CFRG C. Patton
InternetDraft Cloudflare, Inc.
Intended status: Informational R. L. Barnes
Expires: 6 September 2022 Cisco
P. Schoppmann
Google
5 March 2022
Verifiable Distributed Aggregation Functions
draftpattoncfrgvdaf01
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 input that
would result in an incorrect 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
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provisions of BCP 78 and BCP 79.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Conventions and Definitions . . . . . . . . . . . . . . . . . 6
3. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4. Definition of VDAFs . . . . . . . . . . . . . . . . . . . . . 9
4.1. Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2. Sharding . . . . . . . . . . . . . . . . . . . . . . . . 11
4.3. Preparation . . . . . . . . . . . . . . . . . . . . . . . 11
4.4. Aggregation . . . . . . . . . . . . . . . . . . . . . . . 14
4.5. Unsharding . . . . . . . . . . . . . . . . . . . . . . . 15
4.6. Execution of a VDAF . . . . . . . . . . . . . . . . . . . 15
5. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 17
5.1. Finite Fields . . . . . . . . . . . . . . . . . . . . . . 17
5.1.1. Auxiliary Functions . . . . . . . . . . . . . . . . . 18
5.1.2. FFTFriendly Fields . . . . . . . . . . . . . . . . . 19
5.1.3. Parameters . . . . . . . . . . . . . . . . . . . . . 19
5.2. Pseudorandom Generators . . . . . . . . . . . . . . . . . 20
5.2.1. Constructions . . . . . . . . . . . . . . . . . . . . 21
6. Prio3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.1. Fully Linear Proof (FLP) Systems . . . . . . . . . . . . 23
6.1.1. Encoding the Input . . . . . . . . . . . . . . . . . 26
6.2. Construction . . . . . . . . . . . . . . . . . . . . . . 26
6.2.1. Setup . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2.2. Sharding . . . . . . . . . . . . . . . . . . . . . . 28
6.2.3. Preparation . . . . . . . . . . . . . . . . . . . . . 30
6.2.4. Aggregation . . . . . . . . . . . . . . . . . . . . . 32
6.2.5. Unsharding . . . . . . . . . . . . . . . . . . . . . 33
6.2.6. Auxiliary Functions . . . . . . . . . . . . . . . . . 33
6.3. A GeneralPurpose FLP . . . . . . . . . . . . . . . . . . 35
6.3.1. Overview . . . . . . . . . . . . . . . . . . . . . . 35
6.3.2. Validity Circuits . . . . . . . . . . . . . . . . . . 38
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6.3.3. Construction . . . . . . . . . . . . . . . . . . . . 40
6.4. Instantiations . . . . . . . . . . . . . . . . . . . . . 43
6.4.1. Prio3Aes128Count . . . . . . . . . . . . . . . . . . 43
6.4.2. Prio3Aes128Sum . . . . . . . . . . . . . . . . . . . 44
6.4.3. Prio3Aes128Histogram . . . . . . . . . . . . . . . . 46
7. Poplar1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.1. Incremental Distributed Point Functions (IDPFs) . . . . . 49
7.2. Construction . . . . . . . . . . . . . . . . . . . . . . 50
7.2.1. Setup . . . . . . . . . . . . . . . . . . . . . . . . 50
7.2.2. Preparation . . . . . . . . . . . . . . . . . . . . . 52
7.2.3. Aggregation . . . . . . . . . . . . . . . . . . . . . 54
7.2.4. Unsharding . . . . . . . . . . . . . . . . . . . . . 54
7.2.5. Helper Functions . . . . . . . . . . . . . . . . . . 55
8. Security Considerations . . . . . . . . . . . . . . . . . . . 55
9. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 57
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 57
10.1. Normative References . . . . . . . . . . . . . . . . . . 57
10.2. Informative References . . . . . . . . . . . . . . . . . 57
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 58
Test Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Prio3Aes128Count . . . . . . . . . . . . . . . . . . . . . . . 59
Prio3Aes128Sum . . . . . . . . . . . . . . . . . . . . . . . . 60
Prio3Aes128Histogram . . . . . . . . . . . . . . . . . . . . . 61
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 62
1. Introduction
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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Most prior approaches to this problem fall under the rubric 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 well
known distribution and adds it to their input before submitting to
the aggregation server. The aggregation server then adds up the
noisy inputs, and because it knows the distribution from whence the
noise was sampled, it can estimate the true sum with reasonable
precision.
Differentially private systems like RAPPOR are easy to deploy and
provide a useful guarantee. On its own, however, DP falls short of
the strongest privacy property one could hope for. Specifically,
depending on the "amount" of noise a client adds to its input, it may
be possible for a curious aggregator to make a reasonable guess of
the input's true value. Indeed, the more noise the clients add, the
less reliable will be the server's estimate of the output. Thus
systems employing DP techniques alone must strike a delicate balance
between privacy and utility.
The ideal goal for a privacypreserving measurement system is that of
secure multiparty computation: No participant in the protocol should
learn anything about an individual input beyond what it can deduce
from the aggregate. In this document, we describe Verifiable
Distributed Aggregation Functions (VDAFs) as a general class of
protocols that 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 input is ever accessible
to any party besides the client that submitted it. At the same time,
VDAFs are "verifiable" in the sense that malformed inputs that would
otherwise garble the output of the computation can be detected and
removed from the set of inputs.
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
Mozilla Origin Telemetry project [OriginTelemetry] and the Exposure
Notification Private Analytics collaboration among the Internet
Security Research Group (ISRG), Google, Apple, and others [ENPA].
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The VDAF abstraction laid out in Section 4 represents a class of
multiparty protocols for privacypreserving measurement proposed in
the literature. These protocols vary in their operational and
security considerations, sometimes in subtle but consequential ways.
This document therefore has two important goals:
1. Providing applications like [ID.draftgpewprivppm] with a
simple, uniform interface for accessing privacypreserving
measurement schemes, and documenting relevant operational and
security bounds for that interface:
1. General patterns of communications among the various actors
involved in the system (clients, aggregation servers, and
measurement collectors);
2. Capabilities of a malicious coalition of servers attempting
to divulge information about client inputs; and
3. Conditions that are necessary to ensure that malicious
clients cannot corrupt the computation.
2. Providing cryptographers with design criteria that allow new
constructions to be easily used by applications.
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 input 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 combine their additive shares
to obtain the final aggregate.
The difficult part of this system is ensuring that the servers
hold shares of a valid input, e.g., the input is an integer in a
specific range. Thus Prio specifies a multiparty protocol for
accomplishing this task.
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In Section 6 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 inputs that occur at least t times. The core primitive
used in their protocol is a generalization of a 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
input. 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 input.
In Section 7 we describe a VDAF called Poplar1 that implements
this functionality.
The remainder of this document is organized as follows: Section 3
gives a brief overview of VDAFs; Section 4 defines the syntax for
VDAFs; Section 5 defines various functionalities that are common to
our constructions; Section 7 describes the Poplar1 construction;
Section 6 describes the Prio3 construction; and Section 8 enumerates
the security considerations for VDAFs.
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.
Algorithms in this document are written in Python 3. Type hints are
used to define input and output types. A fatal error in a program
(e.g., failure to parse one of the function parameters) is usually
handled by raising an exception.
Some common functionalities:
* zeros(len: Unsigned) > Bytes returns an array of zero bytes. The
length of output MUST be len.
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* gen_rand(len: Unsigned) > Bytes returns an array of random bytes.
The length of output MUST be len.
* byte(int: Unsigned) > Bytes returns the representation of int as
a byte string. The value of int MUST be in [0,256).
* 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.
* I2OSP and OS2IP from [RFC8017], which are used, respectively, to
convert a nonnegative integer to a byte string and convert a byte
string to a nonnegative integer.
* next_power_of_2(n: Unsigned) > Unsigned returns the smallest
integer greater than or equal to n that is also a power of two.
3. Overview
++
+> Aggregator 0 +
 ++ 
 ^ 
  
 V 
 ++ 
 +> Aggregator 1 + 
  ++  
+++  ^  +>++
 Client +  +> Collector > Aggregate
+++ +>++
 ... 
 
  
 V 
 ++ 
+> Aggregator N1 +
++
Input shares Aggregate shares
Figure 1: Overall data flow of a VDAF
In a VDAFbased private measurement system, we distinguish three
types of actors: Clients, Aggregators, and Collectors. The overall
flow of the measurement process is as follows:
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* Clients are configured with public parameters for a set of
Aggregators.
* To submit an individual measurement, the Client shards the
measurement into "input shares" and sends one input share to each
Aggregator.
* The Aggregators verify the validity of the input shares, producing
a set of "output shares".
 Output shares are in onetoone correspondence with the input
shares.
 Just as each Aggregator receives one input share of each input,
at the end of the validation process, each aggregator holds one
output share.
 In most VDAFs, Aggregators will need to exchange information
among themselves as part of the validation process.
* Each Aggregator combines the output shares across inputs in the
batch to compute the "aggregate share" for that batch, i.e., its
share of the desired aggregate result.
* 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 the correctness of the measurements obtained. The privacy
properties of the system are assured by noncollusion among
Aggregators, and Aggregators are the entities that perform validation
of client inputs. Thus clients trust Aggregators not to collude
(typically it is required that at least one Aggregator is honest),
and Collectors trust Aggregators to properly verify Client inputs.
Within the bounds of the noncollusion requirements of a given VDAF
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 Aggregators to augment a basic client
server protocol.
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In this document, we describe the computations performed by the
actors in this system. It is up to applications 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 VDAFs
A concrete VDAF specifies the algorithms involved in evaluating an
aggregation function across a batch of inputs. This section
specifies the interfaces of these algorithms as they would be exposed
to applications.
The overall execution of a VDAF comprises the following steps:
* Setup  Generating shared parameters for the Aggregators
* 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
The setup algorithm is performed once for a given collection of
Aggregators. Sharding and preparation are done once per measurement
input. Aggregation and unsharding are done over a batch of inputs
(more precisely, over the output shares recovered from those inputs).
Note that the preparation step performs two functions: Verification
and conversion. Conversion translates input shares into output
shares that are compatible with the aggregation function.
Verification ensures that aggregating the recovered output shares
will not lead to a garbled aggregate result.
The remainder of this section defines the VDAF interface in terms of
an abstract base class Vdaf. This class defines the set of methods
and attributes a concrete VDAF must provide. The attributes are
listed in [{vdafparam}}; the methods are defined in the subsections
that follow.
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+=============+=============================================+
 Parameter  Description 
+=============+=============================================+
 ROUNDS  Number of rounds of communication during 
  the preparation phase (Section 4.3) 
+++
 SHARES  Number of input shares into which each 
  measurement is sharded (Section 4.2) 
+++
 Measurement  Type of each measurement 
+++
 PublicParam  Type of public parameter used by the Client 
  during the sharding phase (Section 4.2) 
+++
 VerifyParam  Type of verification parameter used by each 
  Aggregator during the preparation phase 
  (Section 4.3) 
+++
 AggParam  Type of aggregation parameter 
+++
 Prep  State of each Aggregator during the 
  preparation phase (Section 4.3) 
+++
 OutShare  Type of each output share 
+++
 AggShare  Type of each aggregate share 
+++
 AggResult  Type of the aggregate result 
+++
Table 1: Constants and types defined by each concrete VDAF.
4.1. Setup
Before execution of the VDAF can begin, it is necessary to distribute
longlived parameters to the Client and Aggregators. The longlived
parameters are generated by the following algorithm:
* Vdaf.setup() > (PublicParam, Vec[VerifyParam]) is the randomized
setup algorithm used to generate the public parameter used by the
Clients and the verification parameters used by the Aggregators.
The length of the latter MUST be equal to SHARES. In general, an
Aggregator's verification parameter is considered secret and MUST
NOT be revealed to the Clients, Collector or other Aggregators.
The parameters MAY be reused across multiple VDAF evaluations.
See Section 8 for a discussion of the security implications this
has depending on the threat model.
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4.2. Sharding
In order to protect the privacy of its measurements, a VDAF Client
splits its measurements into "input shares". The
measurement_to_input_shares method is used for this purpose.
* Vdaf.measurement_to_input_shares(public_param: PublicParam, input:
Measurement) > Vec[Bytes] is the randomized inputdistribution
algorithm run by each Client. It consumes the public parameter
and input measurement and produces a sequence of input shares, one
for each Aggregator. The length of the output MUST be SHARES.
Client
======
measurement

V
++
 measurement_to_input_shares 
++
  ... 
V V V
input_share_0 input_share_1 input_share_[SHARES1]
  ... 
V V V
Aggregator 0 Aggregator 1 Aggregator SHARES1
Figure 2: The Client divides its measurement input into input
shares and distributes them to the Aggregators.
CP The public_param is intended to allow for protocols that
require the Client to use a public key for sharding its
measurement. When rotating the verify_param for such a scheme, it
would be necessary to also update the public_param with which the
clients are configured. For PPM it would be nice if we could
rotate the verify_param without also having to update the clients.
We should consider dropping this at some point. See
https://github.com/cjpatton/vdaf/issues/19.
4.3. 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". (Each of the outbound messages are called "preparation
message shares".) Finally, the preparation message is distributed to
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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 preparation message.
Its output in the last round is its output share and its output in
each of the preceding rounds is a preparationmessage 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 inputs 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_next   prep_next   prep_next  
++ ++ ++ 
  ...  
V V V  x ROUNDS
++ 
 prep_shares_to_prep  
++ 
 
+++ 
  ...  
V V V /
... ... ...
  
V V V
++ ++ ++
 prep_next   prep_next   prep_next 
++ ++ ++
  ... 
V V V
out_share_0 out_share_1 out_share_[SHARES1]
Figure 3: 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.
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To facilitate the preparation process, a concrete VDAF implements the
following class methods:
* Vdaf.prep_init(verify_param: VerifyParam, agg_param: AggParam,
nonce: Bytes, input_share: Bytes) > Prep is the deterministic
preparationstate initialization algorithm run by each Aggregator
to begin processing its input share into an output share. Its
inputs are its verification parameter (verify_param), the
aggregation parameter (agg_param), the nonce provided by the
environment (nonce, see Figure 6), and one of the input shares
generated by the client (input_share). Its output is the
Aggregator's initial preparation state.
* Vdaf.prep_next(prep: Prep, inbound: Optional[Bytes]) >
Union[Tuple[Prep, Bytes], OutShare] is the deterministic
preparationstate update algorithm run by each Aggregator. It
updates the Aggregator's preparation state (prep) 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 or, if this is the first round, None.
* Vdaf.prep_shares_to_prep(agg_param: AggParam, prep_shares:
Vec[Bytes]) > Bytes is the deterministic preparationmessage pre
processing algorithm. It combines the preparationmessage shares
generated by the Aggregators in the previous round into the
preparation message consumed by each in the next round.
In effect, each Aggregator moves through a linear state machine with
ROUNDS+1 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.
TODO Consider how to bake this "linear state machine" condition
into the syntax. Given that Python 3 is used as our pseudocode,
it's easier to specify the preparation state using a class.
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
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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.
Note that it is possible for a VDAF to specify ROUNDS == 0, in which
case each Aggregator runs the preparationstate update algorithm once
and immediately recovers its output share without interacting with
the other Aggregators. However, most, if not all, constructions will
require some amount of interaction in order to ensure validity of the
output shares (while also maintaining privacy).
OPEN ISSUE Depending on what we do for issue#20, we may end up
needing to revise the above paragraph.
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. This algorithm is performed
locally at each Aggregator, without communication with the other
Aggregators.
* Vdaf.out_shares_to_agg_share(agg_param: AggParam, output_shares:
Vec[OutShare]) > agg_share: AggShare is the deterministic
aggregation algorithm. It is run by each Aggregator over the
output shares it has computed over a batch of measurement inputs.
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
++ ++ ++
 out2agg   out2agg  ...  out2agg 
++ ++ ++
  
V V V
agg_share_0 agg_share_1 agg_share_[SHARES1]
Figure 4: Aggregation of output shares. `B` indicates the number of
measurements in the batch.
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For simplicity, we have written this algorithm and the unsharding
algorithm below in "oneshot" form, where all shares for a batch are
provided at the same time. Some VDAFs may also support a "streaming"
form, where shares are processed one at a time.
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:
* Vdaf.agg_shares_to_result(agg_param: AggParam, agg_shares:
Vec[AggShare]) > 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. This algorithm is
deterministic.
Aggregator 0 Aggregator 1 Aggregator SHARES1
============ ============ ===================
agg_share_0 agg_share_1 agg_share_[SHARES1]
  
V V V
++
 agg_shares_to_result 
++

V
agg_result
Collector
=========
Figure 5: Computation of the final aggregate result from
aggregate shares.
QUESTION Maybe the aggregation algorithms should be randomized in
order to allow the Aggregators (or the Collector) to add noise for
differential privacy. (See the security considerations of
[ID.draftgpewprivppm].) Or is this outofscope of this
document?
4.6. Execution of a VDAF
Executing a VDAF involves the concurrent evaluation of the VDAF on
individual inputs and aggregation of the recovered output shares.
This is captured by the following example algorithm:
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def run_vdaf(Vdaf,
agg_param: Vdaf.AggParam,
nonces: Vec[Bytes],
measurements: Vec[Vdaf.Measurement]):
# Distribute longlived parameters.
(public_param, verify_params) = Vdaf.setup()
out_shares = []
for (nonce, measurement) in zip(nonces, measurements):
# Each Client shards its input into shares.
input_shares = Vdaf.measurement_to_input_shares(public_param,
measurement)
# Each Aggregator initializes its preparation state.
prep_states = []
for j in range(Vdaf.SHARES):
state = Vdaf.prep_init(verify_params[j],
agg_param,
nonce,
input_shares[j])
prep_states.append(state)
# Aggregators recover their output shares.
inbound = None
for i in range(Vdaf.ROUNDS+1):
outbound = []
for j in range(Vdaf.SHARES):
out = Vdaf.prep_next(prep_states[j], inbound)
if i < Vdaf.ROUNDS:
(prep_states[j], out) = out
outbound.append(out)
# This is where we would send messages over the
# network in a distributed VDAF computation.
if i < Vdaf.ROUNDS:
inbound = Vdaf.prep_shares_to_prep(agg_param,
outbound)
# The final outputs of prepare phasre are the output shares.
out_shares.append(outbound)
# Each Aggregator aggregates its output shares into an
# aggregate share.
agg_shares = []
for j in range(Vdaf.SHARES):
out_shares_j = [out[j] for out in out_shares]
agg_share_j = Vdaf.out_shares_to_agg_share(agg_param,
out_shares_j)
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agg_shares.append(agg_share_j)
# Collector unshards the aggregate.
agg_result = Vdaf.agg_shares_to_result(agg_param, agg_shares)
return agg_result
Figure 6: Execution of a VDAF.
The inputs to this algorithm are the aggregation parameter agg_param,
a list of nonces nonces, and a batch of Client inputs input_batch.
The aggregation parameter is chosen by the Aggregators prior to
executing the VDAF. This document does not specify how the nonces
are chosen, but security requires that the nonces be unique for each
measurement. See Section 8 for details.
Another important question this document leaves out of scope is how a
VDAF is to be executed by Aggregators distributed over a real
network. Algorithm run_vdaf prescribes the protocol's execution in a
"benign" environment in which there is no adversary and 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 [ID.draftgpewprivppm] that implements
suitable cryptographic functionalities. Moreover, some fraction of
the Aggregators (or Clients) may be malicious and diverge from their
prescribed behaviors. Section 8 describes the execution of the VDAF
in various adversarial environments and what properties the wrapper
protocol needs to provide in each.
5. Preliminaries
This section describes the primitives that are common to the VDAFs
specified in this document.
5.1. 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: Unsigned is the prime modulus that defines the field.
* ENCODED_SIZE: Unsigned is the number of bytes used to encode a
field element as a byte string.
A concrete Field also implements the following class methods:
* Field.zeros(length: Unsigned) > output: Vec[Field] returns a
vector of zeros. The length of output MUST be length.
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* Field.rand_vec(length: Unsigned) > output: Vec[Field] returns a
vector of random field elements. The length of output MUST be
length.
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() > Unsigned 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: Unsigned) returns integer represented as a field
element. If integer >= Field.MODULUS, then integer is first
reduced modulo Field.MODULUS.
Finally, 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.
def encode_vec(Field, data: Vec[Field]) > Bytes:
encoded = Bytes()
for x in data:
encoded += I2OSP(x.as_unsigned(), Field.ENCODED_SIZE)
return encoded
def decode_vec(Field, encoded: Bytes) > Vec[Field]:
L = Field.ENCODED_SIZE
if len(encoded) % L != 0:
raise ERR_DECODE
vec = []
for i in range(0, len(encoded), L):
encoded_x = encoded[i:i+L]
x = Field(OS2IP(encoded_x))
vec.append(x)
return vec
Figure 7: Derived class methods for finite fields.
5.1.1. 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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# Compute the inner product of the operands.
def inner_product(left: Vec[Field], right: Vec[Field]) > Field:
return sum(map(lambda x: x[0] * x[1], zip(left, right)))
# Subtract the right operand from the left and return the result.
def vec_sub(left: Vec[Field], right: Vec[Field]):
return list(map(lambda x: x[0]  x[1], zip(left, right)))
# Add the right operand to the left and return the result.
def vec_add(left: Vec[Field], right: Vec[Field]):
return list(map(lambda x: x[0] + x[1], zip(left, right)))
Figure 8: Common functions for finite fields.
5.1.2. FFTFriendly Fields
Some VDAFs require fields that are suitable for efficient computation
of the discrete Fourier transform. (One example is prio3 (Section 6)
when instantiated with the generic FLP of Section 6.3.3.)
Specifically, a field is said to be "FFTfriendly" if, in addition to
satisfying the interface described in Section 5.1, it implements the
following method:
* Field.gen() > Field returns the generator of a large subgroup of
the multiplicative group.
FFTfriendly fields also define the following parameter:
* GEN_ORDER: Unsigned is the order of a multiplicative subgroup
generated by Field.gen(). This value MUST be a power of 2.
5.1.3. Parameters
The tables below define finite fields used in the remainder of this
document.
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+==============+=======================+
 Parameter  Value 
+==============+=======================+
 MODULUS  2^32 * 4294967295 + 1 
+++
 ENCODED_SIZE  8 
+++
 Generator  7^4294967295 
+++
 GEN_ORDER  2^32 
+++
Table 2: Field64, an FFTfriendly field.
+==============+================================+
 Parameter  Value 
+==============+================================+
 MODULUS  2^66 * 4611686018427387897 + 1 
+++
 ENCODED_SIZE  16 
+++
 Generator  7^4611686018427387897 
+++
 GEN_ORDER  2^66 
+++
Table 3: Field128, an FFTfriendly field.
5.2. Pseudorandom Generators
A pseudorandom generator (PRG) is used to expand a short,
(pseudo)random seed into a long string of pseudorandom bits. A PRG
suitable for this document implements the interface specified in this
section. Concrete constructions are described in Section 5.2.1.
PRGs are defined by a class Prg with the following associated
parameter:
* SEED_SIZE: Unsigned is the size (in bytes) of a seed.
A concrete Prg implements the following class method:
* Prg(seed: Bytes, info: Bytes) constructs an instance of Prg from
the given seed and info string. The seed MUST be of length
SEED_SIZE and MUST be generated securely (i.e., it is either the
output of gen_rand or a previous invocation of the PRG). The info
string is used for domain separation.
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* prg.next(length: Unsigned) returns the next length bytes of output
of PRG. If the seed was securely generated, the output can be
treated as pseudorandom.
Each Prg has two derived class methods. The first is used to derive
a fresh seed from an existing one. The second is used to compute a
sequence of pseudorandom field elements. For each method, the seed
MUST be of length SEED_SIZE and MUST be generated securely (i.e., it
is either the output of gen_rand or a previous invocation of the
PRG).
# Derive a new seed.
def derive_seed(Prg, seed: Bytes, info: Bytes) > bytes:
prg = Prg(seed, info)
return prg.next(Prg.SEED_SIZE)
# Expand a seed into a vector of Field elements.
def expand_into_vec(Prg,
Field,
seed: Bytes,
info: Bytes,
length: Unsigned):
m = next_power_of_2(Field.MODULUS)  1
prg = Prg(seed, info)
vec = []
while len(vec) < length:
x = OS2IP(prg.next(Field.ENCODED_SIZE))
x &= m
if x < Field.MODULUS:
vec.append(Field(x))
return vec
Figure 9: Derived class methods for PRGs.
5.2.1. Constructions
5.2.1.1. PrgAes128
OPEN ISSUE Phillipp points out that a fixedkey mode of AES may be
more performant (https://eprint.iacr.org/2019/074.pdf). See
https://github.com/cjpatton/vdaf/issues/32 for details.
Our first construction, PrgAes128, converts a blockcipher, namely
AES128, into a PRG. Seed expansion involves two steps. In the
first step, CMAC [RFC4493] is applied to the seed and info string to
get a fresh key. In the second step, the fresh key is used in CTR
mode to produce a key stream for generating the output. A fixed
initialization vector (IV) is used.
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class PrgAes128:
SEED_SIZE: Unsigned = 16
def __init__(self, seed, info):
self.length_consumed = 0
# Use CMAC as a pseudorandom function to derive a key.
self.key = AES128CMAC(seed, info)
def next(self, length):
self.length_consumed += length
# CTRmode encryption of the allzero string of the desired
# length and using a fixed, allzero IV.
stream = AES128CTR(key, zeros(16), zeros(self.length_consumed))
return stream[length:]
Figure 10: Definition of PRG PrgBlockCipher(Aes128).
6. Prio3
NOTE This construction has not undergone significant security
analysis.
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.
* Input validity is determined by an arithmetic circuit evaluated
over the encoded input. (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 input is said
to be "valid"; otherwise, if the output is nonzero, then the
input is said to "invalid".
* The aggregate result is obtained by summing up the encoded input
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
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out a multiparty computation to determine if their shares correspond
to a valid input (as determined by the arithmetic circuit). This
computation involves a "proof" of validity generated by the Client.
Next, each Aggregator sums up its input 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 input share by applying a fixed map.
See Section 7 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.
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 inputs. 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 6.1. The generic transformation of
an FLP into Prio3 is specified in Section 6.2. Next, a concrete FLP
suitable for any validity circuit is specified in Section 6.3.
Finally, instantiations of Prio3 for various types of measurements
are specified in Section 6.4. Test vectors can be found in
Appendix "Test Vectors".
6.1. 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 input's validity and distributes shares of the proof to the
Aggregators. Each Aggregator then performs some a computation on its
input 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 input. (See Section 6.2 for
details.)
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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.
+================+==========================================+
 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 
+++
 INPUT_LEN  Length of the encoded measurement 
  (Section 6.1.1) 
+++
 OUTPUT_LEN  Length of the aggregatable output 
  (Section 6.1.1) 
+++
 PROOF_LEN  Length of the proof 
+++
 VERIFIER_LEN  Length of the verifier message generated 
  by querying the input and proof 
+++
 Measurement  Type of the measurement 
+++
 Field  As defined in (Section 5.1) 
+++
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 6.1.1):
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* Flp.prove(input: Vec[Field], prove_rand: Vec[Field], joint_rand:
Vec[Field]) > Vec[Field] is the deterministic proofgeneration
algorithm run by the prover. Its inputs are the encoded input,
the "prover randomness" prove_rand, and the "joint randomness"
joint_rand. The proof randomness is used only by the prover, but
the joint randomness is shared by both the prover and verifier.
* Flp.query(input: Vec[Field], proof: Vec[Field], query_rand:
Vec[Field], joint_rand: Vec[Field]) > Vec[Field] is the query
generation algorithm run by the verifier. This is used to "query"
the input and proof. The result of the query (i.e., the output of
this function) is called the "verifier message". In addition to
the input 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.
* Flp.decide(verifier: Vec[Field]) > Bool is the deterministic
decision algorithm run by the verifier. It takes as input the
verifier message and outputs a boolean indicating if the input
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 input and proof, the
querygeneration algorithm outputs a share of the verifier message.
Furthermore, the "zeroknowledge" property of the FLP system ensures
that the verifier message reveals nothing about the input's validity.
Therefore, to decide if an input 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.
An FLP is executed by the prover and verifier as follows:
def run_flp(Flp, inp: Vec[Flp.Field], num_shares: Unsigned):
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(inp, prove_rand, joint_rand)
# Verifier queries the input and proof.
verifier = Flp.query(inp, proof, query_rand, joint_rand, num_shares)
# Verifier decides if the input is valid.
return Flp.decide(verifier)
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Figure 11: Execution of an FLP.
The proof system is constructed so that, if input is a valid input,
then run_flp(Flp,. input) always returns True. On the other hand, if
input is invalid, then as long as joint_rand and query_rand are
generated uniform randomly, the output is False with overwhelming
probability.
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.
6.1.1. 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) > Vec[Field] encodes a raw
measurement as a vector of field elements. The return value MUST
be of length INPUT_LEN.
For some FLPs, the encoded input 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 (this type is also defined in
Section 6.4). After consuming this vector, all that is needed is the
integer it represents. Thus the FLP defines an algorithm for
truncating the input to the length of the aggregated output:
* Flp.truncate(input: Vec[Field]) > Vec[Field] maps an encoded
input to an aggregatable output. The length of the input MUST be
INPUT_LEN and the length of the output MUST be OUTPUT_LEN.
We remark that, taken together, these two functionalities correspond
roughly to the notion of "Affineaggregatable encodings (AFEs)" from
[CGB17].
6.2. Construction
This section specifies Prio3, an implementation of the Vdaf interface
(Section 4). It has two generic parameters: an Flp (Section 6.1) and
a Prg (Section 5.2). 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.
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+=============+===============================================+
 Parameter  Value 
+=============+===============================================+
 ROUNDS  1 
+++
 SHARES  in [2, 255) 
+++
 Measurement  Flp.Measurement 
+++
 PublicParam  None 
+++
 VerifyParam  Tuple[Unsigned, Bytes] 
+++
 AggParam  None 
+++
 Prep  Tuple[Vec[Flp.Field], Optional[Bytes], Bytes] 
+++
 OutShare  Vec[Flp.Field] 
+++
 AggShare  Vec[Flp.Field] 
+++
 AggResult  Vec[Unsigned] 
+++
Table 5: Associated parameters for the prio3 VDAF.
6.2.1. Setup
The setup algorithm generates a symmetric key shared by all of the
Aggregators. The key is used to derive query randomness for the FLP
querygeneration algorithm run by the Aggregators during preparation.
An Aggregator's verification parameter also includes its "ID", a
unique integer in [0, SHARES).
def setup(Prio3):
k_query_init = gen_rand(Prio3.Prg.SEED_SIZE)
verify_param = [(j, k_query_init) for j in range(Prio3.SHARES)]
return (None, verify_param)
Figure 12: The setup algorithm for prio3.
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6.2.2. Sharding
Recall from Section 6.1 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 input in a way that
allows the Aggregators to reconstruct it from their input shares.
(This idea is based on the FiatShamir heuristic and is described in
Section 6.2.3 of [BBCGGI19].)
The inputdistribution algorithm involves the following steps:
1. Encode the Client's raw measurement as an input for the FLP
2. Shard the input into a sequence of input shares
3. Derive the joint randomness from the input shares
4. Run the FLP proofgeneration algorithm using the derived joint
randomness
5. Shard the proof into a sequence of proof shares
The algorithm is specified below. Notice that only one set input and
proof shares (called the "leader" shares below) are vectors of field
elements. The other shares (called the "helper" shares) are
represented instead by PRG seeds, which are expanded into vectors of
field elements.
The code refers to a pair of auxiliary functions for encoding the
shares. These are called encode_leader_share and encode_helper_share
respectively and they are described in Section 6.2.6.
def measurement_to_input_shares(Prio3, _public_param, measurement):
dst = b"vdaf00 prio3"
inp = Prio3.Flp.encode(measurement)
k_joint_rand = zeros(Prio3.Prg.SEED_SIZE)
# Generate input shares.
leader_input_share = inp
k_helper_input_shares = []
k_helper_blinds = []
k_helper_hints = []
for j in range(Prio3.SHARES1):
k_blind = gen_rand(Prio3.Prg.SEED_SIZE)
k_share = gen_rand(Prio3.Prg.SEED_SIZE)
helper_input_share = Prio3.Prg.expand_into_vec(
Prio3.Flp.Field,
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k_share,
dst + byte(j+1),
Prio3.Flp.INPUT_LEN
)
leader_input_share = vec_sub(leader_input_share,
helper_input_share)
encoded = Prio3.Flp.Field.encode_vec(helper_input_share)
k_hint = Prio3.Prg.derive_seed(k_blind,
byte(j+1) + encoded)
k_joint_rand = xor(k_joint_rand, k_hint)
k_helper_input_shares.append(k_share)
k_helper_blinds.append(k_blind)
k_helper_hints.append(k_hint)
k_leader_blind = gen_rand(Prio3.Prg.SEED_SIZE)
encoded = Prio3.Flp.Field.encode_vec(leader_input_share)
k_leader_hint = Prio3.Prg.derive_seed(k_leader_blind,
byte(0) + encoded)
k_joint_rand = xor(k_joint_rand, k_leader_hint)
# Finish joint randomness hints.
for j in range(Prio3.SHARES1):
k_helper_hints[j] = xor(k_helper_hints[j], k_joint_rand)
k_leader_hint = xor(k_leader_hint, k_joint_rand)
# Generate the proof shares.
prove_rand = Prio3.Prg.expand_into_vec(
Prio3.Flp.Field,
gen_rand(Prio3.Prg.SEED_SIZE),
dst,
Prio3.Flp.PROVE_RAND_LEN
)
joint_rand = Prio3.Prg.expand_into_vec(
Prio3.Flp.Field,
k_joint_rand,
dst,
Prio3.Flp.JOINT_RAND_LEN
)
proof = Prio3.Flp.prove(inp, prove_rand, joint_rand)
leader_proof_share = proof
k_helper_proof_shares = []
for j in range(Prio3.SHARES1):
k_share = gen_rand(Prio3.Prg.SEED_SIZE)
k_helper_proof_shares.append(k_share)
helper_proof_share = Prio3.Prg.expand_into_vec(
Prio3.Flp.Field,
k_share,
dst + byte(j+1),
Prio3.Flp.PROOF_LEN
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)
leader_proof_share = vec_sub(leader_proof_share,
helper_proof_share)
input_shares = []
input_shares.append(Prio3.encode_leader_share(
leader_input_share,
leader_proof_share,
k_leader_blind,
k_leader_hint,
))
for j in range(Prio3.SHARES1):
input_shares.append(Prio3.encode_helper_share(
k_helper_input_shares[j],
k_helper_proof_shares[j],
k_helper_blinds[j],
k_helper_hints[j],
))
return input_shares
Figure 13: Inputdistribution algorithm for prio3.
6.2.3. Preparation
This section describes the process of recovering output shares from
the input shares. The highlevel idea is that each Aggregator first
queries its input and proof share locally, then exchanges its
verifier share with the other Aggregators. The verifier shares are
then combined into the verifier message, which is used to decide
whether to accept.
In addition, the Aggregators must ensure that they have all used the
same joint randomness for the querygeneration algorithm. The joint
randomness is generated by a PRG seed. Each Aggregator derives an
XOR secret share of this seed from its input share and the "blind"
generated by the client. Thus, before running the querygeneration
algorithm, it must first gather the XOR secret shares derived by the
other Aggregators.
In order to avoid extra round of communication, the Client sends each
Aggregator a "hint" equal to the XOR of the other Aggregators' shares
of the joint randomness seed. 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 XOR shares of the PRG
seed along with their verifier shares.
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NOTE This optimization somewhat diverges from Section 6.2.3 of
[BBCGGI19]. Security analysis is needed.
The algorithms required for preparation are defined as follows.
These algorithms make use of encoding and decoding methods defined in
Section 6.2.6.
def prep_init(Prio3, verify_param, _agg_param, nonce, input_share):
dst = b"vdaf00 prio3"
(j, k_query_init) = verify_param
(input_share, proof_share, k_blind, k_hint) = \
Prio3.decode_leader_share(input_share) if j == 0 else \
Prio3.decode_helper_share(dst, j, input_share)
out_share = Prio3.Flp.truncate(input_share)
k_query_rand = Prio3.Prg.derive_seed(k_query_init,
byte(255) + nonce)
query_rand = Prio3.Prg.expand_into_vec(
Prio3.Flp.Field,
k_query_rand,
dst,
Prio3.Flp.QUERY_RAND_LEN
)
joint_rand, k_joint_rand, k_joint_rand_share = [], None, None
if Prio3.Flp.JOINT_RAND_LEN > 0:
encoded = Prio3.Flp.Field.encode_vec(input_share)
k_joint_rand_share = Prio3.Prg.derive_seed(k_blind,
byte(j) + encoded)
k_joint_rand = xor(k_hint, k_joint_rand_share)
joint_rand = Prio3.Prg.expand_into_vec(
Prio3.Flp.Field,
k_joint_rand,
dst,
Prio3.Flp.JOINT_RAND_LEN
)
verifier_share = Prio3.Flp.query(input_share,
proof_share,
query_rand,
joint_rand,
Prio3.SHARES)
prep_msg = Prio3.encode_prepare_message(verifier_share,
k_joint_rand_share)
return (out_share, k_joint_rand, prep_msg)
def prep_next(Prio3, prep, inbound):
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(out_share, k_joint_rand, prep_msg) = prep
if inbound is None:
return (prep, prep_msg)
(verifier, k_joint_rand_check) = \
Prio3.decode_prepare_message(inbound)
if k_joint_rand_check != k_joint_rand or \
not Prio3.Flp.decide(verifier):
raise ERR_VERIFY
return out_share
def prep_shares_to_prep(Prio3, _agg_param, prep_shares):
verifier = Prio3.Flp.Field.zeros(Prio3.Flp.VERIFIER_LEN)
k_joint_rand_check = zeros(Prio3.Prg.SEED_SIZE)
for encoded in prep_shares:
(verifier_share, k_joint_rand_share) = \
Prio3.decode_prepare_message(encoded)
verifier = vec_add(verifier, verifier_share)
if Prio3.Flp.JOINT_RAND_LEN > 0:
k_joint_rand_check = xor(k_joint_rand_check,
k_joint_rand_share)
return Prio3.encode_prepare_message(verifier,
k_joint_rand_check)
Figure 14: Preparation state for prio3.
6.2.4. Aggregation
Aggregating a set of output shares is simply a matter of adding up
the vectors elementwise.
def out_shares_to_agg_share(Prio3, _agg_param, out_shares):
agg_share = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
for out_share in out_shares:
agg_share = vec_add(agg_share, out_share)
return agg_share
Figure 15: Aggregation algorithm for prio3.
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6.2.5. 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 agg_shares_to_result(Prio3, _agg_param, agg_shares):
agg = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
for agg_share in agg_shares:
agg = vec_add(agg, agg_share)
return list(map(lambda x: x.as_unsigned(), agg))
Figure 16: Computation of the aggregate result for prio3.
6.2.6. Auxiliary Functions
def encode_leader_share(Prio3,
input_share,
proof_share,
k_blind,
k_hint):
encoded = Bytes()
encoded += Prio3.Flp.Field.encode_vec(input_share)
encoded += Prio3.Flp.Field.encode_vec(proof_share)
if Prio3.Flp.JOINT_RAND_LEN > 0:
encoded += k_blind
encoded += k_hint
return encoded
def decode_leader_share(Prio3, encoded):
l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.INPUT_LEN
encoded_input_share, encoded = encoded[:l], encoded[l:]
input_share = Prio3.Flp.Field.decode_vec(encoded_input_share)
l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.PROOF_LEN
encoded_proof_share, encoded = encoded[:l], encoded[l:]
proof_share = Prio3.Flp.Field.decode_vec(encoded_proof_share)
l = Prio3.Prg.SEED_SIZE
k_blind, k_hint = None, None
if Prio3.Flp.JOINT_RAND_LEN > 0:
k_blind, encoded = encoded[:l], encoded[l:]
k_hint, encoded = encoded[:l], encoded[l:]
if len(encoded) != 0:
raise ERR_DECODE
return (input_share, proof_share, k_blind, k_hint)
def encode_helper_share(Prio3,
k_input_share,
k_proof_share,
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k_blind,
k_hint):
encoded = Bytes()
encoded += k_input_share
encoded += k_proof_share
if Prio3.Flp.JOINT_RAND_LEN > 0:
encoded += k_blind
encoded += k_hint
return encoded
def decode_helper_share(Prio3, dst, j, encoded):
l = Prio3.Prg.SEED_SIZE
k_input_share, encoded = encoded[:l], encoded[l:]
input_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field,
k_input_share,
dst + byte(j),
Prio3.Flp.INPUT_LEN)
k_proof_share, encoded = encoded[:l], encoded[l:]
proof_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field,
k_proof_share,
dst + byte(j),
Prio3.Flp.PROOF_LEN)
k_blind, k_hint = None, None
if Prio3.Flp.JOINT_RAND_LEN > 0:
k_blind, encoded = encoded[:l], encoded[l:]
k_hint, encoded = encoded[:l], encoded[l:]
if len(encoded) != 0:
raise ERR_DECODE
return (input_share, proof_share, k_blind, k_hint)
def encode_prepare_message(Prio3, verifier, k_joint_rand):
encoded = Bytes()
encoded += Prio3.Flp.Field.encode_vec(verifier)
if Prio3.Flp.JOINT_RAND_LEN > 0:
encoded += k_joint_rand
return encoded
def decode_prepare_message(Prio3, encoded):
l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.VERIFIER_LEN
encoded_verifier, encoded = encoded[:l], encoded[l:]
verifier = Prio3.Flp.Field.decode_vec(encoded_verifier)
k_joint_rand = None
if Prio3.Flp.JOINT_RAND_LEN > 0:
l = Prio3.Prg.SEED_SIZE
k_joint_rand, encoded = encoded[:l], encoded[l:]
if len(encoded) != 0:
raise ERR_DECODE
return (verifier, k_joint_rand)
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Figure 17: Helper functions required for prio3.
6.3. A GeneralPurpose FLP
This section describes an FLP based on the construction from in
[BBCGGI19], Section 4.2. We begin in Section 6.3.1 with an overview
of their proof system and the extensions to their proof system made
here. The construction is specified in Section 6.3.3.
OPEN ISSUE We're not yet sure if specifying this generalpurpose
FLP is desirable. It might be preferable to specify specialized
FLPs for each data type that we want to standardize, for two
reasons. First, clear and concise specifications are likely
easier to write for specialized FLPs rather than the general one.
Second, we may end up tailoring each FLP to the measurement type
in a way that improves performance, but breaks compatibility with
the generalpurpose FLP.
In any case, we can't make this decision until we know which data
types to standardize, so for now, we'll stick with the general
purpose construction. The reference implementation can be found
at https://github.com/cjpatton/vdaf/tree/main/poc.
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.
6.3.1. Overview
In the proof system of [BBCGGI19], validity is defined via an
arithmetic circuit evaluated over the input: If the circuit output is
zero, then the input is deemed valid; otherwise, if the circuit
output is nonzero, then the input is deemed invalid. Thus the goal
of the proof system is merely to allow the verifier to evaluate the
validity circuit over the input. For our application (Section 6),
this computation is distributed among multiple Aggregators, each of
which has only a share of the input.
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 an
input 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
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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 input and proof.
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]
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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 input 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 6.3.1.1; 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.
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
input 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 6.3.3.
6.3.1.1. Extensions
The FLP described in the next section extends the proof system
[BBCGGI19], Section 4.2 in three ways.
First, the validity circuit in our construction includes an
additional, random input (this is the "joint randomness" derived from
the input shares in Prio3; see Section 6.2). 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(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N1])
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(Note that this is a special case of [BBCGGI19], Theorem 5.2.) Here
inp 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 inp is valid
(i.e., each inp[j] is in [0,2)), then the circuit will evaluate to 0
regardless of the value of r; but if inp[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.) For example, the following
circuit is allowed, where Mul and Range2 are the gadgets defined
above (the input has length N+1):
C(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N1]) + \
2^0 * inp[0] + ... + 2^(N1) * inp[N1]  \
Mul(inp[N], inp[N])
Finally, [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 5.1.2.
6.3.2. Validity Circuits
The FLP described in Section 6.3.3 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 
+++
 INPUT_LEN  Length of the input 
+++
 OUTPUT_LEN  Length of the aggregatable output 
+++
 JOINT_RAND_LEN  Length of the random input 
+++
 Measurement  The type of measurement 
+++
 Field  An FFTfriendly finite field as 
  defined in Section 5.1.2 
+++
Table 6: 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 6.3.1 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).
A concrete Valid provides the following methods for encoding a
measurement as an input vector and truncating an input vector to the
length of an aggregatable output:
* Valid.encode(measurement: Measurement) > Vec[Field] returns a
vector of length INPUT_LEN representing a measurement.
* Valid.truncate(input: Vec[Field]) > Vec[Field] returns a vector
of length OUTPUT_LEN representing an aggregatable output.
Finally, the following class methods are derived for each concrete
Valid:
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# Length of the prover randomness.
def prove_rand_len(Valid):
return sum(map(lambda g: g.ARITY, Valid.GADGETS))
# Length of the query randomness.
def query_rand_len(Valid):
return len(Valid.GADGETS)
# Length of the proof.
def proof_len(Valid):
length = 0
for (g, g_calls) in zip(Valid.GADGETS, Valid.GADGET_CALLS):
P = next_power_of_2(1 + g_calls)
length += g.ARITY + g.DEGREE * (P  1) + 1
return length
# Length of the verifier message.
def verifier_len(Valid):
length = 1
for g in Valid.GADGETS:
length += g.ARITY + 1
return length
Figure 18: Derived methods for validity circuits.
6.3.3. Construction
This section specifies FlpGeneric, an implementation of the Flp
interface (Section 6.1). It has as a generic parameter a validity
circuit Valid implementing the interface defined in Section 6.3.2.
NOTE A reference implementation can be found in
https://github.com/cjpatton/vdaf/blob/main/poc/flp_generic.sage.
The FLP parameters for FlpGeneric are defined in Table 7. 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
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 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 6.3.2) 
+++
 QUERY_RAND_LEN  Valid.query_rand_len() (see Section 6.3.2) 
+++
 JOINT_RAND_LEN  Valid.JOINT_RAND_LEN 
+++
 INPUT_LEN  Valid.INPUT_LEN 
+++
 OUTPUT_LEN  Valid.OUTPUT_LEN 
+++
 PROOF_LEN  Valid.proof_len() (see Section 6.3.2) 
+++
 VERIFIER_LEN  Valid.verifier_len() (see Section 6.3.2) 
+++
 Measurement  Valid.Measurement 
+++
 Field  Valid.Field 
+++
Table 7: FLP Parameters of FlpGeneric.
6.3.3.1. Proof Generation
On input inp, prove_rand, and joint_rand, the proof is computed as
follows:
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 inp 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.
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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)).
* 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].
6.3.3.2. Query Generation
On input of inp, 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 6.3.3.1.
3. Evaluate Valid on input of inp 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 v denote
the output of the circuit evaluation.
4. Compute the wire polynomials just as in the prover's step (4.).
5. 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.)
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* 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].
6.3.3.3. Decision
On input of vector verifier, the verifier decides if the input is
valid as follows:
1. Parse verifier into v, x_1, y_1, ..., x_H, y_H as defined in
Section 6.3.3.2.
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.
6.3.3.4. Encoding
The FLP encoding and truncation methods invoke Valid.encode and
Valid.truncate in the natural way.
6.4. Instantiations
This section specifies instantiations of Prio3 for various
measurement types. Each uses FlpGeneric as the FLP (Section 6.3) and
is determined by a validity circuit (Section 6.3.2) and a PRG
(Section 5.2). Test vectors for each can be found in Appendix "Test
Vectors".
NOTE Reference implementations of each of these VDAFs can be found
in https://github.com/cjpatton/vdaf/blob/main/poc/vdaf_prio3.sage.
6.4.1. Prio3Aes128Count
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.
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This instance uses PrgAes128 (see Section 5.2.1) as its PRG. Its
validity circuit, denoted Count, uses Field64 (Table 2) as its finite
field. Its gadget, denoted Mul, is the degree2, arity2 gadget
defined as
def Mul(x, y):
return x * y
The validity circuit is defined as
def Count(inp: Vec[Field64]):
return Mul(inp[0], inp[0])  inp[0]
The measurement is encoded as a singleton vector in the natural way.
The parameters for this circuit are summarized below.
+================+==========================+
 Parameter  Value 
+================+==========================+
 GADGETS  [Mul] 
+++
 GADGET_CALLS  [1] 
+++
 INPUT_LEN  1 
+++
 OUTPUT_LEN  1 
+++
 JOINT_RAND_LEN  0 
+++
 Measurement  Unsigned, in range [0,2) 
+++
 Field  Field64 (Table 2) 
+++
Table 8: Parameters of validity circuit
Count.
6.4.2. Prio3Aes128Sum
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.
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This instance of prio3 uses PrgAes128 (see Section 5.2.1) as its PRG.
Its validity circuit, denoted Sum, uses Field128 (Table 3) as its
finite field. 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(Sum, measurement: Integer):
if 0 > measurement or measurement >= 2^Sum.INPUT_LEN:
raise ERR_INPUT
encoded = []
for l in range(Sum.INPUT_LEN):
encoded.append(Sum.Field((measurement >> l) & 1))
return encoded
def truncate(Sum, inp):
decoded = Sum.Field(0)
for (l, b) in enumerate(inp):
w = Sum.Field(1 << l)
decoded += w * b
return [decoded]
The validity circuit checks that the input comprised of ones and
zeros. Its gadget, denoted Range2, is the degree2, arity1 gadget
defined as
def Range2(x):
return x^2  x
The validity circuit is defined as
def Sum(inp: Vec[Field128], joint_rand: Vec[Field128]):
out = Field128(0)
r = joint_rand[0]
for x in inp:
out += r * Range2(x)
r *= joint_rand[0]
return out
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+================+================================+
 Parameter  Value 
+================+================================+
 GADGETS  [Range2] 
+++
 GADGET_CALLS  [bits] 
+++
 INPUT_LEN  bits 
+++
 OUTPUT_LEN  1 
+++
 JOINT_RAND_LEN  1 
+++
 Measurement  Unsigned, in range [0, 2^bits) 
+++
 Field  Field128 (Table 3) 
+++
Table 9: Parameters of validity circuit Sum.
6.4.3. Prio3Aes128Histogram
This instance of Prio3 allows for estimating the distribution of the
measurements by computing a simple histogram. Each measurement is an
arbitrary integer and the aggregate result counts the number of
measurements that fall in a set of fixed buckets.
This instance of prio3 uses PrgAes128 (see Section 5.2.1) as its PRG.
Its validity circuit, denoted Histogram, uses Field128 (Table 3) as
its finite field. The measurement is encoded as a onehot vector
representing the bucket into which the measurement falls (let bucket
denote a sequence of monotonically increasing integers):
def encode(Histogram, measurement: Integer):
boundaries = buckets + [Infinity]
encoded = [Field128(0) for _ in range(len(boundaries))]
for i in range(len(boundaries)):
if measurement <= boundaries[i]:
encoded[i] = Field128(1)
return encoded
def truncate(Histogram, inp: Vec[Field128]):
return inp
The validity circuit uses Range2 (see Section 6.4.2) as its single
gadget. It checks for onehotness in two steps, as follows:
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def Histogram(inp: Vec[Field128],
joint_rand: Vec[Field128],
num_shares: Unsigned):
# Check that each bucket is one or zero.
range_check = Field128(0)
r = joint_rand[0]
for x in inp:
range_check += r * Range2(x)
r *= joint_rand[0]
# Check that the buckets sum to 1.
sum_check = Field128(1) * Field128(num_shares).inv()
for b in inp:
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
input is sharded. This is provided to the FLP by Prio3.
+================+====================+
 Parameter  Value 
+================+====================+
 GADGETS  [Range2] 
+++
 GADGET_CALLS  [buckets + 1] 
+++
 INPUT_LEN  buckets + 1 
+++
 OUTPUT_LEN  buckets + 1 
+++
 JOINT_RAND_LEN  2 
+++
 Measurement  Integer 
+++
 Field  Field128 (Table 3) 
+++
Table 10: Parameters of validity
circuit Histogram.
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7. Poplar1
NOTE The spec for Poplar1 is still a workinprogress. A partial
implementation can be found at https://github.com/abetterinternet/
libpriors/blob/main/src/vdaf/poplar1.rs. The verification logic
is nearly complete, however as of this draft the code is missing
the IDPF. An implementation of the IDPF can be found at
https://github.com/google/distributed_point_functions/.
This section specifies Poplar1, a VDAF for the following task. Each
Client holds a BITSbit string and the Aggregators hold a set 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 inputs are prefixed by each candidate prefix.
This functionality is the core component of Poplar [BBCGGI21]. At a
high level, the protocol works as follows.
1. Each Clients runs the inputdistribution algorithm on its nbit
string and sends an input 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.
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 as follows. For each p in H, add 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.
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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 path,
then function evaluates to to a nonzero value; otherwise it
evaluates to zero. This structure allows an IDPF to provide the
functionality required for the above protocol, while at the same time
ensuring the same degree of privacy as a DPF.
Our VDAF composes an IDPF with the "secure sketching" protocol of
[BBCGGI21]. 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
one element, which is equal to one.
7.1. Incremental Distributed Point Functions (IDPFs)
An IDPF is defined over a domain of size 2^BITS, where BITS is
constant defined by the IDPF. The Client specifies an index alpha
and values beta, one for each "level" 1 <= l <= BITS. The key
generation generates two IDPF keys, one for each Aggregator. When
evaluated at index 0 <= x < 2^l, each IDPF share returns an additive
share of beta[l] if x is the lbit prefix of alpha and shares of zero
otherwise.
CP What does it mean for x to be the lbit prefix of alpha? We
need to be a bit more precise here.
CP Why isn't the domain size actually 2^(BITS+1), i.e., the number
of nodes in a binary tree of height BITS (excluding the root)?
Each beta[l] is a pair of elements of a finite field. Each level MAY
have different field parameters. Thus a concrete IDPF specifies
associated types Field[1], Field[2], ..., and Field[BITS] defining,
respectively, the field parameters at level 1, level 2, ..., and
level BITS.
An IDPF is comprised of the following algorithms (let type Value[l]
denote (Field[l], Field[l]) for each level l):
* idpf_gen(alpha: Unsigned, beta: (Value[1], ..., Value[BITS])) >
key: (IDPFKey, IDPFKey) is the randomized keygeneration algorithm
run by the client. Its inputs are the index alpha and the values
beta. The value of alpha MUST be in range [0, 2^BITS).
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* IDPFKey.eval(l: Unsigned, x: Unsigned) > value: Value[l]) is
deterministic, stateless keyevaluation algorithm run by each
Aggregator. It returns the value corresponding to index x. The
value of l MUST be in [1, BITS] and the value of x MUST be in
range [2^(l1), 2^l).
A concrete IDPF specifies a single associated constant:
* BITS: Unsigned is the length of each Client input.
A concrete IDPF also specifies the following associated types:
* Field[l] for each level 1 <= l <= BITS. Each defines the same
methods and associated constants as Field in Section 6.
Note that IDPF construction of [BBCGGI21] uses one field for the
inner nodes of the tree and a different, larger field for the leaf
nodes. See [BBCGGI21], Section 4.3.
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.
7.2. Construction
The VDAF involves two rounds of communication (ROUNDS == 2) and is
defined for two Aggregators (SHARES == 2).
7.2.1. Setup
The verification parameter is a symmetric key shared by both
Aggregators. This VDAF has no public parameter.
def vdaf_setup():
k_verify_init = gen_rand(SEED_SIZE)
return (None, [(0, k_verify_init), (1, k_verify_init)])
Figure 19: The setup algorithm for poplar1.
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7.2.1.1. Client
The client's input is an IDPF index, denoted alpha. The values are
pairs of field elements (1, k) where each k is chosen at random.
This random value is used as part of the secure sketching protocol of
[BBCGGI21]. After evaluating their IDPF key shares on the set 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*a + b  k*a + c
and sends additive shares of a, b, c, A and B to the Aggregators.
Putting everything together, the inputdistribution algorithm is
defined as follows. Function encode_input_share is defined in
Section 7.2.5.
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def measurement_to_input_shares(_, alpha):
if alpha < 2**BITS: raise ERR_INVALID_INPUT
# Prepare IDPF values.
beta = []
correlation_shares_0, correlation_shares_1 = [], []
for l in range(1,BITS+1):
(k, a, b, c) = Field[l].rand_vec(4)
# Construct values of the form (1, k), where k
# is a random field element.
beta += [(1, k)]
# Create secret shares of correlations to aid
# the Aggregators' computation.
A = 2*a+k
B = a*a + b  a * k + c
correlation_share = Field[l].rand_vec(5)
correlation_shares_1.append(correlation_share)
correlation_shares_0.append(
[a, b, c, A, B]  correlation_share)
# Generate IDPF shares.
(key_0, key_1) = idpf_gen(input, beta)
input_shares = [
encode_input_share(key_0, correlation_shares_0),
encode_input_share(key_1, correlation_shares_1),
]
return input_shares
Figure 20: The inputdistribution algorithm for poplar1.
TODO It would be more efficient to represent the correlation
shares using PRG seeds as suggested in [BBCGGI21].
7.2.2. 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
pair of vectors of field elements data_share and auth_share, The
Aggregators use auth_share and the correlation shares provided by the
Client to verify that their data_share vectors are additive shares of
a onehot vector.
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CP Consider adding aggregation parameter as input to k_verify_rand
derivation.
class PrepState:
def __init__(verify_param, agg_param, nonce, input_share):
(self.l, self.candidate_prefixes) = decode_indexes(agg_param)
(self.idpf_key,
self.correlation_shares) = decode_input_share(input_share)
(self.party_id, k_verify_init) = verify_param
self.k_verify_rand = get_key(k_verify_init, nonce)
self.step = "ready"
def next(self, inbound: Optional[Bytes]):
l = self.l
(a_share, b_share, c_share,
A_share, B_share) = correlation_shares[l1]
if self.step == "ready" and inbound == None:
# Evaluate IDPF on candidate prefixes.
data_share, auth_share = [], []
for x in self.candidate_prefixes:
value = kdpf_key.eval(l, x)
data_share.append(value[0])
auth_share.append(value[1])
# Prepare first sketch verification message.
r = Prg.expand_into_vec(Field[l], self.k_verify_rand, len(data_share))
verifier_share_1 = [
a_share + inner_product(data_share, r),
b_share + inner_product(data_share, r * r),
c_share + inner_product(auth_share, r),
]
self.output_share = data_share
self.step = "sketch round 1"
return verifier_share_1
elif self.step == "sketch round 1" and inbound != None:
verifier_1 = Field[l].decode_vec(inbound)
verifier_share_2 = [
(verifier_1[0] * verifier_1[0] \
 verifier_1[1] \
 verifier_1[2]) * self.party_id \
+ A_share * verifier_1[0] \
+ B_share
]
self.step = "sketch round 2"
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return Field[l].encode_vec(verifier_share_2)
elif self.step == "sketch round 2" and inbound != None:
verifier_2 = Field[l].decode_vec(inbound)
if verifier_2 != 0: raise ERR_INVALID
return Field[l].encode_vec(self.output_share)
else: raise ERR_INVALID_STATE
def prep_shares_to_prep(agg_param, inbound: Vec[Bytes]):
if len(inbound) != 2:
raise ERR_INVALID_INPUT
(l, _) = decode_indexes(agg_param)
verifier = Field[l].decode_vec(inbound[0]) + \
Field[l].decode_vec(inbound[1])
return Field[l].encode_vec(verifier)
Figure 21: Preparation state for poplar1.
7.2.3. Aggregation
def out_shares_to_agg_share(agg_param, output_shares: Vec[Bytes]):
(l, candidate_prefixes) = decode_indexes(agg_param)
if len(output_shares) != len(candidate_prefixes):
raise ERR_INVALID_INPUT
agg_share = Field[l].zeros(len(candidate_prefixes))
for output_share in output_shares:
agg_share += Field[l].decode_vec(output_share)
return Field[l].encode_vec(agg_share)
Figure 22: Aggregation algorithm for poplar1.
7.2.4. Unsharding
def agg_shares_to_result(agg_param, agg_shares: Vec[Bytes]):
(l, _) = decode_indexes(agg_param)
if len(agg_shares) != 2:
raise ERR_INVALID_INPUT
agg = Field[l].decode_vec(agg_shares[0]) + \
Field[l].decode_vec(agg_shares[1]J)
return Field[l].encode_vec(agg)
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Figure 23: Computation of the aggregate result for poplar1.
7.2.5. Helper Functions
TODO Specify the following functionalities:
* encode_input_share is used to encode an input share, consisting of
an IDPF key share and correlation shares.
* decode_input_share is used to decode an input share.
* decode_indexes(encoded: Bytes) > (l: Unsigned, indexes:
Vec[Unsigned]) decodes a sequence of indexes, i.e., candidate
indexes for IDPF evaluation. The value of l MUST be in range [1,
BITS] and indexes[i] MUST be in range [2^(l1), 2^l) for all i.
An error is raised if encoded cannot be decoded.
8. Security Considerations
NOTE: This is a brief outline of the security considerations.
This section will be filled out more as the draft matures and
security analyses are completed.
VDAFs have two essential security goals:
1. Privacy: An attacker that controls the network, 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.
2. Robustness: An attacker that controls the network and a subset of
Clients cannot cause the Collector to compute anything other than
the aggregate of the measurements of honest Clients.
Note that, to achieve robustness, it is important to ensure that the
verification parameters distributed to the Aggregators
(verify_params, see Section 6.2.1) is never revealed to the Clients.
It is also possible to consider a stronger form of robustness, where
the attacker also controls a subset of Aggregators (see [BBCGGI19],
Section 6.3). To satisfy this stronger notion of robustness, it is
necessary to prevent the attacker from sharing the verification
parameters with the Clients. It is therefore RECOMMENDED that the
Aggregators generate verify_params only after a set of Client inputs
has been collected for verification, and regenerate them for each
such set of inputs.
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In order to achieve robustness, the Aggregators MUST ensure that the
nonces used to process the measurements in a batch are all unique.
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.
* 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 mitigate 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 inputs for the remaining
Clients. Applications can guard against these risks by adding
additional controls on measurement submission, such as client
authentication and rate limits.
VDAFs do not inherently provide differential privacy [Dwo06]. The
VDAF approach to private measurement can be viewed as complementary
to differential privacy, relying on noncollusion instead of
statistical noise to protect the privacy of the inputs. It is
possible that a future VDAF could incorporate differential privacy
features, e.g., by injecting noise before the sharding stage and
removing it after unsharding.
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9. IANA Considerations
This document makes no request of IANA.
10. References
10.1. 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,
.
[RFC4493] Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The
AESCMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June
2006, .
[RFC8017] Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch,
"PKCS #1: RSA Cryptography Specifications Version 2.2",
RFC 8017, DOI 10.17487/RFC8017, November 2016,
.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, .
10.2. 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, .
[CGB17] CorriganGibbs, H. and D. Boneh, "Prio: Private, Robust,
and Scalable Computation of Aggregate Statistics", NSDI
2017 , 2017,
.
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[Dou02] Douceur, J., "The Sybil Attack", IPTPS 2002 , 2002,
.
[Dwo06] Dwork, C., "Differential Privacy", ICALP 2006 , 2006,
.
[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,
.
[ID.draftgpewprivppm]
Geoghegan, T., Patton, C., Rescorla, E., and C. A. Wood,
"Privacy Preserving Measurement", Work in Progress,
InternetDraft, draftgpewprivppm00, 25 October 2021,
.
[OriginTelemetry]
"Origin Telemetry", 2020, .
[Vad16] Vadhan, S., "The Complexity of Differential Privacy",
2016, .
Acknowledgments
Thanks to Henry CorriganGibbs, Armando FazHernandez, Mariana
Raykova, and Christopher Wood for useful feedback on and
contributions to the spec.
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Test Vectors
Test vectors cover the generation input shares and the conversion of
input shares into output shares. Vectors specify the public and
verification parameters, the measurement, the aggregation parameter,
the expected input shares, the prepare messages, and the expected
output shares.
Test vectors are encoded in JSON. Input shares and prepare messages
are represented as hexadecimal streams. To make the tests
deterministic, gen_rand() was replaced with a function that returns
the requested number of 0x01 octets.
Prio3Aes128Count
For this test, the value of SHARES is 2.
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{
"public_param": null,
"verify_params": [
[
0,
"01010101010101010101010101010101"
],
[
1,
"01010101010101010101010101010101"
]
],
"agg_param": null,
"prep": [
{
"measurement": 1,
"nonce": "01010101010101010101010101010101",
"input_shares": [
"05ac22db75e9b262e9642de9ec1ec37990625f92bf426c52e12c88d7c6e53ed673a3a8a3c7944170e09a52b96573259d",
"0101010101010101010101010101010101010101010101010101010101010101"
],
"prep_shares": [
[
"48771012eeda70a056cf2fd53022cf7b2edf45090eaa765c2b6cefb7a4abc524",
"b788efec11258f61fa53dd238a164da076bfd3f2e4bd966634b24bb64c2fa160"
]
],
"out_shares": [
[
408739992155304546
],
[
18038004077259279776
]
]
}
]
}
Prio3Aes128Sum
For this test:
* The value of SHARES is 2.
* The value of bits is 8.
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{
"public_param": null,
"verify_params": [
[
0,
"01010101010101010101010101010101"
],
[
1,
"01010101010101010101010101010101"
]
],
"agg_param": null,
"prep": [
{
"measurement": 100,
"nonce": "01010101010101010101010101010101",
"input_shares": [
"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",
"0101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010b9e7430cc7a71a356d6bd09d36d1fdb"
],
"prep_shares": [
[
"2bcc0144c56fcf120a7aab22d57cde99fd2ee2301be4c59983d5e68e79f04cd8978b2c4598eafab7b1e8b0af8ba4bda20b9e7430cc7a71a356d6bd09d36d1fdb",
"d433febb3a9030d1f58554dd2a8321682eff019a7a26e88471ed622fbd956e5e0af1c04b5573400ed13ae90b325aed4edfed32c071cc6899645ab72c36bd3670"
]
],
"out_shares": [
[
178602842237398423407215704739732627917
],
[
161679524683540039539650068628168138392
]
]
}
]
}
Prio3Aes128Histogram
For this test:
* The value of SHARES is 2.
* The value of buckets is [1, 10, 100].
Patton, et al. Expires 6 September 2022 [Page 61]
InternetDraft VDAF March 2022
{
"public_param": null,
"verify_params": [
[
0,
"01010101010101010101010101010101"
],
[
1,
"01010101010101010101010101010101"
]
],
"agg_param": null,
"prep": [
{
"measurement": 50,
"nonce": "01010101010101010101010101010101",
"input_shares": [
"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",
"0101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101014a3951bdbf7a4564e422499e59351ba7"
],
"prep_shares": [
[
"03faf09f79624d05e3a1e17bf1e5117ea6eec816dbd047641506a0c1b1e817fa1ebb27507cea09b00b47e0615ba82ff64a3951bdbf7a4564e422499e59351ba7",
"fc050f60869db2de1c5e1e840e1aee830a99091f9a820099b6b9591984046f48b6b6d58ddb1675ef43101aa8773f3025688dfdc50bd6d3a9ecf1613c58b7ced1"
]
],
"out_shares": [
[
7539922107207114695252505926366364067,
198783809130402957557687312006462666532,
261868461448231140209796284667530078285,
19075760356742656327154126012204712008
],
[
332742444813731348251613267441534402142,
141498557790535505389178461361438099677,
78413905472707322737069488700370687925,
321206606564195806619711647355696054201
]
]
}
]
}
Authors' Addresses
Patton, et al. Expires 6 September 2022 [Page 62]
InternetDraft VDAF March 2022
Christopher Patton
Cloudflare, Inc.
Email: cpatton@cloudflare.com
Richard L. Barnes
Cisco
Email: rlb@ipv.sx
Phillipp Schoppmann
Google
Email: schoppmann@google.com
Patton, et al. Expires 6 September 2022 [Page 63]