst), 1) + dst + seed + binder
def next(self, length: int) > bytes:
self.l += length
# Function `TurboSHAKE128(M, D, L)` is as defined in
# Section 2.2 of [TurboSHAKE].
#
# Implementation note: Rather than regenerate the output
# stream each time `next()` is invoked, most implementations
# of TurboSHAKE128 will expose an "absorbthensqueeze" API
# that allows stateful handling of the stream.
stream = TurboSHAKE128(self.m, 1, self.l)
return stream[length:]
Figure 13: Definition of XOF XofTurboShake128.
6.2.2. XofFixedKeyAes128
While XofTurboShake128 as described above can be securely used in all
cases where a XOF is needed in the VDAFs described in this document,
there are some cases where a more efficient instantiation based on
fixedkey AES is possible. For now, this is limited to the XOF used
inside the Idpf Section 8.1 implementation in Poplar1 Section 8.3.
It is NOT RECOMMENDED to use this XOF anywhere else. The length of
the domain separation string dst passed to XofFixedKeyAes128 MUST NOT
exceed 255 bytes. See Security Considerations Section 9 for a more
detailed discussion.
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class XofFixedKeyAes128(Xof):
"""
XOF based on a circular collisionresistant hash function from
fixedkey AES.
"""
# Associated parameters
SEED_SIZE = 16
def __init__(self, seed: bytes, dst: bytes, binder: bytes):
if len(seed) != self.SEED_SIZE:
raise ValueError("incorrect seed size")
self.length_consumed = 0
# Use TurboSHAKE128 to derive a key from the binder string
# and domain separation tag. Note that the AES key does not
# need to be kept secret from any party. However, when used
# with an IDPF, we require the binder to be a random nonce.
#
# Implementation note: This step can be cached across XOF
# evaluations with many different seeds.
dst_length = to_le_bytes(len(dst), 1)
self.fixed_key = TurboSHAKE128(
dst_length + dst + binder,
2,
16,
)
self.seed = seed
def next(self, length: int) > bytes:
offset = self.length_consumed % 16
new_length = self.length_consumed + length
block_range = range(
self.length_consumed // 16,
new_length // 16 + 1
)
self.length_consumed = new_length
hashed_blocks = [
self.hash_block(xor(self.seed, to_le_bytes(i, 16)))
for i in block_range
]
return concat(hashed_blocks)[offset:offset+length]
def hash_block(self, block: bytes) > bytes:
"""
The multiinstance tweakable circular correlationrobust hash
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function of [GKWWY20] (Section 4.2). The tweak here is the
key that stays constant for all XOF evaluations of the same
Client, but differs between Clients.
Function `AES128(key, block)` is the AES128 blockcipher.
"""
lo, hi = block[:8], block[8:]
sigma_block = concat([hi, xor(hi, lo)])
return xor(AES128(self.fixed_key, sigma_block), sigma_block)
6.2.3. The Domain Separation Tag and Binder String
XOFs are used to map a seed to a finite domain, e.g., a fresh seed or
a vector of field elements. To ensure domain separation, the
derivation is needs to be bound to some distinguished domain
separation tag. The domain separation tag encodes the following
values:
1. The document version (i.e.,VERSION);
2. The "class" of the algorithm using the output (e.g., VDAF);
3. A unique identifier for the algorithm; and
4. Some indication of how the output is used (e.g., for deriving the
measurement shares in Prio3 Section 7).
The following algorithm is used in the remainder of this document in
order to format the domain separation tag:
def format_dst(algo_class: int,
algo: int,
usage: int) > bytes:
"""
Format XOF domain separation tag for use within a (V)DAF.
Preconditions:
 `algo_class` in `range(0, 2 ** 8)`
 `algo` in `range(0, 2 ** 32)`
 `usage` in `range(0, 2 ** 16)`
"""
return concat([
to_be_bytes(VERSION, 1),
to_be_bytes(algo_class, 1),
to_be_bytes(algo, 4),
to_be_bytes(usage, 2),
])
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It is also sometimes necessary to bind the output to some ephemeral
value that multiple parties need to agree on. We call this input the
"binder string".
7. Prio3
This section describes Prio3, a VDAF for Prio [CGB17]. Prio is
suitable for a wide variety of aggregation functions, including (but
not limited to) sum, mean, standard deviation, estimation of
quantiles (e.g., median), and linear regression. In fact, the scheme
described in this section is compatible with any aggregation function
that has the following structure:
* Each measurement is encoded as a vector over some finite field.
* Measurement validity is determined by an arithmetic circuit
evaluated over the encoded measurement. (An "arithmetic circuit"
is a function comprised of arithmetic operations in the field.)
The circuit's output is a single field element: if zero, then the
measurement is said to be "valid"; otherwise, if the output is
nonzero, then the measurement is said to be "invalid".
* The aggregate result is obtained by summing up the encoded
measurement vectors and computing some function of the sum.
At a high level, Prio3 distributes this computation as follows. Each
Client first shards its measurement by first encoding it, then
splitting the vector into secret shares and sending a share to each
Aggregator. Next, in the preparation phase, the Aggregators carry
out a multiparty computation to determine if their shares correspond
to a valid measurement (as determined by the arithmetic circuit).
This computation involves a "proof" of validity generated by the
Client. Next, each Aggregator sums up its shares locally. Finally,
the Collector sums up the aggregate shares and computes the aggregate
result.
This VDAF does not have an aggregation parameter. Instead, the
output share is derived from the measurement share by applying a
fixed map. See Section 8 for an example of a VDAF that makes
meaningful use of the aggregation parameter.
As the name implies, Prio3 is a descendant of the original Prio
construction. A second iteration was deployed in the [ENPA] system,
and like the VDAF described here, the ENPA system was built from
techniques introduced in [BBCGGI19] that significantly improve
communication cost. That system was specialized for a particular
aggregation function; the goal of Prio3 is to provide the same level
of generality as the original construction.
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The core component of Prio3 is a "Fully Linear Proof (FLP)" system.
Introduced by [BBCGGI19], the FLP encapsulates the functionality
required for encoding and validating measurements. Prio3 can be
thought of as a transformation of a particular class of FLPs into a
VDAF.
The remainder of this section is structured as follows. The syntax
for FLPs is described in Section 7.1. The generic transformation of
an FLP into Prio3 is specified in Section 7.2. Next, a concrete FLP
suitable for any validity circuit is specified in Section 7.3.
Finally, instantiations of Prio3 for various types of measurements
are specified in Section 7.4. Test vectors can be found in
Appendix "Test Vectors".
7.1. Fully Linear Proof (FLP) Systems
Conceptually, an FLP is a twoparty protocol executed by a prover and
a verifier. In actual use, however, the prover's computation is
carried out by the Client, and the verifier's computation is
distributed among the Aggregators. The Client generates a "proof" of
its measurement's validity and distributes shares of the proof to the
Aggregators. Each Aggregator then performs some computation on its
measurement share and proof share locally and sends the result to the
other Aggregators. Combining the exchanged messages allows each
Aggregator to decide if it holds a share of a valid measurement.
(See Section 7.2 for details.)
As usual, we will describe the interface implemented by a concrete
FLP in terms of an abstract base class Flp that specifies the set of
methods and parameters a concrete FLP must provide.
The parameters provided by a concrete FLP are listed in Table 4.
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+================+==========================================+
 Parameter  Description 
+================+==========================================+
 PROVE_RAND_LEN  Length of the prover randomness, the 
  number of random field elements consumed 
  by the prover when generating a proof 
+++
 QUERY_RAND_LEN  Length of the query randomness, the 
  number of random field elements consumed 
  by the verifier 
+++
 JOINT_RAND_LEN  Length of the joint randomness, the 
  number of random field elements consumed 
  by both the prover and verifier 
+++
 MEAS_LEN  Length of the encoded measurement 
  (Section 7.1.1) 
+++
 OUTPUT_LEN  Length of the aggregatable output 
  (Section 7.1.1) 
+++
 PROOF_LEN  Length of the proof 
+++
 VERIFIER_LEN  Length of the verifier message generated 
  by querying the measurement and proof 
+++
 Measurement  Type of the measurement 
+++
 AggResult  Type of the aggregate result 
+++
 field  Class object for the field (Section 6.1) 
+++
Table 4: Constants and types defined by a concrete FLP.
An FLP specifies the following algorithms for generating and
verifying proofs of validity (encoding is described below in
Section 7.1.1):
* flp.prove(meas: list[F], prove_rand: list[F], joint_rand: list[F])
> list[F] is the deterministic proofgeneration algorithm run by
the prover. Its inputs are the encoded measurement, the "prover
randomness" prove_rand, and the "joint randomness" joint_rand.
The prover randomness is used only by the prover, but the joint
randomness is shared by both the prover and verifier.
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* flp.query(meas: list[F], proof: list[F], query_rand: list[F],
joint_rand: list[F], num_shares: int) > list[F] is the query
generation algorithm run by the verifier. This is used to "query"
the measurement and proof. The result of the query (i.e., the
output of this function) is called the "verifier message". In
addition to the measurement and proof, this algorithm takes as
input the query randomness query_rand and the joint randomness
joint_rand. The former is used only by the verifier. num_shares
specifies how many shares were generated.
* flp.decide(verifier: list[F]) > bool is the deterministic
decision algorithm run by the verifier. It takes as input the
verifier message and outputs a boolean indicating if the
measurement from which it was generated is valid.
Our application requires that the FLP is "fully linear" in the sense
defined in [BBCGGI19]. As a practical matter, what this property
implies is that, when run on a share of the measurement and proof,
the querygeneration algorithm outputs a share of the verifier
message. Furthermore, the privacy property of the FLP system ensures
that the verifier message reveals nothing about the measurement other
than whether it is valid. Therefore, to decide if a measurement is
valid, the Aggregators will run the querygeneration algorithm
locally, exchange verifier shares, combine them to recover the
verifier message, and run the decision algorithm.
The querygeneration algorithm includes a parameter num_shares that
specifies the number of shares that were generated. If these data
are not secret shared, then num_shares == 1. This parameter is
useful for certain FLP constructions. For example, the FLP in
Section 7.3 is defined in terms of an arithmetic circuit; when the
circuit contains constants, it is sometimes necessary to normalize
those constants to ensure that the circuit's output, when run on a
valid measurement, is the same regardless of the number of shares.
An FLP is executed by the prover and verifier as follows:
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def run_flp(
flp: Flp[Measurement, AggResult, F],
meas: list[F],
num_shares: int) > bool:
"""Run the FLP on an encoded measurement."""
joint_rand = flp.field.rand_vec(flp.JOINT_RAND_LEN)
prove_rand = flp.field.rand_vec(flp.PROVE_RAND_LEN)
query_rand = flp.field.rand_vec(flp.QUERY_RAND_LEN)
# Prover generates the proof.
proof = flp.prove(meas, prove_rand, joint_rand)
# Shard the measurement and the proof.
meas_shares = additive_secret_share(
meas,
num_shares,
flp.field,
)
proof_shares = additive_secret_share(
proof,
num_shares,
flp.field,
)
# Verifier queries the meas shares and proof shares.
verifier_shares = [
flp.query(
meas_share,
proof_share,
query_rand,
joint_rand,
num_shares,
)
for meas_share, proof_share in zip(meas_shares, proof_shares)
]
# Combine the verifier shares into the verifier.
verifier = flp.field.zeros(len(verifier_shares[0]))
for verifier_share in verifier_shares:
verifier = vec_add(verifier, verifier_share)
# Verifier decides if the measurement is valid.
return flp.decide(verifier)
Figure 14: Execution of an FLP.
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The proof system is constructed so that, if meas is valid, then
run_flp(flp, meas, 1) always returns True. On the other hand, if
meas is invalid, then as long as joint_rand and query_rand are
generated uniform randomly, the output is False with high
probability. False positives are possible: there is a small
probability that a verifier accepts an invalid input as valid. An
FLP is said to be "sound" if this probability is sufficiently small.
The soundness of the FLP depends on a variety of parameters, like the
length of the input and the size of the field. See Section 7.3 for
details.
Note that soundness of an FLP system is not the same as robustness
for a VDAF In particular, soundness of the FLP is necessary, but
insufficient for robusntess of Prio3 (Section 7). See Section 9.6
for details.
We remark that [BBCGGI19] defines a much larger class of fully linear
proof systems than we consider here. In particular, what is called
an "FLP" here is called a 1.5round, publiccoin, interactive oracle
proof system in their paper.
7.1.1. Encoding the Input
The type of measurement being aggregated is defined by the FLP.
Hence, the FLP also specifies a method of encoding raw measurements
as a vector of field elements:
* flp.encode(measurement: Measurement) > list[F] encodes a raw
measurement as a vector of field elements. The return value MUST
be of length MEAS_LEN.
For some FLPs, the encoded measurement also includes redundant field
elements that are useful for checking the proof, but which are not
needed after the proof has been checked. An example is the "integer
sum" data type from [CGB17] in which an integer in range [0, 2^k) is
encoded as a vector of k field elements, each representing a bit of
the integer (this type is also defined in Section 7.4.2). After
consuming this vector, all that is needed is the integer it
represents. Thus the FLP defines an algorithm for truncating the
encoded measurement to the length of the aggregated output:
* flp.truncate(meas: list[F]) > list[F] maps an encoded measurement
(e.g., the bitencoding of the measurement) to an aggregatable
output (e.g., the singleton vector containing the measurement).
The length of the input MUST be MEAS_LEN and the length of the
output MUST be OUTPUT_LEN.
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Once the aggregate shares have been computed and combined together,
their sum can be converted into the aggregate result. This could be
a projection from the FLP's field to the integers, or it could
include additional postprocessing.
* flp.decode(output: list[F], num_measurements: int) > AggResult
maps a sum of aggregate shares to an aggregate result.
Preconditions:
 The length of output MUST be OUTPUT_LEN.
 num_measurements MUST equal the number of measurements that
contributed to the output.
We remark that, taken together, these three functionalities
correspond roughly to the notion of "Affineaggregatable encodings
(AFEs)" from [CGB17].
7.1.2. Multiple Proofs
It is sometimes desirable to generate and verify multiple independent
proofs for the same input. First, this improves the soundness of the
proof system without having to change any of its parameters. Second,
it allows a smaller field to be used (e.g., replace Field128 with
Field64, see Section 7.3) without sacrificing soundness. Generally,
choosing a smaller field can significantly reduce communication cost.
(This is a tradeoff, of course, since generating and verifying more
proofs requires more time.) Given these benefits, this feature is
implemented by Prio3 (Section 7).
To generate these proofs for a specific measurement, the prover calls
flp.prove multiple times, each time using an independently generated
prover and joint randomness string. The verifier checks each proof
independently, each time with an independently generated query
randomness string. It accepts the measurement only if all the
decision algorithm accepts on each proof.
See Section 9.6 below for discussions on choosing the right number of
proofs.
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7.2. Construction
This section specifies Prio3, an implementation of the Vdaf interface
(Section 5). It has three generic parameters: an FftField ({{fft
field}}), anFlp({{flp}}) and aXof({{xof}}). It also has an
associated constant,PROOFS, with a value within the range of[1,
256)`, denoting the number of FLPs generated by the Client
(Section 7.1.2).
The associated constants and types required by the Vdaf interface are
defined in Table 5. The methods required for sharding, preparation,
aggregation, and unsharding are described in the remaining
subsections. These methods refer to constants enumerated in Table 6.
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+=================+============================================+
 Parameter  Value 
+=================+============================================+
 VERIFY_KEY_SIZE  Xof.SEED_SIZE 
+++
 RAND_SIZE  Xof.SEED_SIZE * (1 + 2 * (SHARES  1)) if 
  flp.JOINT_RAND_LEN == 0 else Xof.SEED_SIZE 
  * (1 + 2 * (SHARES  1) + SHARES) 
+++
 NONCE_SIZE  16 
+++
 ROUNDS  1 
+++
 SHARES  in [2, 256) 
+++
 Measurement  Flp.Measurement 
+++
 AggParam  None 
+++
 PublicShare  Optional[list[bytes]] 
+++
 InputShare  Union[tuple[list[F], list[F], 
  Optional[bytes]], tuple[bytes, bytes, 
  Optional[bytes]]] 
+++
 OutShare  list[F] 
+++
 AggShare  list[F] 
+++
 AggResult  Flp.AggResult 
+++
 PrepState  tuple[list[F], Optional[bytes]] 
+++
 PrepShare  tuple[list[F], Optional[bytes]] 
+++
 PrepMessage  Optional[bytes] 
+++
Table 5: VDAF parameters for Prio3.
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+=============================+=======+
 Variable  Value 
+=============================+=======+
 USAGE_MEAS_SHARE: int  1 
+++
 USAGE_PROOF_SHARE: int  2 
+++
 USAGE_JOINT_RANDOMNESS: int  3 
+++
 USAGE_PROVE_RANDOMNESS: int  4 
+++
 USAGE_QUERY_RANDOMNESS: int  5 
+++
 USAGE_JOINT_RAND_SEED: int  6 
+++
 USAGE_JOINT_RAND_PART: int  7 
+++
Table 6: Constants used by Prio3.
7.2.1. Sharding
Recall from Section 7.1 that the FLP syntax calls for "joint
randomness" shared by the prover (i.e., the Client) and the verifier
(i.e., the Aggregators). VDAFs have no such notion. Instead, the
Client derives the joint randomness from its measurement in a way
that allows the Aggregators to reconstruct it from their shares.
(This idea is based on the FiatShamir heuristic and is described in
Section 6.2.3 of [BBCGGI19].)
The sharding algorithm involves the following steps:
1. Encode the Client's measurement for the FLP
2. Shard the measurement into a sequence of measurement shares
3. Derive the joint randomness from the measurement shares and nonce
4. Run the FLP proofgeneration algorithm using the derived joint
randomness
5. Shard the proof into a sequence of proof shares
6. Return the public share, consisting of the joint randomness
parts, and the input shares, each consisting of the measurement
share, proof share, and blind of one of the Aggregators
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As described in Section 7.1.2, the soundness of the FLP can be
amplified by generating and verifying multiple FLPs. (This in turn
improves the robustness of Prio3.) To support this, in Prio3:
* In step 3, derive as much joint randomness as required by PROOFS
proofs
* Repeat step 4 PROOFS times, each time with a unique joint
randomness
Depending on the FLP, joint randomness may not be required. In
particular, when flp.JOINT_RAND_LEN == 0, the Client does not derive
the joint randomness (Step 3). The sharding algorithm is specified
below.
def shard(
self,
measurement: Measurement,
nonce: bytes,
rand: bytes) > tuple[
Optional[list[bytes]],
list[Prio3InputShare]]:
if len(nonce) != self.NONCE_SIZE:
raise ValueError("incorrect nonce size")
if len(rand) != self.RAND_SIZE:
raise ValueError("incorrect size of random bytes argument")
l = self.xof.SEED_SIZE
seeds = [rand[i:i + l] for i in range(0, self.RAND_SIZE, l)]
meas = self.flp.encode(measurement)
if self.flp.JOINT_RAND_LEN > 0:
return self.shard_with_joint_rand(meas, nonce, seeds)
else:
return self.shard_without_joint_rand(meas, seeds)
Figure 15: Inputdistribution algorithm for Prio3.
It starts by splitting the randomness into seeds. It then encodes
the measurement as prescribed by the FLP and calls one of two
methods, depending on whether joint randomness is required by the
FLP. The methods are defined in the subsections below.
7.2.1.1. FLPs without joint randomness
The following method is used for FLPs that do not require joint
randomness, i.e., when flp.JOINT_RAND_LEN == 0:
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def shard_without_joint_rand(
self,
meas: list[F],
seeds: list[bytes]) > tuple[
Optional[list[bytes]],
list[Prio3InputShare[F]]]:
k_helper_seeds, seeds = front((self.SHARES  1) * 2, seeds)
k_helper_meas_shares = [
k_helper_seeds[i]
for i in range(0, (self.SHARES  1) * 2, 2)
]
k_helper_proofs_shares = [
k_helper_seeds[i]
for i in range(1, (self.SHARES  1) * 2, 2)
]
(k_prove,), seeds = front(1, seeds)
# Shard the encoded measurement into shares.
leader_meas_share = meas
for j in range(self.SHARES  1):
leader_meas_share = vec_sub(
leader_meas_share,
self.helper_meas_share(j + 1, k_helper_meas_shares[j]),
)
# Generate and shard each proof into shares.
prove_rands = self.prove_rands(k_prove)
leader_proofs_share = []
for _ in range(self.PROOFS):
prove_rand, prove_rands = front(
self.flp.PROVE_RAND_LEN, prove_rands)
leader_proofs_share += self.flp.prove(meas, prove_rand, [])
for j in range(self.SHARES  1):
leader_proofs_share = vec_sub(
leader_proofs_share,
self.helper_proofs_share(
j + 1,
k_helper_proofs_shares[j],
),
)
# Each Aggregator's input share contains its measurement share
# and share of proof(s).
input_shares: list[Prio3InputShare[F]] = []
input_shares.append((
leader_meas_share,
leader_proofs_share,
None,
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))
for j in range(self.SHARES  1):
input_shares.append((
k_helper_meas_shares[j],
k_helper_proofs_shares[j],
None,
))
return (None, input_shares)
Figure 16: Sharding an encoded measurement without joint randomness.
The steps in this method are as follows:
1. Shard the encoded measurement into shares
2. Generate and shard each proof into shares
3. Encode each measurement and shares of each proof into an input
share
Notice that only one pair of measurement and proof(s) share (called
the "leader" shares above) are vectors of field elements. The other
shares (called the "helper" shares) are represented instead by XOF
seeds, which are expanded into vectors of field elements.
The methods on Prio3 for deriving the prover randomness, measurement
shares, and proof shares and the methods for encoding the input
shares are defined in Section 7.2.6.
7.2.1.2. FLPs with joint randomness
The following method is used for FLPs that require joint randomness,
i.e., for which flp.JOINT_RAND_LEN > 0:
def shard_with_joint_rand(
self,
meas: list[F],
nonce: bytes,
seeds: list[bytes]) > tuple[
Optional[list[bytes]],
list[Prio3InputShare[F]]]:
k_helper_seeds, seeds = front((self.SHARES  1) * 3, seeds)
k_helper_meas_shares = [
k_helper_seeds[i]
for i in range(0, (self.SHARES  1) * 3, 3)
]
k_helper_proofs_shares = [
k_helper_seeds[i]
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for i in range(1, (self.SHARES  1) * 3, 3)
]
k_helper_blinds = [
k_helper_seeds[i]
for i in range(2, (self.SHARES  1) * 3, 3)
]
(k_leader_blind, k_prove), seeds = front(2, seeds)
# Shard the encoded measurement into shares and compute the
# joint randomness parts.
leader_meas_share = meas
k_joint_rand_parts = []
for j in range(self.SHARES  1):
helper_meas_share = self.helper_meas_share(
j + 1, k_helper_meas_shares[j])
leader_meas_share = vec_sub(leader_meas_share,
helper_meas_share)
k_joint_rand_parts.append(self.joint_rand_part(
j + 1, k_helper_blinds[j], helper_meas_share, nonce))
k_joint_rand_parts.insert(0, self.joint_rand_part(
0, k_leader_blind, leader_meas_share, nonce))
# Generate the proof and shard it into proof shares.
prove_rands = self.prove_rands(k_prove)
joint_rands = self.joint_rands(
self.joint_rand_seed(k_joint_rand_parts))
leader_proofs_share = []
for _ in range(self.PROOFS):
prove_rand, prove_rands = front(
self.flp.PROVE_RAND_LEN, prove_rands)
joint_rand, joint_rands = front(
self.flp.JOINT_RAND_LEN, joint_rands)
leader_proofs_share += self.flp.prove(
meas,
prove_rand,
joint_rand,
)
for j in range(self.SHARES  1):
leader_proofs_share = vec_sub(
leader_proofs_share,
self.helper_proofs_share(
j + 1,
k_helper_proofs_shares[j],
),
)
# Each Aggregator's input share contains its measurement share,
# share of proof(s), and blind. The public share contains the
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# Aggregators' joint randomness parts.
input_shares: list[Prio3InputShare[F]] = []
input_shares.append((
leader_meas_share,
leader_proofs_share,
k_leader_blind,
))
for j in range(self.SHARES  1):
input_shares.append((
k_helper_meas_shares[j],
k_helper_proofs_shares[j],
k_helper_blinds[j],
))
return (k_joint_rand_parts, input_shares)
Figure 17: Sharding an encoded measurement with joint randomness.
The difference between this procedure and previous one is that here
we compute joint randomnesses joint_rands, split it into multiple
joint_rand, and pass each joint_rand to the proof generationg
algorithm. (In Figure 16 the joint randomness is the empty vector,
[].) This requires generating an additional value, called the
"blind", that is incorporated into each input share.
The joint randomness computation involves the following steps:
1. Compute a "joint randomness part" from each measurement share and
blind
2. Compute a "joint randomness seed" from the joint randomness parts
3. Compute the joint randomness for each proof evaluation from the
joint randomness seed
This threestep process is designed to ensure that the joint
randomness does not leak the measurement to the Aggregators while
preventing a malicious Client from tampering with the joint
randomness in a way that allows it to break robustness. To bootstrap
the required check, the Client encodes the joint randomness parts in
the public share. (See Section 7.2.2 for details.)
The methods used in this computation are defined in Section 7.2.6.
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7.2.2. Preparation
This section describes the process of recovering output shares from
the input shares. The highlevel idea is that each Aggregator first
queries its measurement and share of proof(s) locally, then exchanges
its share of verifier(s) with the other Aggregators. The shares of
verifier(s) are then combined into the verifier message(s) used to
decide whether to accept.
In addition, for FLPs that require joint randomness, the Aggregators
must ensure that they have all used the same joint randomness for the
querygeneration algorithm. To do so, they collectively rederive
the joint randomness from their measurement shares just as the Client
did during sharding.
In order to avoid extra round of communication, the Client sends each
Aggregator a "hint" consisting of the joint randomness parts. This
leaves open the possibility that the Client cheated by, say, forcing
the Aggregators to use joint randomness that biases the proof check
procedure some way in its favor. To mitigate this, the Aggregators
also check that they have all computed the same joint randomness seed
before accepting their output shares. To do so, they exchange their
parts of the joint randomness along with their shares of verifier(s).
The definitions of constants and a few auxiliary functions are
defined in Section 7.2.6.
def prep_init(
self,
verify_key: bytes,
agg_id: int,
_agg_param: None,
nonce: bytes,
public_share: Optional[list[bytes]],
input_share: Prio3InputShare[F]) > tuple[
Prio3PrepState[F],
Prio3PrepShare[F]]:
k_joint_rand_parts = public_share
(meas_share, proofs_share, k_blind) = \
self.expand_input_share(agg_id, input_share)
out_share = self.flp.truncate(meas_share)
# Compute the joint randomness.
joint_rand: list[F] = []
k_corrected_joint_rand, k_joint_rand_part = None, None
if self.flp.JOINT_RAND_LEN > 0:
assert k_blind is not None
assert k_joint_rand_parts is not None
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k_joint_rand_part = self.joint_rand_part(
agg_id, k_blind, meas_share, nonce)
k_joint_rand_parts[agg_id] = k_joint_rand_part
k_corrected_joint_rand = self.joint_rand_seed(
k_joint_rand_parts)
joint_rands = self.joint_rands(k_corrected_joint_rand)
# Query the measurement and proof share.
query_rands = self.query_rands(verify_key, nonce)
verifiers_share = []
for _ in range(self.PROOFS):
proof_share, proofs_share = front(
self.flp.PROOF_LEN, proofs_share)
query_rand, query_rands = front(
self.flp.QUERY_RAND_LEN, query_rands)
if self.flp.JOINT_RAND_LEN > 0:
joint_rand, joint_rands = front(
self.flp.JOINT_RAND_LEN, joint_rands)
verifiers_share += self.flp.query(
meas_share,
proof_share,
query_rand,
joint_rand,
self.SHARES,
)
prep_state = (out_share, k_corrected_joint_rand)
prep_share = (verifiers_share, k_joint_rand_part)
return (prep_state, prep_share)
def prep_next(
self,
prep_state: Prio3PrepState[F],
prep_msg: Optional[bytes]) > Union[
tuple[Prio3PrepState[F], Prio3PrepShare[F]],
list[F]]:
k_joint_rand = prep_msg
(out_share, k_corrected_joint_rand) = prep_state
# If joint randomness was used, check that the value computed by
# the Aggregators matches the value indicated by the Client.
if k_joint_rand != k_corrected_joint_rand:
raise ValueError('joint randomness check failed')
return out_share
def prep_shares_to_prep(
self,
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_agg_param: None,
prep_shares: list[Prio3PrepShare[F]]) > Optional[bytes]:
# Unshard the verifier shares into the verifier message.
verifiers = self.flp.field.zeros(
self.flp.VERIFIER_LEN * self.PROOFS)
k_joint_rand_parts = []
for (verifiers_share, k_joint_rand_part) in prep_shares:
verifiers = vec_add(verifiers, verifiers_share)
if self.flp.JOINT_RAND_LEN > 0:
assert k_joint_rand_part is not None
k_joint_rand_parts.append(k_joint_rand_part)
# Verify that each proof is wellformed and input is valid
for _ in range(self.PROOFS):
verifier, verifiers = front(self.flp.VERIFIER_LEN, verifiers)
if not self.flp.decide(verifier):
raise ValueError('proof verifier check failed')
# Combine the joint randomness parts computed by the
# Aggregators into the true joint randomness seed. This is
# used in the last step.
k_joint_rand = None
if self.flp.JOINT_RAND_LEN > 0:
k_joint_rand = self.joint_rand_seed(k_joint_rand_parts)
return k_joint_rand
Figure 18: Preparation state for Prio3.
7.2.3. Validity of Aggregation Parameters
Every input share MUST only be used once, regardless of the
aggregation parameters used.
def is_valid(
self,
_agg_param: None,
previous_agg_params: list[None]) > bool:
"""
Checks if `previous_agg_params` is empty, as input shares in
Prio3 may only be used once.
"""
return len(previous_agg_params) == 0
Figure 19: Validity of aggregation parameters for Prio3.
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7.2.4. Aggregation
Aggregating a set of output shares is simply a matter of adding up
the vectors elementwise.
def aggregate(
self,
_agg_param: None,
out_shares: list[list[F]]) > list[F]:
agg_share = self.flp.field.zeros(self.flp.OUTPUT_LEN)
for out_share in out_shares:
agg_share = vec_add(agg_share, out_share)
return agg_share
Figure 20: Aggregation algorithm for Prio3.
7.2.5. Unsharding
To unshard a set of aggregate shares, the Collector first adds up the
vectors elementwise. It then converts each element of the vector
into an integer.
def unshard(
self,
_agg_param: None,
agg_shares: list[list[F]],
num_measurements: int) > AggResult:
agg = self.flp.field.zeros(self.flp.OUTPUT_LEN)
for agg_share in agg_shares:
agg = vec_add(agg, agg_share)
return self.flp.decode(agg, num_measurements)
Figure 21: Computation of the aggregate result for Prio3.
7.2.6. Auxiliary Functions
This section defines a number of auxiliary functions referenced by
the main algorithms for Prio3 in the preceding sections.
The following methods are called by the sharding and preparation
algorithms.
def helper_meas_share(self, agg_id: int, k_share: bytes) > list[F]:
return self.xof.expand_into_vec(
self.flp.field,
k_share,
self.domain_separation_tag(USAGE_MEAS_SHARE),
byte(agg_id),
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self.flp.MEAS_LEN,
)
def helper_proofs_share(
self,
agg_id: int,
k_share: bytes) > list[F]:
return self.xof.expand_into_vec(
self.flp.field,
k_share,
self.domain_separation_tag(USAGE_PROOF_SHARE),
byte(self.PROOFS) + byte(agg_id),
self.flp.PROOF_LEN * self.PROOFS,
)
def expand_input_share(
self,
agg_id: int,
input_share: Prio3InputShare[F]) > tuple[
list[F],
list[F],
Optional[bytes]]:
(meas_share, proofs_share, k_blind) = input_share
if agg_id > 0:
assert isinstance(meas_share, bytes)
assert isinstance(proofs_share, bytes)
meas_share = self.helper_meas_share(agg_id, meas_share)
proofs_share = self.helper_proofs_share(agg_id, proofs_share)
else:
assert isinstance(meas_share, list)
assert isinstance(proofs_share, list)
return (meas_share, proofs_share, k_blind)
def prove_rands(self, k_prove: bytes) > list[F]:
return self.xof.expand_into_vec(
self.flp.field,
k_prove,
self.domain_separation_tag(USAGE_PROVE_RANDOMNESS),
byte(self.PROOFS),
self.flp.PROVE_RAND_LEN * self.PROOFS,
)
def query_rands(self, verify_key: bytes, nonce: bytes) > list[F]:
return self.xof.expand_into_vec(
self.flp.field,
verify_key,
self.domain_separation_tag(USAGE_QUERY_RANDOMNESS),
byte(self.PROOFS) + nonce,
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self.flp.QUERY_RAND_LEN * self.PROOFS,
)
def joint_rand_part(
self,
agg_id: int,
k_blind: bytes,
meas_share: list[F],
nonce: bytes) > bytes:
return self.xof.derive_seed(
k_blind,
self.domain_separation_tag(USAGE_JOINT_RAND_PART),
byte(agg_id) + nonce + self.flp.field.encode_vec(meas_share),
)
def joint_rand_seed(self, k_joint_rand_parts: list[bytes]) > bytes:
"""Derive the joint randomness seed from its parts."""
return self.xof.derive_seed(
zeros(self.xof.SEED_SIZE),
self.domain_separation_tag(USAGE_JOINT_RAND_SEED),
concat(k_joint_rand_parts),
)
def joint_rands(self, k_joint_rand_seed: bytes) > list[F]:
"""Derive the joint randomness from its seed."""
return self.xof.expand_into_vec(
self.flp.field,
k_joint_rand_seed,
self.domain_separation_tag(USAGE_JOINT_RANDOMNESS),
byte(self.PROOFS),
self.flp.JOINT_RAND_LEN * self.PROOFS,
)
7.2.7. Message Serialization
This section defines serialization formats for messages exchanged
over the network while executing Prio3. It is RECOMMENDED that
implementations provide serialization methods for them.
Message structures are defined following Section 3 of [RFC8446]). In
the remainder we use S as an alias for Prio3.xof.SEED_SIZE and F as
an alias for Prio3.field.ENCODED_SIZE. XOF seeds are represented as
follows:
opaque Prio3Seed[S];
Field elements are encoded in littleendian byte order (as defined in
Section 6.1) and represented as follows:
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opaque Prio3Field[F];
7.2.7.1. Public Share
The encoding of the public share depends on whether joint randomness
is required for the underlying FLP (i.e., Prio3.flp.JOINT_RAND_LEN >
0). If joint randomness is not used, then the public share is the
empty string. If joint randomness is used, then the public share
encodes the joint randomness parts as follows:
struct {
Prio3Seed k_joint_rand_parts[S * Prio3.SHARES];
} Prio3PublicShareWithJointRand;
7.2.7.2. Input share
Just as for the public share, the encoding of the input shares
depends on whether joint randomness is used. If so, then each input
share includes the Aggregator's blind for generating its joint
randomness part.
In addition, the encoding of the input shares depends on which
aggregator is receiving the message. If the aggregator ID is 0, then
the input share includes the full measurement and share of proof(s).
Otherwise, if the aggregator ID is greater than 0, then the
measurement and shares of proof(s) are represented by XOF seeds. We
shall call the former the "Leader" and the latter the "Helpers".
In total there are four variants of the input share. When joint
randomness is not used, the Leader's share is structured as follows:
struct {
Prio3Field meas_share[F * Prio3.flp.MEAS_LEN];
Prio3Field proofs_share[F * Prio3.flp.PROOF_LEN * Prio3.PROOFS];
} Prio3LeaderShare;
When joint randomness is not used, the Helpers' shares are structured
as follows:
struct {
Prio3Seed k_meas_share;
Prio3Seed k_proofs_share;
} Prio3HelperShare;
When joint randomness is used, the Leader's input share is structured
as follows:
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struct {
Prio3LeaderShare inner;
Prio3Seed k_blind;
} Prio3LeaderShareWithJointRand;
Finally, when joint randomness is used, the Helpers' shares are
structured as follows:
struct {
Prio3HelperShare inner;
Prio3Seed k_blind;
} Prio3HelperShareWithJointRand;
7.2.7.3. Prep Share
When joint randomness is not used, the prep share is structured as
follows:
struct {
Prio3Field verifiers_share[
F * Prio3.flp.VERIFIER_LEN * Prio3.PROOFS
];
} Prio3PrepShare;
When joint randomness is used, the prep share includes the
Aggregator's joint randomness part and is structured as follows:
struct {
Prio3Field verifiers_share[
F * Prio3.flp.VERIFIER_LEN * Prio3.PROOFS
];
Prio3Seed k_joint_rand_part;
} Prio3PrepShareWithJointRand;
7.2.7.4. Prep Message
When joint randomness is not used, the prep message is the empty
string. Otherwise the prep message consists of the joint randomness
seed computed by the Aggregators:
struct {
Prio3Seed k_joint_rand;
} Prio3PrepMessageWithJointRand;
7.2.7.5. Aggregation
Aggregate shares are structured as follows:
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struct {
Prio3Field agg_share[F * Prio3.flp.OUTPUT_LEN];
} Prio3AggShare;
7.3. The FLP of [BBCGGI19]
This section describes an FLP based on the construction from in
[BBCGGI19], Section 4.2. We begin in Section 7.3.1 with an overview
of their proof system and the extensions to their proof system made
here. The construction is specified in Section 7.3.3.
OPEN ISSUE Chris Wood points out that the this section reads more
like a paper than a standard. Eventually we'll want to work this
into something that is readily consumable by the CFRG.
7.3.1. Overview
In the proof system of [BBCGGI19], validity is defined via an
arithmetic circuit evaluated over the encoded measurement: If the
circuit output is zero, then the measurement is deemed valid;
otherwise, if the circuit output is nonzero, then the measurement is
deemed invalid. Thus the goal of the proof system is merely to allow
the verifier to evaluate the validity circuit over the measurement.
For our application (Section 7), this computation is distributed
among multiple Aggregators, each of which has only a share of the
measurement.
Suppose for a moment that the validity circuit C is affine, meaning
its only operations are addition and multiplicationbyconstant. In
particular, suppose the circuit does not contain a multiplication
gate whose operands are both nonconstant. Then to decide if a
measurement x is valid, each Aggregator could evaluate C on its share
of x locally, broadcast the output share to its peers, then combine
the output shares locally to recover C(x). This is true because for
any SHARESway secret sharing of x it holds that
C(x_shares[0] + ... + x_shares[SHARES1]) =
C(x_shares[0]) + ... + C(x_shares[SHARES1])
(Note that, for this equality to hold, it may be necessary to scale
any constants in the circuit by SHARES.) However this is not the
case if C is notaffine (i.e., it contains at least one
multiplication gate whose operands are nonconstant). In the proof
system of [BBCGGI19], the proof is designed to allow the
(distributed) verifier to compute the nonaffine operations using
only linear operations on (its share of) the measurement and proof.
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To make this work, the proof system is restricted to validity
circuits that exhibit a special structure. Specifically, an
arithmetic circuit with "Ggates" (see [BBCGGI19], Definition 5.2) is
composed of affine gates and any number of instances of a
distinguished gate G, which may be nonaffine. We will refer to this
class of circuits as 'gadget circuits' and to G as the "gadget".
As an illustrative example, consider a validity circuit C that
recognizes the set L = set([0], [1]). That is, C takes as input a
length1 vector x and returns 0 if x[0] is in [0,2) and outputs
something else otherwise. This circuit can be expressed as the
following degree2 polynomial:
C(x) = (x[0]  1) * x[0] = x[0]^2  x[0]
This polynomial recognizes L because x[0]^2 = x[0] is only true if
x[0] == 0 or x[0] == 1. Notice that the polynomial involves a non
affine operation, x[0]^2. In order to apply [BBCGGI19], Theorem 4.3,
the circuit needs to be rewritten in terms of a gadget that subsumes
this nonaffine operation. For example, the gadget might be
multiplication:
Mul(left, right) = left * right
The validity circuit can then be rewritten in terms of Mul like so:
C(x[0]) = Mul(x[0], x[0])  x[0]
The proof system of [BBCGGI19] allows the verifier to evaluate each
instance of the gadget (i.e., Mul(x[0], x[0]) in our example) using a
linear function of the measurement and proof. The proof is
constructed roughly as follows. Let C be the validity circuit and
suppose the gadget is arityL (i.e., it has L input wires.). Let
wire[j1,k1] denote the value of the jth wire of the kth call to the
gadget during the evaluation of C(x). Suppose there are M such calls
and fix distinct field elements alpha[0], ..., alpha[M1]. (We will
require these points to have a special property, as we'll discuss in
Section 7.3.1.1; but for the moment it is only important that they
are distinct.)
The prover constructs from wire and alpha a polynomial that, when
evaluated at alpha[k1], produces the output of the kth call to the
gadget. Let us call this the "gadget polynomial". Polynomial
evaluation is linear, which means that, in the distributed setting,
the Client can disseminate additive shares of the gadget polynomial
that the Aggregators then use to compute additive shares of each
gadget output, allowing each Aggregator to compute its share of C(x)
locally.
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There is one more wrinkle, however: It is still possible for a
malicious prover to produce a gadget polynomial that would result in
C(x) being computed incorrectly, potentially resulting in an invalid
measurement being accepted. To prevent this, the verifier performs a
probabilistic test to check that the gadget polynomial is well
formed. This test, and the procedure for constructing the gadget
polynomial, are described in detail in Section 7.3.3.
7.3.1.1. Extensions
The FLP described in the next section extends the proof system of
[BBCGGI19], Section 4.2 in a few ways.
First, the validity circuit in our construction includes an
additional, random input (this is the "joint randomness" derived from
the measurement shares in Prio3; see Section 7.2). This allows for
circuit optimizations that trade a small soundness error for a
shorter proof. For example, consider a circuit that recognizes the
set of lengthN vectors for which each element is either one or zero.
A deterministic circuit could be constructed for this language, but
it would involve a large number of multiplications that would result
in a large proof. (See the discussion in [BBCGGI19], Section 5.2 for
details). A much shorter proof can be constructed for the following
randomized circuit:
C(meas, r) = r * Range2(meas[0]) + ... + r^N * Range2(meas[N1])
(Note that this is a special case of [BBCGGI19], Theorem 5.2.) Here
meas is the lengthN input and r is a random field element. The
gadget circuit Range2 is the "rangecheck" polynomial described
above, i.e., Range2(x) = x^2  x. The idea is that, if meas is valid
(i.e., each meas[j] is in [0,2)), then the circuit will evaluate to 0
regardless of the value of r; but if meas[j] is not in [0,2) for some
j, the output will be nonzero with high probability.
The second extension implemented by our FLP allows the validity
circuit to contain multiple gadget types. (This generalization was
suggested in [BBCGGI19], Remark 4.5.) This provides additional
flexibility for designing circuits by allowing multiple, nonaffine
subcomponents. For example, the following circuit is allowed:
C(meas, r) = r * Range2(meas[0]) + ... + r^L * Range2(meas[L1]) + \
r^(L+1) * Range3(meas[L]) + ... + r^N * Range3(meas[N1])
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where Range3(x) = x^3  3x^2 + 2x. This circuit checks that the
first L inputs are in range [0,2) and the last NL inputs are in
range [0,3). Of course, the same circuit can be expressed using a
subcomponent that the gadgets have in common, namely Mul, but the
resulting proof would be longer.
Third, [BBCGGI19], Theorem 4.3 makes no restrictions on the choice of
the fixed points alpha[0], ..., alpha[M1], other than to require
that the points are distinct. In this document, the fixed points are
chosen so that the gadget polynomial can be constructed efficiently
using the CooleyTukey FFT ("Fast Fourier Transform") algorithm.
Note that this requires the field to be "FFTfriendly" as defined in
Section 6.1.2.
Finally, the validity circuit in our FLP may have any number of
outputs (at least one). The input is said to be valid if each of the
outputs is zero. To save bandwidth, we take a random linear
combination of the outputs. If each of the outputs is zero, then the
reduced output will be zero; but if one of the outputs is nonzero,
then the reduced output will be nonzero with high probability.
7.3.2. Validity Circuits
The FLP described in Section 7.3.3 is defined in terms of a validity
circuit Valid that implements the interface described here.
A concrete Valid defines the following parameters:
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+=================+=======================================+
 Parameter  Description 
+=================+=======================================+
 GADGETS  A list of gadgets 
+++
 GADGET_CALLS  Number of times each gadget is called 
+++
 MEAS_LEN  Length of the measurement 
+++
 OUTPUT_LEN  Length of the aggregatable output 
+++
 JOINT_RAND_LEN  Length of the random input 
+++
 EVAL_OUTPUT_LEN  Length of the circuit output 
+++
 Measurement  The type of measurement 
+++
 AggResult  Type of the aggregate result 
+++
 field  Class object for the field 
+++
Table 7: Validity circuit parameters.
Each gadget G in GADGETS defines a constant DEGREE that specifies the
circuit's "arithmetic degree". This is defined to be the degree of
the polynomial that computes it. For example, the Mul circuit in
Section 7.3.1 is defined by the polynomial Mul(x) = x * x, which has
degree 2. Hence, the arithmetic degree of this gadget is 2.
Each gadget also defines a parameter ARITY that specifies the
circuit's arity (i.e., the number of input wires).
Gadgets provide a method to evaluate their circuit on a list of
inputs, eval(). The inputs can either belong to the validity
circuit's field, or the polynomial ring over that field.
A concrete Valid provides the following methods for encoding a
measurement as an input vector, truncating an input vector to the
length of an aggregatable output, and converting an aggregated output
to an aggregate result:
* valid.encode(measurement: Measurement) > list[F] returns a vector
of length MEAS_LEN representing a measurement.
* valid.truncate(meas: list[F]) > list[F] returns a vector of
length OUTPUT_LEN representing an aggregatable output.
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* valid.decode(output: list[F], num_measurements: int) > AggResult
returns an aggregate result.
Finally, the following methods are derived for each concrete Valid:
def prove_rand_len(self) > int:
"""Length of the prover randomness."""
return sum(g.ARITY for g in self.GADGETS)
def query_rand_len(self) > int:
"""Length of the query randomness."""
query_rand_len = len(self.GADGETS)
if self.EVAL_OUTPUT_LEN > 1:
query_rand_len += 1
return query_rand_len
def proof_len(self) > int:
"""Length of the proof."""
length = 0
for (g, g_calls) in zip(self.GADGETS, self.GADGET_CALLS):
P = next_power_of_2(1 + g_calls)
length += g.ARITY + g.DEGREE * (P  1) + 1
return length
def verifier_len(self) > int:
"""Length of the verifier message."""
length = 1
for g in self.GADGETS:
length += g.ARITY + 1
return length
def check_valid_eval(
self,
meas: list[F],
joint_rand: list[F]) > None:
if len(meas) != self.MEAS_LEN:
raise ValueError('incorrect measurement length')
if len(joint_rand) != self.JOINT_RAND_LEN:
raise ValueError('incorrect joint randomness length')
Figure 22: Derived methods for validity circuits.
7.3.3. Construction
This section specifies an implementation of the Flp interface
(Section 7.1). It has as a generic parameter a validity circuit
Valid implementing the interface defined in Section 7.3.2.
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The parameters are defined in Table 8. The required methods for
generating the proof, generating the verifier, and deciding validity
are specified in the remaining subsections.
In the remainder, we let [n] denote the set {1, ..., n} for positive
integer n. We also define the following constants:
* Let H = len(valid.GADGETS)
* For each i in [H]:
 Let G_i = valid.GADGETS[i]
 Let L_i = valid.GADGETS[i].ARITY
 Let M_i = valid.GADGET_CALLS[i]
 Let P_i = next_power_of_2(M_i+1)
 Let alpha_i = field.gen()^(field.GEN_ORDER / P_i)
+================+============================================+
 Parameter  Value 
+================+============================================+
 PROVE_RAND_LEN  valid.prove_rand_len() (see Section 7.3.2) 
+++
 QUERY_RAND_LEN  valid.query_rand_len() (see Section 7.3.2) 
+++
 JOINT_RAND_LEN  valid.JOINT_RAND_LEN 
+++
 MEAS_LEN  valid.MEAS_LEN 
+++
 OUTPUT_LEN  valid.OUTPUT_LEN 
+++
 PROOF_LEN  valid.proof_len() (see Section 7.3.2) 
+++
 VERIFIER_LEN  valid.verifier_len() (see Section 7.3.2) 
+++
 Measurement  valid.Measurement 
+++
Table 8: Parameters of FLP of BBCGGI19.
7.3.3.1. Proof Generation
On input of meas, prove_rand, and joint_rand, the proof is computed
as follows:
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1. For each i in [H] create an empty table wire_i.
2. Partition the prover randomness prove_rand into subvectors
seed_1, ..., seed_H where len(seed_i) == L_i for all i in [H].
Let us call these the "wire seeds" of each gadget.
3. Evaluate Valid on input of meas and joint_rand, recording the
inputs of each gadget in the corresponding table. Specifically,
for every i in [H], set wire_i[j1,k1] to the value on the jth
wire into the kth call to gadget G_i.
4. Compute the "wire polynomials". That is, for every i in [H] and
j in [L_i], construct poly_wire_i[j1], the jth wire polynomial
for the ith gadget, as follows:
* Let w = [seed_i[j1], wire_i[j1,0], ..., wire_i[j1,M_i1]].
* Let padded_w = w + field.zeros(P_i  len(w)).
NOTE We pad w to the nearest power of 2 so that we can use FFT
for interpolating the wire polynomials. Perhaps there is some
clever math for picking wire_inp in a way that avoids having
to pad.
* Let poly_wire_i[j1] be the lowest degree polynomial for which
poly_wire_i[j1](alpha_i^k) == padded_w[k] for all k in [P_i].
5. Compute the "gadget polynomials". That is, for every i in [H]:
* Let poly_gadget_i = G_i(poly_wire_i[0], ..., poly_wire_i[L_i
1]). That is, evaluate the circuit G_i on the wire
polynomials for the ith gadget. (Arithmetic is in the ring of
polynomials over field.)
The proof is the vector proof = seed_1 + coeff_1 + ... + seed_H +
coeff_H, where coeff_i is the vector of coefficients of poly_gadget_i
for each i in [H].
7.3.3.2. Query Generation
On input of meas, proof, query_rand, and joint_rand, the verifier
message is generated as follows:
1. For every i in [H] create an empty table wire_i.
2. Partition proof into the subvectors seed_1, coeff_1, ...,
seed_H, coeff_H defined in Section 7.3.3.1.
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3. Evaluate Valid on input of meas and joint_rand, recording the
inputs of each gadget in the corresponding table. This step is
similar to the prover's step (3.) except the verifier does not
evaluate the gadgets. Instead, it computes the output of the kth
call to G_i by evaluating poly_gadget_i(alpha_i^k). Let out
denote the output of the circuit evaluation.
4. Next, reduce out as follows. If EVAL_OUTPUT_LEN > 1, then
consume the first element of query_rand by letting [r],
query_rand = front(1, query_rand). Then let v = r*out[0] +
r**2*out[1] + r**3*out[2] + .... That is, interpret the outputs
as coefficients of a polynomial f(x) and evaluate polynomial
f(x)*x at a random point r.
5. Compute the wire polynomials just as in the prover's step (4.).
6. Compute the tests for wellformedness of the gadget polynomials.
That is, for every i in [H]:
* Let t = query_rand[i]. Check if t^(P_i) == 1: If so, then
raise ERR_ABORT and halt. (This prevents the verifier from
inadvertently leaking a gadget output in the verifier
message.)
* Let y_i = poly_gadget_i(t).
* For each j in [0,L_i) let x_i[j1] = poly_wire_i[j1](t).
The verifier message is the vector verifier = [v] + x_1 + [y_1] + ...
+ x_H + [y_H].
7.3.3.3. Decision
On input of vector verifier, the verifier decides if the measurement
is valid as follows:
1. Parse verifier into v, x_1, y_1, ..., x_H, y_H as defined in
Section 7.3.3.2.
2. Check for wellformedness of the gadget polynomials. For every i
in [H]:
* Let z = G_i(x_i). That is, evaluate the circuit G_i on x_i
and set z to the output.
* If z != y_i, then return False and halt.
3. Return True if v == 0 and False otherwise.
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7.3.3.4. Encoding
The FLP encoding and truncation methods invoke valid.encode,
valid.truncate, and valid.decode in the natural way.
7.4. Instantiations
This section specifies instantiations of Prio3 for various
measurement types. Each is determined by a field (Section 6.1), a
validity circuit (Section 7.3.2), an XOF (Section 6.2). and the
number of proofs to generate and verify. Test vectors for each can
be found in Appendix "Test Vectors".
7.4.1. Prio3Count
+===========+==================================+
 Parameter  Value 
+===========+==================================+
 Valid  Count(Field64) (this section) 
+++
 Field  Field64 (Table 3) 
+++
 PROOFS  1 
+++
 Xof  XofTurboShake128 (Section 6.2.1) 
+++
Table 9: Parameters for Prio3Count.
Our first instance of Prio3 is for a simple counter: Each measurement
is either one or zero and the aggregate result is the sum of the
measurements.
Its validity circuit, denoted Count, uses the following degree2,
arity2 gadget, denoted Mul:
def eval(self, _field: type[F], inp: list[F]) > F:
self.check_gadget_eval(inp)
return inp[0] * inp[1]
The call to check_gadget_eval() raises an error if the length of the
input is not equal to the gadget's ARITY parameter.
The Count validity circuit is defined as
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def eval(
self,
meas: list[F],
joint_rand: list[F],
_num_shares: int) > list[F]:
self.check_valid_eval(meas, joint_rand)
squared = self.GADGETS[0].eval(self.field, [meas[0], meas[0]])
return [squared  meas[0]]
The measurement is encoded and decoded as a singleton vector in the
natural way. The parameters for this circuit are summarized below.
+=================+=================+
 Parameter  Value 
+=================+=================+
 GADGETS  [Mul] 
+++
 GADGET_CALLS  [1] 
+++
 MEAS_LEN  1 
+++
 OUTPUT_LEN  1 
+++
 JOINT_RAND_LEN  0 
+++
 EVAL_OUTPUT_LEN  1 
+++
 Measurement  int in range(2) 
+++
 AggResult  int 
+++
Table 10: Parameters of validity
circuit Count.
7.4.2. Prio3Sum
+===========+====================================+
 Parameter  Value 
+===========+====================================+
 Valid  Sum(Field128, bits) (this section) 
+++
 Field  Field128 (Table 3) 
+++
 PROOFS  1 
+++
 Xof  XofTurboShake128 (Section 6.2.1) 
+++
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Table 11: Parameters for Prio3Sum.
The next instance of Prio3 supports summing of integers in a pre
determined range. Each measurement is an integer in range [0,
2^bits), where bits is an associated parameter.
The validity circuit is denoted Sum. The measurement is encoded as a
lengthbits vector of field elements, where the lth element of the
vector represents the lth bit of the summand:
def encode(self, measurement: int) > list[F]:
if 0 > measurement or measurement >= 2 ** self.MEAS_LEN:
raise ValueError('measurement out of range')
return self.field.encode_into_bit_vector(measurement,
self.MEAS_LEN)
def truncate(self, meas: list[F]) > list[F]:
return [self.field.decode_from_bit_vector(meas)]
def decode(
self,
output: list[F],
_num_measurements: int) > int:
return output[0].as_unsigned()
The validity circuit checks that the input consists of ones and
zeros. Its gadget, denoted Range2, is the degree2, arity1 gadget
defined as
def eval(self, _field: type[F], inp: list[F]) > F:
self.check_gadget_eval(inp)
return inp[0] * inp[0]  inp[0]
The Sum validity circuit is defined as
def eval(
self,
meas: list[F],
joint_rand: list[F],
_num_shares: int) > list[F]:
self.check_valid_eval(meas, joint_rand)
out = self.field(0)
r = joint_rand[0]
for b in meas:
out += r * self.GADGETS[0].eval(self.field, [b])
r *= joint_rand[0]
return [out]
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+=================+=======================+
 Parameter  Value 
+=================+=======================+
 GADGETS  [Range2] 
+++
 GADGET_CALLS  [bits] 
+++
 MEAS_LEN  bits 
+++
 OUTPUT_LEN  1 
+++
 JOINT_RAND_LEN  1 
+++
 EVAL_OUTPUT_LEN  1 
+++
 Measurement  int in range(2**bits) 
+++
 AggResult  int 
+++
Table 12: Parameters of validity
circuit Sum.
7.4.3. Prio3SumVec
+===========+=============================+
 Parameter  Value 
+===========+=============================+
 Valid  Sum(Field128, length, bits, 
  chunk_lengh) (this section) 
+++
 Field  Field128 (Table 3) 
+++
 PROOFS  1 
+++
 Xof  XofTurboShake128 
  (Section 6.2.1) 
+++
Table 13: Parameters for Prio3SumVec.
This instance of Prio3 supports summing a vector of integers. It has
three parameters, length, bits, and chunk_length. Each measurement
is a vector of positive integers with length equal to the length
parameter. Each element of the measurement is an integer in the
range [0, 2^bits). It is RECOMMENDED to set chunk_length to an
integer near the square root of length * bits (see Section 7.4.3.1).
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The validity circuit is denoted SumVec. Measurements are encoded as
a vector of field elements with length length * bits. The field
elements in the encoded vector represent all the bits of the
measurement vector's elements, consecutively, in LSB to MSB order:
def encode(self, measurement: list[int]) > list[F]:
if len(measurement) != self.length:
raise ValueError('incorrect measurement length')
encoded = []
for val in measurement:
if val not in range(2**self.bits):
raise ValueError(
'entry of measurement vector is out of range'
)
encoded += self.field.encode_into_bit_vector(val, self.bits)
return encoded
def truncate(self, meas: list[F]) > list[F]:
truncated = []
for i in range(self.length):
truncated.append(self.field.decode_from_bit_vector(
meas[i * self.bits: (i + 1) * self.bits]
))
return truncated
def decode(
self,
output: list[F],
_num_measurements: int) > list[int]:
return [x.as_unsigned() for x in output]
This validity circuit uses a ParallelSum gadget to achieve a smaller
proof size. This optimization for "parallelsum circuits" is
described in [BBCGGI19], section 4.4. Briefly, for circuits that add
up the output of multiple identical subcircuits, it is possible to
achieve smaller proof sizes (on the order of O(sqrt(MEAS_LEN))
instead of O(MEAS_LEN)) by packaging more than one such subcircuit
into a gadget.
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The ParallelSum gadget is parameterized with an arithmetic
subcircuit, and a count of how many times it evaluates that
subcircuit. It takes in a list of inputs and passes them through to
instances of the subcircuit in the same order. It returns the sum of
the subcircuit outputs. Note that only the ParallelSum gadget
itself, and not its subcircuit, participates in the FLP's wire
recording during evaluation, gadget consistency proofs, and proof
validation, even though the subcircuit is provided to ParallelSum as
an implementation of the Gadget interface.
def eval(self, field: type[F], inp: list[F]) > F:
self.check_gadget_eval(inp)
out = field(0)
for i in range(self.count):
start_index = i * self.subcircuit.ARITY
end_index = (i + 1) * self.subcircuit.ARITY
out += self.subcircuit.eval(
field,
inp[start_index:end_index],
)
return out
The SumVec validity circuit checks that the encoded measurement
consists of ones and zeros. Rather than use the Range2 gadget on
each element, as in the Sum validity circuit, it instead uses Mul
subcircuits and "free" constant multiplication and addition gates to
simultaneously evaluate the same range check polynomial on each
element, and multiply by a constant. One of the two Mul subcircuit
inputs is equal to a measurement element multiplied by a power of the
joint randomness value, and the other is equal to the same
measurement element minus one. These Mul subcircuits are evaluated
by a ParallelSum gadget, and the results are added up both within the
ParallelSum gadget and after it.
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def eval(
self,
meas: list[F],
joint_rand: list[F],
num_shares: int) > list[F]:
self.check_valid_eval(meas, joint_rand)
out = self.field(0)
r = joint_rand[0]
r_power = r
shares_inv = self.field(num_shares).inv()
for i in range(self.GADGET_CALLS[0]):
inputs: list[Optional[F]]
inputs = [None] * (2 * self.chunk_length)
for j in range(self.chunk_length):
index = i * self.chunk_length + j
if index < len(meas):
meas_elem = meas[index]
else:
meas_elem = self.field(0)
inputs[j * 2] = r_power * meas_elem
inputs[j * 2 + 1] = meas_elem  shares_inv
r_power *= r
out += self.GADGETS[0].eval(
self.field,
cast(list[F], inputs),
)
return [out]
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+=================+====================================+
 Parameter  Value 
+=================+====================================+
 GADGETS  [ParallelSum(Mul(), chunk_length)] 
+++
 GADGET_CALLS  [(length * bits + chunk_length  
  1) // chunk_length] 
+++
 MEAS_LEN  length * bits 
+++
 OUTPUT_LEN  length 
+++
 JOINT_RAND_LEN  1 
+++
 EVAL_OUTPUT_LEN  1 
+++
 Measurement  list[int], each element in 
  range(2**bits) 
+++
 AggResult  list[int] 
+++
Table 14: Parameters of validity circuit SumVec.
7.4.3.1. Selection of ParallelSum chunk length
The chunk_length parameter provides a tradeoff between the arity of
the ParallelSum gadget and the number of times the gadget is called.
The proof length is asymptotically minimized when the chunk length is
near the square root of the length of the measurement. However, the
relationship between VDAF parameters and proof length is complicated,
involving two forms of rounding (the circuit pads the inputs to its
last ParallelSum gadget call, up to the chunk length, and proof
system rounds the degree of wire polynomials  determined by the
number of times a gadget is called  up to the next power of two).
Therefore, the optimal choice of chunk_length for a concrete
measurement size will vary, and must be found through trial and
error. Setting chunk_length equal to the square root of the
appropriate measurement length will result in proofs up to 50% larger
than the optimal proof size.
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7.4.4. Prio3Histogram
+===========+===================================================+
 Parameter  Value 
+===========+===================================================+
 Valid  Sum(Field128, length, chunk_lengh) (this section) 
+++
 Field  Field128 (Table 3) 
+++
 PROOFS  1 
+++
 Xof  XofTurboShake128 (Section 6.2.1) 
+++
Table 15: Parameters for Prio3Histogram.
This instance of Prio3 allows for estimating the distribution of some
quantity by computing a simple histogram. Each measurement
increments one histogram bucket, out of a set of fixed buckets.
(Bucket indexing begins at 0.) For example, the buckets might
quantize the real numbers, and each measurement would report the
bucket that the corresponding client's realnumbered value falls
into. The aggregate result counts the number of measurements in each
bucket.
The validity circuit is denoted Histogram. It has two parameters,
length, the number of histogram buckets, and chunk_length, which is
used by by a circuit optimization described below. It is RECOMMENDED
to set chunk_length to an integer near the square root of length (see
Section 7.4.3.1).
The measurement is encoded as a onehot vector representing the
bucket into which the measurement falls:
def encode(self, measurement: int) > list[F]:
encoded = [self.field(0)] * self.length
encoded[measurement] = self.field(1)
return encoded
def truncate(self, meas: list[F]) > list[F]:
return meas
def decode(
self,
output: list[F],
_num_measurements: int) > list[int]:
return [bucket_count.as_unsigned() for bucket_count in output]
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The Histogram validity circuit checks for onehotness in two steps,
by checking that the encoded measurement consists of ones and zeros,
and by checking that the sum of all elements in the encoded
measurement is equal to one. All the individual checks are combined
together in a random linear combination.
As in the SumVec validity circuit (Section 7.4.3), the first part of
the validity circuit uses the ParallelSum gadget to perform range
checks while achieving a smaller proof size. The ParallelSum gadget
uses Mul subcircuits to evaluate a range check polynomial on each
element, and includes an additional constant multiplication. One of
the two Mul subcircuit inputs is equal to a measurement element
multiplied by a power of the first joint randomness value, and the
other is equal to the same measurement element minus one. The
results are added up both within the ParallelSum gadget and after it.
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def eval(
self,
meas: list[F],
joint_rand: list[F],
num_shares: int) > list[F]:
self.check_valid_eval(meas, joint_rand)
# Check that each bucket is one or zero.
range_check = self.field(0)
r = joint_rand[0]
r_power = r
shares_inv = self.field(num_shares).inv()
for i in range(self.GADGET_CALLS[0]):
inputs: list[Optional[F]]
inputs = [None] * (2 * self.chunk_length)
for j in range(self.chunk_length):
index = i * self.chunk_length + j
if index < len(meas):
meas_elem = meas[index]
else:
meas_elem = self.field(0)
inputs[j * 2] = r_power * meas_elem
inputs[j * 2 + 1] = meas_elem  shares_inv
r_power *= r
range_check += self.GADGETS[0].eval(
self.field,
cast(list[F], inputs),
)
# Check that the buckets sum to 1.
sum_check = shares_inv
for b in meas:
sum_check += b
out = joint_rand[1] * range_check + \
joint_rand[1] ** 2 * sum_check
return [out]
Note that this circuit depends on the number of shares into which the
measurement is sharded. This is provided to the FLP by Prio3.
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+=================+===============================================+
 Parameter  Value 
+=================+===============================================+
 GADGETS  [ParallelSum(Mul(), chunk_length)] 
+++
 GADGET_CALLS  [(length + chunk_length  1) // chunk_length] 
+++
 MEAS_LEN  length 
+++
 OUTPUT_LEN  length 
+++
 JOINT_RAND_LEN  2 
+++
 EVAL_OUTPUT_LEN  1 
+++
 Measurement  int 
+++
 AggResult  list[int] 
+++
Table 16: Parameters of validity circuit Histogram.
7.4.5. Prio3MultihotCountVec
+===========+=================================+
 Parameter  Value 
+===========+=================================+
 Valid  Sum(Field128, length, max_eight 
  chunk_lengh) (this section) 
+++
 Field  Field128 (Table 3) 
+++
 PROOFS  1 
+++
 Xof  XofTurboShake128 
  (Section 6.2.1) 
+++
Table 17: Parameters for Prio3MultihotCountVec.
For this instance of Prio3, each measurement is a vector of ones and
zeros, where the number of ones is bounded. This provides a
functionality similar to Prio3Histogram except that more than one
entry may be nonzero. This allows Prio3MultihotCountVec to be
composed with a randomized response mechanism, like [EPK14], for
providing differential privacy. (For example, each Client would set
each entry to one with some small probability.)
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Prio3MultihotCountVec uses XofTurboShake128 (Section 6.2.1) as its
XOF. Its validity circuit is denoted MultihotCountVec. It has three
parameters: length, the number of of entries in the count vector;
max_weight, the maximum number of nonzero entries (i.e., the weight
must be at most max_weight); and chunk_length, used the same way as
in Section 7.4.3 and Section 7.4.4.
Validation works as follows. Let
* bits_for_weight = max_weight.bit_length()
* offset = 2**bits_for_weight  1  max_weight
The Client reports the weight of the count vector by adding offset to
it and bitencoding the result. Observe that only a weight of at
most max_weight can be encoded with bits_for_weight bits.
The verifier checks that each entry of the encoded measurement is a
bit (i.e., either one or zero). It then decodes the reported weight
and subtracts it from offset + sum(count_vec), where count_vec is the
count vector. The result is zero if and only if the reported weight
is equal to the true weight.
Encoding, truncation, and decoding are defined as follows:
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def encode(self, measurement: list[int]) > list[F]:
if len(measurement) != self.length:
raise ValueError('invalid Client measurement length')
# The first part is the vector of counters.
count_vec = list(map(self.field, measurement))
# The second part is the reported weight.
weight_reported = sum(count_vec, self.field(0))
encoded = []
encoded += count_vec
encoded += self.field.encode_into_bit_vector(
(self.offset + weight_reported).as_unsigned(),
self.bits_for_weight)
return encoded
def truncate(self, meas: list[F]) > list[F]:
return meas[:self.length]
def decode(
self,
output: list[F],
_num_measurements: int) > list[int]:
return [bucket_count.as_unsigned() for bucket_count in output]
Circuit evaluation is defined as follows:
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def eval(
self,
meas: list[F],
joint_rand: list[F],
num_shares: int) > list[F]:
self.check_valid_eval(meas, joint_rand)
# Check that each entry in the input vector is one or zero.
range_check = self.field(0)
r = joint_rand[0]
r_power = r
shares_inv = self.field(num_shares).inv()
for i in range(self.GADGET_CALLS[0]):
inputs: list[Optional[F]]
inputs = [None] * (2 * self.chunk_length)
for j in range(self.chunk_length):
index = i * self.chunk_length + j
if index < len(meas):
meas_elem = meas[index]
else:
meas_elem = self.field(0)
inputs[j * 2] = r_power * meas_elem
inputs[j * 2 + 1] = meas_elem  shares_inv
r_power *= r
range_check += self.GADGETS[0].eval(
self.field,
cast(list[F], inputs),
)
# Check that the weight `offset` plus the sum of the counters
# is equal to the value claimed by the Client.
count_vec = meas[:self.length]
weight = sum(count_vec, self.field(0))
weight_reported = \
self.field.decode_from_bit_vector(meas[self.length:])
weight_check = self.offset*shares_inv + weight  \
weight_reported
out = joint_rand[1] * range_check + \
joint_rand[1] ** 2 * weight_check
return [out]
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+================+====================================+
 Parameter  Value 
+================+====================================+
 GADGETS  [ParallelSum(Mul(), chunk_length)] 
+++
 GADGET_CALLS  [(length + bits_for_weight + 
  chunk_length  1) // chunk_length] 
+++
 MEAS_LEN  length + bits_for_weight 
+++
 OUTPUT_LEN  length 
+++
 JOINT_RAND_LEN  2 
+++
 Measurement  list[int] 
+++
 AggResult  list[int] 
+++
Table 18: Parameters of validity circuit
MultihotCountVec.
8. Poplar1
This section specifies Poplar1, a VDAF for the following task. Each
Client holds a bitstring of length BITS and the Aggregators hold a
sequence of Lbit strings, where L <= BITS. We will refer to the
latter as the set of "candidate prefixes". The Aggregators' goal is
to count how many measurements are prefixed by each candidate prefix.
This functionality is the core component of the Poplar protocol
[BBCGGI21], which was designed to compute the heavy hitters over a
set of input strings. At a high level, the protocol works as
follows.
1. Each Client splits its string into input shares and sends one
share to each Aggregator.
2. The Aggregators agree on an initial set of candidate prefixes,
say 0 and 1.
3. The Aggregators evaluate the VDAF on each set of input shares and
aggregate the recovered output shares. The aggregation parameter
is the set of candidate prefixes.
4. The Aggregators send their aggregate shares to the Collector, who
combines them to recover the counts of each candidate prefix.
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5. Let H denote the set of prefixes that occurred at least t times.
If the prefixes all have length BITS, then H is the set of t
heavyhitters. Otherwise compute the next set of candidate
prefixes, e.g., for each p in H, add p  0 and p  1 to the
set. Repeat step 3 with the new set of candidate prefixes.
Poplar1 is constructed from an "Incremental Distributed Point
Function (IDPF)", a primitive described by [BBCGGI21] that
generalizes the notion of a Distributed Point Function (DPF) [GI14].
Briefly, a DPF is used to distribute the computation of a "point
function", a function that evaluates to zero on every input except at
a programmable "point". The computation is distributed in such a way
that no one party knows either the point or what it evaluates to.
An IDPF generalizes this "point" to a path on a full binary tree from
the root to one of the leaves. It is evaluated on an "index"
representing a unique node of the tree. If the node is on the
programmed path, then the function evaluates to a nonzero value;
otherwise it evaluates to zero. This structure allows an IDPF to
provide the functionality required for the above protocol: To compute
the hit count for an index, just evaluate each set of IDPF shares at
that index and add up the results.
Consider the subtree constructed from a set of input strings and a
target threshold t by including all indices that prefix at least t of
the input strings. We shall refer to this structure as the "prefix
tree" for the batch of inputs and target threshold. To compute the
theavy hitters for a set of inputs, the Aggregators and Collector
first compute the prefix tree, then extract the heavy hitters from
the leaves of this tree. (Note that the prefix tree may leak more
information about the set than the heavy hitters themselves; see
Section 9.4.1 for details.)
Poplar1 composes an IDPF with the arithmetic sketch of [BBCGGI21],
Section 4.2. (The paper calls this a "secure sketch", but the
underlying technique was later generalized in [BBCGGI23], where it is
called "arithmetic sketching".) This protocol ensures that
evaluating a set of input shares on a unique set of candidate
prefixes results in shares of a "onehot" vector, i.e., a vector that
is zero everywhere except for at most one element, which is equal to
one.
The remainder of this section is structured as follows. IDPFs are
defined in Section 8.1; a concrete instantiation is given
Section 8.3. The Poplar1 VDAF is defined in Section 8.2 in terms of
a generic IDPF. Finally, a concrete instantiation of Poplar1 is
specified in Section 8.4; test vectors can be found in Appendix "Test
Vectors".
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8.1. Incremental Distributed Point Functions (IDPFs)
An IDPF is defined over a domain of size 2^BITS, where BITS is a
constant. Indices into the IDPF tree are encoded as integers in
range [0, 2^BITS). (In Poplar1, each Client's bit string is encoded
as an index; see Section 8.1.1 for details.) The Client specifies an
index alpha and a vector of values beta, one for each "level" L in
range [0, BITS). The key generation algorithm generates one IDPF
"key" for each Aggregator. When evaluated at level L and index 0 <=
prefix < 2^L, each IDPF key returns an additive share of beta[L] if
prefix is the Lbit prefix of alpha and shares of zero otherwise.
An index x is defined to be a prefix of another index y as follows.
Let LSB(x, L) denote the least significant L bits of positive integer
x. A positive integer 0 <= x < 2^L is defined to be the lengthL
prefix of positive integer 0 <= y < 2^BITS if LSB(x, L) is equal to
the most significant L bits of LSB(y, BITS), For example, 6 (110 in
binary) is the length3 prefix of 25 (11001), but 7 (111) is not.
Each of the programmed points beta is a vector of elements of some
finite field. We distinguish two types of fields: One for inner
nodes (denoted FieldInner), and one for leaf nodes (FieldLeaf). (Our
instantiation of Poplar1 (Section 8.4) will use a much larger field
for leaf nodes than for inner nodes. This is to ensure the IDPF is
"extractable" as defined in [BBCGGI21], Definition 1.)
A concrete IDPF defines the types and constants enumerated in
Table 19. In the remainder we write Output as shorthand for the type
Union[list[list[FieldInner]], list[list[FieldLeaf]]]. (This type
denotes either a vector of inner node field elements or leaf node
field elements.) The scheme is comprised of the following
algorithms:
* idpf.gen(alpha: int, beta_inner: list[list[FieldInner]],
beta_leaf: list[FieldLeaf], binder: bytes, rand: bytes) >
tuple[bytes, list[bytes]] is the randomized IDPFkey generation
algorithm. Its inputs are the index alpha the values beta, and a
binder string.
The output is a public part that is sent to all Aggregators and a
vector of private IDPF keys, one for each aggregator. The binder
string is used to derive the key in the underlying
XofFixedKeyAes128 XOF that is used for expanding seeds at each
level.
Preconditions:
 alpha MUST be in range(2**BITS).
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 beta_inner MUST have length BITS  1.
 beta_inner[level] MUST have length VALUE_LEN for each level in
range(BITS  1).
 beta_leaf MUST have length VALUE_LEN.
 rand MUST be generated by a CSPRNG and have length RAND_SIZE.
 binder MUST be chosen uniformly at random by the Client (see
Section 9.2).
TODO If the binder needs to be random, then we need to
specify its length so that the user knows how many bytes to
sample.
TODO(issue #255) Decide whether to treat the public share as an
opaque byte string or to replace it with an explicit type.
* idpf.eval(agg_id: int, public_share: bytes, key: bytes, level:
int, prefixes: tuple[int, ...], binder: bytes) > Output is the
deterministic, stateless IDPFkey evaluation algorithm run by each
Aggregator. Its inputs are the Aggregator's unique identifier,
the public share distributed to all of the Aggregators, the
Aggregator's IDPF key, the "level" at which to evaluate the IDPF,
the sequence of candidate prefixes, and a binder string. It
returns the share of the value corresponding to each candidate
prefix.
The output type (i.e., Output) depends on the value of level: If
level < BITS1, the output is the value for an inner node, which
has type list[list[FieldInner]]; otherwise, if level == BITS1,
then the output is the value for a leaf node, which has type
list[list[FieldLeaf]].
Preconditions:
 agg_id MUST be in range(SHARES) and match the index of key in
the sequence of IDPF keys output by the Client.
 level MUST be in range(0, BITS).
 Each prefix in prefixes MUST be distinct and in
range(2**level).
In addition, the following method is derived for each concrete Idpf:
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def current_field(
self,
level: int) > Union[type[FieldInner], type[FieldLeaf]]:
if level < self.BITS  1:
return self.field_inner
return self.field_leaf
Finally, an implementation note. The interface for IDPFs specified
here is stateless, in the sense that there is no state carried
between IDPF evaluations. This is to align the IDPF syntax with the
VDAF abstraction boundary, which does not include shared state across
across VDAF evaluations. In practice, of course, it will often be
beneficial to expose a stateful API for IDPFs and carry the state
across evaluations. See Section 8.3 for details.
+============+====================================================+
 Parameter  Description 
+============+====================================================+
 SHARES  Number of IDPF keys output by IDPFkey generator 
+++
 BITS  Length in bits of each input string 
+++
 VALUE_LEN  Number of field elements of each output value 
+++
 RAND_SIZE  Size of the random string consumed by the IDPFkey 
  generator. Equal to twice the XOF's seed size. 
+++
 KEY_SIZE  Size in bytes of each IDPF key 
+++
 FieldInner  Implementation of Field (Section 6.1) used for 
  values of inner nodes 
+++
 FieldLeaf  Implementation of Field used for values of leaf 
  nodes 
+++
 Output  Alias of Union[list[list[FieldInner]], 
  list[list[FieldLeaf]]] 
+++
 FieldVec  Alias of Union[list[FieldInner], list[FieldLeaf]] 
+++
Table 19: Constants and types defined by a concrete IDPF.
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8.1.1. Encoding inputs as indices
How data are represented as IDPF indices is up to the application.
When the inputs are fixedlength byte strings, the most natural
choice of encoder is from_be_bytes(). This ensures that, when a
string is a prefix of another, so too is its index. (Index prefixes
are defined in Section 8.1). For example,
from_be_bytes(b"\x01\x02") == 0x0102
is a prefix of
from_be_bytes(b"\x01\x02\x03") == 0x010203
When the inputs are variable length, it is necessary to pad each
input to some fixed length. Further, the padding scheme must be non
ambiguous. For example, each input could be padded with b"\x01"
followed by as many b"\x00" bytes as needed.
8.2. Construction
This section specifies Poplar1, an implementation of the Vdaf
interface (Section 5). It is defined in terms of any Idpf
(Section 8.1) for which SHARES == 2 and VALUE_LEN == 2 and an
implementation of Xof (Section 6.2). The associated constants and
types required by the Vdaf interface are defined in Table 20. The
methods required for sharding, preparation, aggregation, and
unsharding are described in the remaining subsections. These methods
make use of constants defined in Table 21.
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+=================+====================================+
 Parameter  Value 
+=================+====================================+
 VERIFY_KEY_SIZE  Xof.SEED_SIZE 
+++
 RAND_SIZE  Xof.SEED_SIZE * 3 + Idpf.RAND_SIZE 
+++
 NONCE_SIZE  16 
+++
 ROUNDS  2 
+++
 SHARES  2 
+++
 Measurement  int 
+++
 AggParam  tuple[int, Sequence[int]] 
+++
 PublicShare  bytes (IDPF public share) 
+++
 InputShare  tuple[bytes, bytes, 
  list[FieldInner], list[FieldLeaf]] 
+++
 OutShare  FieldVec 
+++
 AggShare  FieldVec 
+++
 AggResult  list[int] 
+++
 PrepState  tuple[bytes, int, FieldVec] 
+++
 PrepShare  FieldVec 
+++
 PrepMessage  Optional[FieldVec] 
+++
Table 20: VDAF parameters for Poplar1.
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+========================+=======+
 Variable  Value 
+========================+=======+
 USAGE_SHARD_RAND: int  1 
+++
 USAGE_CORR_INNER: int  2 
+++
 USAGE_CORR_LEAF: int  3 
+++
 USAGE_VERIFY_RAND: int  4 
+++
Table 21: Constants used by
Poplar1.
8.2.1. Sharding
The Client's measurement is an IDPF index, denoted alpha. (See
Section 8.1.1 for guidelines on index encoding.) The programmed IDPF
values are pairs of field elements (1, k) where each k is chosen at
random. This random value is used as part of the arithmetic
sketching protocol of [BBCGGI21], Appendix C.4. After evaluating
their IDPF key shares on a given sequence of candidate prefixes, the
sketching protocol is used by the Aggregators to verify that they
hold shares of a onehot vector. In addition, for each level of the
tree, the prover generates random elements a, b, and c and computes
A = 2*a + k
B = a^2 + b  k*a + c
and sends additive shares of a, b, c, A and B to the Aggregators.
Putting everything together, the sharding algorithm is defined as
follows.
def shard(
self,
measurement: int,
nonce: bytes,
rand: bytes) > tuple[bytes, list[Poplar1InputShare]]:
if len(nonce) != self.NONCE_SIZE:
raise ValueError("incorrect nonce size")
if len(rand) != self.RAND_SIZE:
raise ValueError("incorrect size of random bytes argument")
l = self.xof.SEED_SIZE
# Split the random input into the random input for IDPF key
# generation, correlated randomness, and sharding.
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if len(rand) != self.RAND_SIZE:
raise ValueError('incorrect rand size')
idpf_rand, rand = front(self.idpf.RAND_SIZE, rand)
seeds = [rand[i:i + l] for i in range(0, 3 * l, l)]
corr_seed, seeds = front(2, seeds)
(k_shard,), seeds = front(1, seeds)
xof = self.xof(
k_shard,
self.domain_separation_tag(USAGE_SHARD_RAND),
nonce,
)
# Construct the IDPF values for each level of the IDPF tree.
# Each "data" value is 1; in addition, the Client generates
# a random "authenticator" value used by the Aggregators to
# evaluate the sketch during preparation. This sketch is used
# to verify the onehotness of their output shares.
beta_inner = [
[self.idpf.field_inner(1), k]
for k in xof.next_vec(self.idpf.field_inner,
self.idpf.BITS  1)
]
beta_leaf = [self.idpf.field_leaf(1)] + \
xof.next_vec(self.idpf.field_leaf, 1)
# Generate the IDPF keys.
(public_share, keys) = self.idpf.gen(
measurement,
beta_inner,
beta_leaf,
nonce,
idpf_rand,
)
# Generate correlated randomness used by the Aggregators to
# evaluate the sketch over their output shares. Seeds are used
# to encode shares of the `(a, b, c)` triples. (See [BBCGGI21,
# Appendix C.4].)
corr_offsets: list[Field] = vec_add(
self.xof.expand_into_vec(
self.idpf.field_inner,
corr_seed[0],
self.domain_separation_tag(USAGE_CORR_INNER),
byte(0) + nonce,
3 * (self.idpf.BITS  1),
),
self.xof.expand_into_vec(
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self.idpf.field_inner,
corr_seed[1],
self.domain_separation_tag(USAGE_CORR_INNER),
byte(1) + nonce,
3 * (self.idpf.BITS  1),
),
)
corr_offsets += vec_add(
self.xof.expand_into_vec(
self.idpf.field_leaf,
corr_seed[0],
self.domain_separation_tag(USAGE_CORR_LEAF),
byte(0) + nonce,
3,
),
self.xof.expand_into_vec(
self.idpf.field_leaf,
corr_seed[1],
self.domain_separation_tag(USAGE_CORR_LEAF),
byte(1) + nonce,
3,
),
)
# For each level of the IDPF tree, shares of the `(A, B)`
# pairs are computed from the corresponding `(a, b, c)`
# triple and authenticator value `k`.
corr_inner: list[list[Field64]] = [[], []]
for level in range(self.idpf.BITS):
field = cast(type[Field], self.idpf.current_field(level))
k = beta_inner[level][1] if level < self.idpf.BITS  1 \
else beta_leaf[1]
(a, b, c), corr_offsets = corr_offsets[:3], corr_offsets[3:]
A = field(2) * a + k
B = a ** 2 + b  a * k + c
corr1 = xof.next_vec(field, 2)
corr0 = vec_sub([A, B], corr1)
if level < self.idpf.BITS  1:
corr_inner[0] += cast(list[Field64], corr0)
corr_inner[1] += cast(list[Field64], corr1)
else:
corr_leaf = [
cast(list[Field255], corr0),
cast(list[Field255], corr1),
]
# Each input share consists of the Aggregator's IDPF key
# and a share of the correlated randomness.
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input_shares = list(zip(keys, corr_seed, corr_inner, corr_leaf))
return (public_share, input_shares)
Figure 23: The sharding algorithm for Poplar1.
8.2.2. Preparation
The aggregation parameter encodes a sequence of candidate prefixes.
When an Aggregator receives an input share from the Client, it begins
by evaluating its IDPF share on each candidate prefix, recovering a
data_share and auth_share for each. The Aggregators use these and
the correlation shares provided by the Client to verify that the
sequence of data_share values are additive shares of a onehot
vector.
Aggregators MUST ensure the candidate prefixes are all unique and
appear in lexicographic order. (This is enforced in the definition
of prep_init() below.) Uniqueness is necessary to ensure the refined
measurement (i.e., the sum of the output shares) is in fact a onehot
vector. Otherwise, sketch verification might fail, causing the
Aggregators to erroneously reject a report that is actually valid.
Note that enforcing the order is not strictly necessary, but this
does allow uniqueness to be determined more efficiently.
def prep_init(
self,
verify_key: bytes,
agg_id: int,
agg_param: Poplar1AggParam,
nonce: bytes,
public_share: bytes,
input_share: Poplar1InputShare) > tuple[
Poplar1PrepState,
FieldVec]:
(level, prefixes) = agg_param
(key, corr_seed, corr_inner, corr_leaf) = input_share
field = self.idpf.current_field(level)
# Ensure that candidate prefixes are all unique and appear in
# lexicographic order.
for i in range(1, len(prefixes)):
if prefixes[i  1] >= prefixes[i]:
raise ValueError('out of order prefix')
# Evaluate the IDPF key at the given set of prefixes.
value = self.idpf.eval(
agg_id, public_share, key, level, prefixes, nonce)
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# Get shares of the correlated randomness for evaluating the
# Aggregator's share of the sketch.
if level < self.idpf.BITS  1:
corr_xof = self.xof(
corr_seed,
self.domain_separation_tag(USAGE_CORR_INNER),
byte(agg_id) + nonce,
)
# Fastforward the XOF state to the current level.
corr_xof.next_vec(field, 3 * level)
else:
corr_xof = self.xof(
corr_seed,
self.domain_separation_tag(USAGE_CORR_LEAF),
byte(agg_id) + nonce,
)
(a_share, b_share, c_share) = corr_xof.next_vec(field, 3)
if level < self.idpf.BITS  1:
(A_share, B_share) = cast(
list[Field],
corr_inner[2 * level:2 * (level + 1)],
)
else:
(A_share, B_share) = cast(list[Field], corr_leaf)
# Evaluate the Aggregator's share of the sketch. These are
# called the "masked input values" [BBCGGI21, Appendix C.4].
verify_rand_xof = self.xof(
verify_key,
self.domain_separation_tag(USAGE_VERIFY_RAND),
nonce + to_be_bytes(level, 2),
)
verify_rand = cast(
list[Field],
verify_rand_xof.next_vec(field, len(prefixes)),
)
sketch_share = [a_share, b_share, c_share]
out_share = []
for (i, r) in enumerate(verify_rand):
data_share = cast(Field, value[i][0])
auth_share = cast(Field, value[i][1])
sketch_share[0] += data_share * r
sketch_share[1] += data_share * r ** 2
sketch_share[2] += auth_share * r
out_share.append(data_share)
prep_mem = [A_share, B_share, field(agg_id)] + out_share
return (
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(
b'evaluate sketch',
level,
cast(FieldVec, prep_mem),
),
cast(FieldVec, sketch_share),
)
def prep_next(
self,
prep_state: Poplar1PrepState,
prep_msg: Optional[FieldVec]) > Union[
tuple[Poplar1PrepState, FieldVec],
FieldVec]:
prev_sketch = cast(list[Field], prep_msg)
(step, level, prep_mem) = prep_state
if step == b'evaluate sketch':
if prev_sketch is None:
raise ValueError('expected value, got none')
elif len(prev_sketch) != 3:
raise ValueError('incorrect sketch length')
A_share = cast(Field, prep_mem[0])
B_share = cast(Field, prep_mem[1])
agg_id = cast(Field, prep_mem[2])
prep_mem = prep_mem[3:]
sketch_share = [
agg_id * (prev_sketch[0] ** 2
 prev_sketch[1]
 prev_sketch[2])
+ A_share * prev_sketch[0]
+ B_share
]
return cast(
tuple[Poplar1PrepState, FieldVec],
(
(
b'reveal sketch',
level,
prep_mem,
),
sketch_share,
)
)
elif step == b'reveal sketch':
if prev_sketch is None:
return prep_mem # Output shares
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else:
raise ValueError('invalid prep message')
raise ValueError('invalid prep state')
def prep_shares_to_prep(
self,
agg_param: Poplar1AggParam,
prep_shares: list[FieldVec]) > Optional[FieldVec]:
if len(prep_shares) != 2:
raise ValueError('incorrect number of prep shares')
(level, _) = agg_param
field = self.idpf.current_field(level)
sketch = vec_add(
cast(list[Field], prep_shares[0]),
cast(list[Field], prep_shares[1]),
)
if len(sketch) == 3:
return cast(FieldVec, sketch)
elif len(sketch) == 1:
if sketch == field.zeros(1):
# In order to reduce communication overhead, let `None`
# denote a successful sketch verification.
return None
else:
raise ValueError('sketch verification failed')
else:
raise ValueError('incorrect sketch length')
Figure 24: Preparation state for Poplar1.
8.2.3. Validity of Aggregation Parameters
Aggregation parameters are valid for a given input share if no
aggregation parameter with the same level has been used with the same
input share before. The whole preparation phase MUST NOT be run more
than once for a given combination of input share and level. This
function checks that levels are increasing between calls, and also
enforces that the prefixes at each level are suffixes of the previous
level's prefixes.
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def get_ancestor(
index: int,
this_level: int,
last_level: int) > int:
"""
Helper function to determine the prefix of `index` at
`last_level`.
"""
return index >> (this_level  last_level)
def is_valid(
self,
agg_param: Poplar1AggParam,
previous_agg_params: list[Poplar1AggParam]) > bool:
"""
Checks that levels are increasing between calls, and also
enforces that the prefixes at each level are suffixes of the
previous level's prefixes.
"""
if len(previous_agg_params) < 1:
return True
(level, prefixes) = agg_param
(last_level, last_prefixes) = previous_agg_params[1]
last_prefixes_set = set(last_prefixes)
# Check that level increased.
if level <= last_level:
return False
# Check that prefixes are suffixes of the last level's prefixes.
for prefix in prefixes:
last_prefix = get_ancestor(prefix, level, last_level)
if last_prefix not in last_prefixes_set:
# Current prefix not a suffix of last level's prefixes.
return False
return True
Figure 25: Validity of aggregation parameters for Poplar1.
8.2.4. Aggregation
Aggregation involves simply adding up the output shares.
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def aggregate(
self,
agg_param: Poplar1AggParam,
out_shares: list[FieldVec]) > FieldVec:
(level, prefixes) = agg_param
field = self.idpf.current_field(level)
agg_share = cast(list[Field], field.zeros(len(prefixes)))
for out_share in out_shares:
agg_share = vec_add(agg_share, cast(list[Field], out_share))
return cast(FieldVec, agg_share)
Figure 26: Aggregation algorithm for Poplar1.
8.2.5. Unsharding
Finally, the Collector unshards the aggregate result by adding up the
aggregate shares.
def unshard(
self,
agg_param: Poplar1AggParam,
agg_shares: list[FieldVec],
_num_measurements: int) > list[int]:
(level, prefixes) = agg_param
field = self.idpf.current_field(level)
agg = cast(list[Field], field.zeros(len(prefixes)))
for agg_share in agg_shares:
agg = vec_add(agg, cast(list[Field], agg_share))
return [x.as_unsigned() for x in agg]
Figure 27: Computation of the aggregate result for Poplar1.
8.2.6. Message Serialization
This section defines serialization formats for messages exchanged
over the network while executing Poplar1. It is RECOMMENDED that
implementations provide serialization methods for them.
Message structures are defined following Section 3 of [RFC8446]). In
the remainder we use S as an alias for Poplar1.xof.SEED_SIZE, Fi as
an alias for Poplar1.idpf.field_inner and Fl as an alias for
Poplar1.idpf.field_leaf. XOF seeds are represented as follows:
opaque Poplar1Seed[S];
Elements of the inner field are encoded in littleendian byte order
(as defined in Section 6.1) and are represented as follows:
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opaque Poplar1FieldInner[Fi];
Likewise, elements of the leaf field are encoded in littleendian
byte order (as defined in Section 6.1) and are represented as
follows:
opaque Poplar1FieldLeaf[Fl];
8.2.6.1. Public Share
The public share is equal to the IDPF public share, which is a byte
string. (See Section 8.1.)
8.2.6.2. Input Share
Each input share is structured as follows:
struct {
opaque idpf_key[Poplar1.Idpf.KEY_SIZE];
Poplar1Seed corr_seed;
Poplar1FieldInner corr_inner[Fi * 2 * (Poplar1.Idpf.BITS  1)];
Poplar1FieldLeaf corr_leaf[Fl * 2];
} Poplar1InputShare;
8.2.6.3. Prep Share
Encoding of the prep share depends on the round of sketching: if the
first round, then each sketch share has three field elements; if the
second round, then each sketch share has one field element. The
field that is used depends on the level of the IDPF tree specified by
the aggregation parameter, either the inner field or the leaf field.
For the first round and inner field:
struct {
Poplar1FieldInner sketch_share[Fi * 3];
} Poplar1PrepShareRoundOneInner;
For the first round and leaf field:
struct {
Poplar1FieldLeaf sketch_share[Fl * 3];
} Poplar1PrepShareRoundOneLeaf;
For the second round and inner field:
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struct {
Poplar1FieldInner sketch_share;
} Poplar1PrepShareRoundTwoInner;
For the second round and leaf field:
struct {
Poplar1FieldLeaf sketch_share;
} Poplar1PrepShareRoundTwoLeaf;
8.2.6.4. Prep Message
Likewise, the structure of the prep message for Poplar1 depends on
the sketching round and field. For the first round and inner field:
struct {
Poplar1FieldInner[Fi * 3];
} Poplar1PrepMessageRoundOneInner;
For the first round and leaf field:
struct {
Poplar1FieldLeaf sketch[Fl * 3];
} Poplar1PrepMessageRoundOneLeaf;
Note that these messages have the same structures as the prep shares
for the first round.
The secondround prep message is the empty string. This is because
the sketch shares are expected to sum to a particular value if the
output shares are valid; we represent a successful preparation with
the empty string and otherwise return an error.
8.2.6.5. Aggregate Share
The encoding of the aggregate share depends on whether the inner or
leaf field is used, and the number of candidate prefixes. Both of
these are determined by the aggregation parameter.
Let prefix_count denote the number of candidate prefixes. For the
inner field:
struct {
Poplar1FieldInner agg_share[Fi * prefix_count];
} Poplar1AggShareInner;
For the leaf field:
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struct {
Poplar1FieldLeaf agg_share[Fl * prefix_count];
} Poplar1AggShareLeaf;
8.2.6.6. Aggregation Parameter
The aggregation parameter is encoded as follows:
TODO(issue #255) Express the aggregation parameter encoding in TLS
syntax. Decide whether to RECOMMEND this encoding, and if so, add
it to test vectors.
def encode_agg_param(self, agg_param: Poplar1AggParam) > bytes:
level, prefixes = agg_param
if level not in range(2 ** 16):
raise ValueError('level out of range')
if len(prefixes) not in range(2 ** 32):
raise ValueError('number of prefixes out of range')
encoded = bytes()
encoded += to_be_bytes(level, 2)
encoded += to_be_bytes(len(prefixes), 4)
packed = 0
for (i, prefix) in enumerate(prefixes):
packed = prefix << ((level + 1) * i)
l = ((level + 1) * len(prefixes) + 7) // 8
encoded += to_be_bytes(packed, l)
return encoded
def decode_agg_param(self, encoded: bytes) > Poplar1AggParam:
encoded_level, encoded = encoded[:2], encoded[2:]
level = from_be_bytes(encoded_level)
encoded_prefix_count, encoded = encoded[:4], encoded[4:]
prefix_count = from_be_bytes(encoded_prefix_count)
l = ((level + 1) * prefix_count + 7) // 8
encoded_packed, encoded = encoded[:l], encoded[l:]
packed = from_be_bytes(encoded_packed)
prefixes = []
m = 2 ** (level + 1)  1
for i in range(prefix_count):
prefixes.append(packed >> ((level + 1) * i) & m)
if len(encoded) != 0:
raise ValueError('trailing bytes')
return (level, tuple(prefixes))
Implementation note: The aggregation parameter includes the level of
the IDPF tree and the sequence of indices to evaluate. For
implementations that perform perreport caching across executions of
the VDAF, this may be more information than is strictly needed. In
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particular, it may be sufficient to convey which indices from the
previous execution will have their children included in the next.
This would help reduce communication overhead.
8.3. The IDPF scheme of [BBCGGI21]
In this section we specify a concrete IDPF suitable for instantiating
Poplar1. The scheme gets its name from the name of the protocol of
[BBCGGI21].
The constant and type definitions required by the Idpf interface are
given in Table 22.
Our IDPF requires an XOF for deriving the output shares, as well as a
variety of other artifacts used internally. For performance reasons,
we instantiate this object using XofFixedKeyAes128 (Section 6.2.2).
See Section 9.5 for justification of this choice.
+============+======================+
 Parameter  Value 
+============+======================+
 SHARES  2 
+++
 BITS  any positive integer 
+++
 VALUE_LEN  any positive integer 
+++
 KEY_SIZE  Xof.SEED_SIZE 
+++
 FieldInner  Field64 (Table 3) 
+++
 FieldLeaf  Field255 (Table 3) 
+++
Table 22: Constants and type
definitions for the IDPF of
BBCGGI21.
8.3.1. Key Generation
TODO Describe the construction in prose, beginning with a gentle
introduction to the high level idea.
The description of the IDPFkey generation algorithm makes use of
auxiliary functions extend(), convert(), and encode_public_share()
defined in Section 8.3.3. In the following, we let Field2 denote the
field GF(2).
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def gen(
self,
alpha: int,
beta_inner: list[list[Field64]],
beta_leaf: list[Field255],
binder: bytes,
rand: bytes) > tuple[bytes, list[bytes]]:
if alpha not in range(2 ** self.BITS):
raise ValueError("alpha out of range")
if len(beta_inner) != self.BITS  1:
raise ValueError("incorrect beta_inner length")
if len(rand) != self.RAND_SIZE:
raise ValueError("incorrect rand size")
key = [
rand[:XofFixedKeyAes128.SEED_SIZE],
rand[XofFixedKeyAes128.SEED_SIZE:],
]
seed = key.copy()
ctrl = [Field2(0), Field2(1)]
correction_words = []
for level in range(self.BITS):
field: type[Field]
field = cast(type[Field], self.current_field(level))
keep = (alpha >> (self.BITS  level  1)) & 1
lose = 1  keep
bit = Field2(keep)
(s0, t0) = self.extend(seed[0], binder)
(s1, t1) = self.extend(seed[1], binder)
seed_cw = xor(s0[lose], s1[lose])
ctrl_cw = (
t0[0] + t1[0] + bit + Field2(1),
t0[1] + t1[1] + bit,
)
x0 = xor(s0[keep], ctrl[0].conditional_select(seed_cw))
x1 = xor(s1[keep], ctrl[1].conditional_select(seed_cw))
(seed[0], w0) = self.convert(level, x0, binder)
(seed[1], w1) = self.convert(level, x1, binder)
ctrl[0] = t0[keep] + ctrl[0] * ctrl_cw[keep]
ctrl[1] = t1[keep] + ctrl[1] * ctrl_cw[keep]
if level < self.BITS  1:
b = cast(list[Field], beta_inner[level])
else:
b = cast(list[Field], beta_leaf)
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if len(b) != self.VALUE_LEN:
raise ValueError(
"length of beta must match the value length"
)
w_cw = vec_add(vec_sub(b, w0), w1)
# Implementation note: Here we negate the correction word if
# the control bit `ctrl[1]` is set. We avoid branching on the
# value in order to reduce leakage via timing side channels.
mask = field(1)  field(2) * field(ctrl[1].as_unsigned())
for i in range(len(w_cw)):
w_cw[i] *= mask
correction_words.append((seed_cw, ctrl_cw, w_cw))
public_share = self.encode_public_share(correction_words)
return (public_share, key)
Figure 28: IDPFkey generation algorithm of BBCGGI21.
8.3.2. Key Evaluation
TODO Describe in prose how IDPFkey evaluation algorithm works.
The description of the IDPFevaluation algorithm makes use of
auxiliary functions extend(), convert(), and decode_public_share()
defined in Section 8.3.3.
def eval(
self,
agg_id: int,
public_share: bytes,
key: bytes,
level: int,
prefixes: Sequence[int],
binder: bytes) > Union[
list[list[Field64]],
list[list[Field255]]]:
if agg_id not in range(self.SHARES):
raise ValueError('aggregator id out of range')
if level not in range(self.BITS):
raise ValueError('level out of range')
if len(set(prefixes)) != len(prefixes):
raise ValueError('prefixes must be unique')
correction_words = self.decode_public_share(public_share)
out_share = []
for prefix in prefixes:
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if prefix not in range(2 ** (level + 1)):
raise ValueError('prefix out of range')
# The Aggregator's output share is the value of a node of
# the IDPF tree at the given `level`. The node's value is
# computed by traversing the path defined by the candidate
# `prefix`. Each node in the tree is represented by a seed
# (`seed`) and a control bit (`ctrl`).
seed = key
ctrl = Field2(agg_id)
y: FieldVec
for current_level in range(level + 1):
bit = (prefix >> (level  current_level)) & 1
# Implementation note: Typically the current round of
# candidate prefixes would have been derived from
# aggregate results computed during previous rounds.
# For example, when using the IDPF to compute heavy
# hitters, a string whose hit count exceeded the
# given threshold in the last round would be the
# prefix of each `prefix` in the current round. (See
# [BBCGGI21, Section 5.1].) In this case, part of the
# path would have already been traversed.
#
# Recomputing nodes along previously traversed paths is
# wasteful. Implementations can eliminate this added
# complexity by caching nodes (i.e., `(seed, ctrl)`
# pairs) output by previous calls to `eval_next()`.
(seed, ctrl, y) = self.eval_next(
seed,
ctrl,
correction_words[current_level],
current_level,
bit,
binder,
)
if agg_id == 0:
out_share.append(cast(list[Field], y))
else:
out_share.append(vec_neg(cast(list[Field], y)))
return cast(
Union[list[list[Field64]], list[list[Field255]]],
out_share,
)
def eval_next(
self,
prev_seed: bytes,
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prev_ctrl: Field2,
correction_word: CorrectionWordTuple,
level: int,
bit: int,
binder: bytes) > tuple[bytes, Field2, FieldVec]:
"""
Compute the next node in the IDPF tree along the path determined
by a candidate prefix. The next node is determined by `bit`, the
bit of the prefix corresponding to the next level of the tree.
"""
field = self.current_field(level)
seed_cw = correction_word[0]
ctrl_cw = correction_word[1]
w_cw = cast(list[Field], correction_word[2])
(s, t) = self.extend(prev_seed, binder)
s[0] = xor(s[0], prev_ctrl.conditional_select(seed_cw))
s[1] = xor(s[1], prev_ctrl.conditional_select(seed_cw))
t[0] += ctrl_cw[0] * prev_ctrl
t[1] += ctrl_cw[1] * prev_ctrl
next_ctrl = t[bit]
convert_output = self.convert(level, s[bit], binder)
next_seed = convert_output[0]
y = cast(list[Field], convert_output[1])
# Implementation note: Here we add the correction word to the
# output if `next_ctrl` is set. We avoid branching on the value
# of the control bit in order to reduce side channel leakage.
mask = cast(Field, field(next_ctrl.as_unsigned()))
for i in range(len(y)):
y[i] += w_cw[i] * mask
return (next_seed, next_ctrl, cast(FieldVec, y))
Figure 29: IDPFevaluation generation algorithm of BBCGGI21.
8.3.3. Auxiliary Functions
def extend(
self,
seed: bytes,
binder: bytes) > tuple[list[bytes], list[Field2]]:
xof = XofFixedKeyAes128(seed, format_dst(1, 0, 0), binder)
s = [
bytearray(xof.next(XofFixedKeyAes128.SEED_SIZE)),
bytearray(xof.next(XofFixedKeyAes128.SEED_SIZE)),
]
# Use the least significant bits as the control bit correction,
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# and then zero it out. This gives effectively 127 bits of
# security, but reduces the number of AES calls needed by 1/3.
t = [Field2(s[0][0] & 1), Field2(s[1][0] & 1)]
s[0][0] &= 0xFE
s[1][0] &= 0xFE
return ([bytes(s[0]), bytes(s[1])], t)
def convert(
self,
level: int,
seed: bytes,
binder: bytes) > tuple[bytes, FieldVec]:
xof = XofFixedKeyAes128(seed, format_dst(1, 0, 1), binder)
next_seed = xof.next(XofFixedKeyAes128.SEED_SIZE)
field = self.current_field(level)
w = xof.next_vec(field, self.VALUE_LEN)
return (next_seed, cast(FieldVec, w))
def encode_public_share(
self,
correction_words: list[CorrectionWordTuple]) > bytes:
encoded = bytes()
control_bits = list(itertools.chain.from_iterable(
cw[1] for cw in correction_words
))
encoded += pack_bits(control_bits)
for (level, (seed_cw, _, w_cw)) \
in enumerate(correction_words):
field = cast(type[Field], self.current_field(level))
encoded += seed_cw
encoded += field.encode_vec(cast(list[Field], w_cw))
return encoded
def decode_public_share(
self,
encoded: bytes) > list[CorrectionWordTuple]:
l = (2 * self.BITS + 7) // 8
encoded_ctrl, encoded = encoded[:l], encoded[l:]
control_bits = unpack_bits(encoded_ctrl, 2 * self.BITS)
correction_words = []
for level in range(self.BITS):
field = self.current_field(level)
ctrl_cw = (
control_bits[level * 2],
control_bits[level * 2 + 1],
)
l = XofFixedKeyAes128.SEED_SIZE
seed_cw, encoded = encoded[:l], encoded[l:]
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l = field.ENCODED_SIZE * self.VALUE_LEN
encoded_w_cw, encoded = encoded[:l], encoded[l:]
w_cw = field.decode_vec(encoded_w_cw)
correction_words.append((seed_cw, ctrl_cw, w_cw))
if len(encoded) != 0:
raise ValueError('trailing bytes')
return correction_words
Figure 30: Helper functions for the IDPF.
Here, pack_bits() takes a list of bits, packs each group of eight
bits into a byte, in LSB to MSB order, padding the most significant
bits of the last byte with zeros as necessary, and returns the byte
array. unpack_bits() performs the reverse operation: it takes in a
byte array and a number of bits, and returns a list of bits,
extracting eight bits from each byte in turn, in LSB to MSB order,
and stopping after the requested number of bits. If the byte array
has an incorrect length, or if unused bits in the last bytes are not
zero, it throws an error.
8.4. Instantiation
By default, Poplar1 is instantiated with the IDPF in Section 8.3
(VALUE_LEN == 2) and XofTurboShake128 (Section 6.2.1). This VDAF is
suitable for any positive value of BITS. Test vectors can be found
in Appendix "Test Vectors".
9. Security Considerations
VDAFs have two essential security goals:
1. Privacy: An attacker that controls the Collector and a subset of
Clients and Aggregators learns nothing about the measurements of
honest Clients beyond what it can deduce from the aggregate
result. We assume the attacker controls the entire network
except for channels between honest Clients and honest
Aggregators. In particular, it cannot forge or prevent
transmission of messages on these channels.
2. Robustness: An attacker that controls a subset of Clients cannot
cause the Collector to compute anything other than the aggregate
of the measurements of honest Clients. We assume the attacker
eavesdrops on the network but does not control transmission of
messages between honest parties.
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Formal definitions of privacy and robustness can be found in
[DPRS23]. A VDAF is the core cryptographic primitive of a protocol
that achieves the above privacy and robustness goals. It is not
sufficient on its own, however. The application will need to assure
a few security properties, for example:
* Securely distributing the longlived parameters, in particular the
verification key.
* Establishing secure channels:
 Confidential and authentic channels among Aggregators, and
between the Aggregators and the Collector; and
 Confidential and Aggregatorauthenticated channels between
Clients and Aggregators.
* Enforcing the noncollusion properties required of the specific
VDAF in use.
In such an environment, a VDAF provides the highlevel privacy
property described above: The Collector learns only the aggregate
measurement, and nothing about individual measurements aside from
what can be inferred from the aggregate result. The Aggregators
learn neither individual measurements nor the aggregate result. The
Collector is assured that the aggregate statistic accurately reflects
the inputs as long as the Aggregators correctly executed their role
in the VDAF.
On their own, VDAFs do not provide:
1. Mitigation of Sybil attacks [Dou02]. In this attack, the
adversary observes a subset of input shares transmitted by a
Client it is interested in. It allows the input shares to be
processed, but corrupts and picks bogus measurements for the
remaining Clients. Applications can guard against these risks by
adding additional controls on report submission, such as Client
authentication and rate limits.
2. Differential privacy [Dwo06]. Depending on the distribution of
the measurements, the aggregate result itself can still leak a
significant amount of information about an individual measurement
or the person that generated it.
3. Robustness in the presence of a malicious Aggregator. An
Aggregator can, without detection, manipulate the aggregate
result by modifying its own aggregate share.
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4. Guaranteed output delivery [GSZ20]. An attacker that controls
transmission of messages between honest parties can prevent
computation of the aggregate result by dropping messages.
9.1. Requirements for the Verification Key
The Aggregators are responsible for exchanging the verification key
in advance of executing the VDAF. Any procedure is acceptable as
long as the following conditions are met:
1. To ensure robustness of the computation, the Aggregators MUST NOT
reveal the verification key to the Clients. Otherwise, a
malicious Client might be able to exploit knowledge of this key
to craft an invalid report that would be accepted by the
Aggregators.
2. To ensure privacy of the measurements, the Aggregators MUST
commit to the verification key prior to processing reports
generated by Clients. Otherwise, a malicious Aggregator may be
able to craft a verification key that, for a given report, causes
an honest Aggregator to leak information about the measurement
during preparation.
Meeting these conditions is required in order to leverage security
analysis in the framework of [DPRS23]. Their definition of
robustness allows the attacker, playing the role of a cohort of
malicious Clients, to submit arbitrary reports to the Aggregators and
eavesdrop on their communications as they process them. Security in
this model is achievable as long as the verification key is kept
secret from the attacker.
The privacy definition of [DPRS23] considers an active attacker that
controls the network and a subset of Aggregators; in addition, the
attacker is allowed to choose the verification key used by each
honest Aggregator over the course of the experiment. Security is
achievable in this model as long as the key is picked at the start of
the experiment, prior to any reports being generated. (The model
also requires nonces to be generated at random; see Section 9.2
below.)
Meeting these requirements is relatively straightforward. For
example, the Aggregators may designate one of their peers to generate
the verification key and distribute it to the others. To assure
Clients of key commitment, the Clients and (honest) Aggregators could
bind reports to a shared context string derived from the key. For
instance, the "task ID" of DAP [DAP] could be set to the hash of the
verification key; then as long as honest Aggregators only consume
reports for the task indicated by the Client, forging a new key after
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the fact would reduce to finding collisions in the underlying hash
function. (Keeping the key secret from the Clients would require the
hash function to be oneway.) However, since rotating the key
implies rotating the task ID, this scheme would not allow key
rotation over the lifetime of a task.
9.2. Requirements for the Nonce
The sharding and preparation steps of VDAF execution depend on a
nonce associated with the Client's report. To ensure privacy of the
underlying measurement, the Client MUST generate this nonce using a
CSPRNG. This is required in order to leverage security analysis for
the privacy definition of [DPRS23], which assumes the nonce is chosen
at random prior to generating the report.
Other security considerations may require the nonce to be non
repeating. For example, to achieve differential privacy it is
necessary to avoid "over exposing" a report by including it too many
times in a single batch or across multiple batches. It is
RECOMMENDED that the nonce generated by the Client be used by the
Aggregators for replay protection.
9.3. Requirements for the Public Share
The Aggregators MUST ensure they have both received the same public
share from the Client. It is sufficient, for example, to exchange a
hash of the public share over a secure channel.
9.4. Requirements for Aggregation Parameters
As described in Section 4.3 and Section 5.3 respectively, DAFs and
VDAFs may impose restrictions on the reuse of input shares. This is
to ensure that correlated randomness provided by the Client through
the input share is not used more than once, which might compromise
confidentiality of the Client's measurements.
Protocols that make use of VDAFs therefore MUST call vdaf.is_valid on
the set of all aggregation parameters used for a Client's input
share, and only proceed with the preparation and aggregation phases
if that function call returns True.
9.4.1. Additional Privacy Considerations
Aggregating a batch of reports multiple times, each time with a
different aggregation parameter, could result in information leakage
beyond what is used by the application.
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For example, when Poplar1 is used for heavy hitters, the Aggregators
learn not only the heavy hitters themselves, but also the prefix tree
(as defined in Section 8) computed along the way. Indeed, this
leakage is inherent to any construction that uses an IDPF
(Section 8.1) in the same way. Depending on the distribution of the
measurements, the prefix tree can leak a significant amount of
information about unpopular inputs. For instance, it is possible
(though perhaps unlikely) for a large set of nonheavyhitter values
to share a common prefix, which would be leaked by a prefix tree with
a sufficiently small threshold.
A malicious adversary controlling the Collector and one of the
Aggregators can further turn arbitrary nonheavy prefixes into heavy
ones by tampering with the IDPF output at any position. While our
construction ensures that the nodes evaluated at one level are
children of the nodes evaluated at the previous level, this still may
allow an adversary to discover individual nonheavy strings.
The only practical, generalpurpose approach to mitigating these
leakages is via differential privacy, which is RECOMMENDED for all
protocols using Poplar1 for heavyhitter type applications.
9.4.2. Safe Usage of IDPF Outputs
The arithmetic sketch described in Section 8 is used by the
Aggregators to check that the shares of the vector obtained by
evaluating a Client's IDPF at a sequence of candidate prefixes has at
most one nonzero value, and that the nonzero value is 1. Depending
on how the values are used, the arithmetic sketch on its own may not
be sufficient for robustness of the application. In particular, a
malicious Client may attempt to influence the computation by choosing
an IDPF that evaluates to 1 at more than one node at a given level of
the tree.
This issue can be mitigated by using an IDPF that is extractable as
defined in in Appendix D of [BBCGGI21]. Extractability ensures that,
for a particular level of the tree, it is infeasible for an attacker
to control values of the IDPF such that it takes on chosen nonzero
values at more than one node. (It can practically only achieve the
zero function, a point function, or a pseudorandom function.)
The IDPF specified in Section 8.1 only guarantees extractability at
the last level of the tree. (This is by virtue of using a larger
field for the leaves than for the inner nodes.) For intermediate
levels, it is feasible for a client to produce IDPF shares with two
controlled nonzero nodes.
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This is not an issue for running heavy hitters, since (1) each node
in the prefix tree is a child of a previously traversed node, (2) the
arithmetic sketch would detect double voting at every level of the
prefix tree, and (3) the IDPF is extractable at the last level of the
tree. However, the lack of extractability at intermediate levels may
result in attacks on the robustness of certain applications.
Thus applications SHOULD NOT use prefix counts for intermediate
levels for any purpose beyond the heavyhitters tree traversal.
9.5. Requirements for XOFs
As described in Section 6.2, our constructions rely on eXtendable
Output Functions (XOFs). In the security analyses of our protocols,
these are usually modeled as random oracles. XofTurboShake128 is
designed to be indifferentiable from a random oracle [MRH04], making
it a suitable choice for most situations.
The one exception is the IDPF of Section 8.3. Here, a random oracle
is not needed to prove privacy, since the analysis of [BBCGGI21],
Proposition 1, only requires a Pseudorandom Generator (PRG). As
observed in [GKWY20], a PRG can be instantiated from a correlation
robust hash function H. Informally, correlation robustness requires
that for a random r, H(xor(r, x)) is computationally
indistinguishable from a random function of x. A PRG can therefore
be constructed as
PRG(r) = H(xor(r, 1))  H(xor(r, 2))  ...
since each individual hash function evaluation is indistinguishable
from a random function.
Our construction at Section 6.2.2 implements a correlationrobust
hash function using fixedkey AES. For security, it assumes that AES
with a fixed key can be modeled as a random permutation [GKWY20].
Additionally, we use a different AES key for every client, which in
the ideal cipher model leads to better concrete security [GKWWY20].
We note that for robustness, the analysis of [BBCGGI21] still assumes
a random oracle to make the Idpf extractable. While
XofFixedKeyAes128 has been shown to be differentiable from a random
oracle [GKWWY20], there are no known attacks exploiting this
difference. We also stress that even if the Idpf is not extractable,
Poplar1 guarantees that every client can contribute to at most one
prefix among the ones being evaluated by the helpers.
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9.6. Choosing the Field Size
Prio3 and other systems built from FLPs (Section 7.3 in particular)
may benefit from choosing a field size that is as small as possible.
Generally speaking, a smaller field results in lower communication
and storage costs. Care must be taken, however, since a smaller
field also results in degraded (or even vacuous) robustness.
Different variants of Prio3 (Section 7) use different field sizes:
Prio3Count uses Field64; but Prio3Sum, Prio3SumVec, and
Prio3Histogram use Field128, a field that is twice as large as
Field64. This is due to the use of joint randomness (Section 7.1) in
the latter variants. Joint randomness allows for more flexible
circuit design (see Section 7.3.1.1), but opens up Prio3 to
precomputation attacks, which the larger field mitigates. (See
[DPRS23], Theorem 1.) Note that privacy is not susceptible to such
attacks.
Another way to mitigate this issue (or improve robustness in general)
is to generate and verify multiple, independent proofs. (See
Section 7.1.2.) For Prio3, the PROOFS parameter controls the number
of proofs (at least one) that are generated and verified.
In general, Field128 is RECOMMENDED for use in Prio3 when the circuit
uses joint randomness (JOINT_RAND_LEN > 0) and PROOFS == 1. Field64
MAY be used instead, but PROOFS MUST be set to at least 3. Breaking
robustness for PROOFS == 2 is feasible, if impractical; but PROOFS ==
1 is completely broken for such a small field.
10. IANA Considerations
A codepoint for each (V)DAF in this document is defined in the table
below. Note that 0xFFFF0000 through 0xFFFFFFFF are reserved for
private use.
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+===============+=======================+======+===============+
 Value  Scheme  Type  Reference 
+===============+=======================+======+===============+
 0x00000000  Prio3Count  VDAF  Section 7.4.1 
+++++
 0x00000001  Prio3Sum  VDAF  Section 7.4.2 
+++++
 0x00000002  Prio3SumVec  VDAF  Section 7.4.3 
+++++
 0x00000003  Prio3Histogram  VDAF  Section 7.4.4 
+++++
 0x00000004  Prio3MultihotCountVec  VDAF  Section 7.4.5 
+++++
 0x00000005 to  reserved for Prio3  VDAF  n/a 
 0x00000FFF    
+++++
 0x00001000  Poplar1  VDAF  Section 8.4 
+++++
 0xFFFF0000 to  reserved  n/a  n/a 
 0xFFFFFFFF    
+++++
Table 23: Unique identifiers for (V)DAFs.
TODO Add IANA considerations for the codepoints summarized in
Table 23.
11. References
11.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, .
[RFC8446] Rescorla, E., "The Transport Layer Security (TLS) Protocol
Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
.
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[TurboSHAKE]
Viguier, B., Wong, D., Van Assche, G., Dang, Q., and J.
Daemen, "KangarooTwelve and TurboSHAKE", Work in Progress,
InternetDraft, draftirtfcfrgkangarootwelve14, 9 May
2024, .
11.2. Informative References
[AGJOP21] Addanki, S., Garbe, K., Jaffe, E., Ostrovsky, R., and A.
Polychroniadou, "Prio+: Privacy Preserving Aggregate
Statistics via Boolean Shares", 2021,
.
[BBCGGI19] Boneh, D., Boyle, E., CorriganGibbs, H., Gilboa, N., and
Y. Ishai, "ZeroKnowledge Proofs on SecretShared Data via
Fully Linear PCPs", CRYPTO 2019 , 2019,
.
[BBCGGI21] Boneh, D., Boyle, E., CorriganGibbs, H., Gilboa, N., and
Y. Ishai, "Lightweight Techniques for Private Heavy
Hitters", IEEE S&P 2021 , 2021, .
[BBCGGI23] Boneh, D., Boyle, E., CorriganGibbs, H., Gilboa, N., and
Y. Ishai, "Arithmetic Sketching", CRYPTO 2023 , 2023,
.
[CGB17] CorriganGibbs, H. and D. Boneh, "Prio: Private, Robust,
and Scalable Computation of Aggregate Statistics", NSDI
2017 , 2017,
.
[DAP] Geoghegan, T., Patton, C., Pitman, B., Rescorla, E., and
C. A. Wood, "Distributed Aggregation Protocol for Privacy
Preserving Measurement", Work in Progress, InternetDraft,
draftietfppmdap11, 21 May 2024,
.
[Dou02] Douceur, J., "The Sybil Attack", IPTPS 2002 , 2002,
.
[DPRS23] Davis, H., Patton, C., Rosulek, M., and P. Schoppmann,
"Verifiable Distributed Aggregation Functions", n.d.,
.
[Dwo06] Dwork, C., "Differential Privacy", ICALP 2006 , 2006,
.
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[ENPA] "Exposure Notification Privacypreserving Analytics (ENPA)
White Paper", 2021, .
[EPK14] Erlingsson, Ú., Pihur, V., and A. Korolova, "RAPPOR:
Randomized Aggregatable PrivacyPreserving Ordinal
Response", CCS 2014 , 2014,
.
[GI14] Gilboa, N. and Y. Ishai, "Distributed Point Functions and
Their Applications", EUROCRYPT 2014 , 2014,
.
[GKWWY20] Guo, C., Katz, J., Wang, X., Weng, C., and Y. Yu, "Better
concrete security for halfgates garbling (in the multi
instance setting)", CRYPTO 2020 , 2020,
.
[GKWY20] Guo, C., Katz, J., Wang, X., and Y. Yu, "Efficient and
Secure Multiparty Computation from FixedKey Block
Ciphers", S&P 2020 , 2020,
.
[GSZ20] Goyal, V., Song, Y., and C. Zhu, "Guaranteed Output
Delivery Comes Free in Honest Majority MPC", CRYPTO 2020 ,
2020, .
[MPRV09] Mironov, I., Pandey, O., Reingold, O., and S. Vadhan,
"Computational Differential Privacy", CRYPTO 2009 , n.d.,
.
[MRH04] Maurer, U., Renner, R., and C. Holenstein,
"Indifferentiability, impossibility results on reductions,
and applications to the random oracle methodology", In TCC
2004: Theory of Cryptography, pages 2139,
DOI 10.1007/9783540246381_2, February 2004,
.
[OriginTelemetry]
"Origin Telemetry", 2020, .
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Acknowledgments
The security considerations in Section 9 are based largely on the
security analysis of [DPRS23]. Thanks to Hannah Davis and Mike
Rosulek, who lent their time to developing definitions and security
proofs.
Thanks to Junye Chen, Henry CorriganGibbs, Armando FazHernández,
Simon Friedberger, Tim Geoghegan, Albert Liu, Brandon Pitman, Mariana
Raykova, Jacob Rothstein, Shan Wang, Xiao Wang, Bas Westerbaan, and
Christopher Wood for useful feedback on and contributions to the
spec.
Test Vectors
(TO BE REMOVED BY RFC EDITOR: Machinereadable test vectors can be
found at https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc/
test_vec.)
Test vectors cover the generation of input shares and the conversion
of input shares into output shares. Vectors specify the verification
key, measurements, aggregation parameter, and any parameters needed
to construct the VDAF. (For example, for Prio3Sum, the user
specifies the number of bits for representing each summand.)
Byte strings are encoded in hexadecimal. To make the tests
deterministic, the random inputs of randomized algorithms were fixed
to the byte sequence starting with 0, incrementing by 1, and wrapping
at 256:
0, 1, 2, ..., 255, 0, 1, 2, ...
Prio3Count
TODO Copy the machine readable vectors from the source repository
(https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc/
test_vec) and format them for humans.
Prio3Sum
TODO Copy the machine readable vectors from the source repository
(https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc/
test_vec) and format them for humans.
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Prio3SumVec
TODO Copy the machine readable vectors from the source repository
(https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc/
test_vec) and format them for humans.
Prio3Histogram
TODO Copy the machine readable vectors from the source repository
(https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc/
test_vec) and format them for humans.
Poplar1
TODO Copy the machine readable vectors from the source repository
(https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc/
test_vec) and format them for humans.
Authors' Addresses
Richard L. Barnes
Cisco
Email: rlb@ipv.sx
David Cook
ISRG
Email: divergentdave@gmail.com
Christopher Patton
Cloudflare
Email: chrispatton+ietf@gmail.com
Phillipp Schoppmann
Google
Email: schoppmann@google.com
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