CFRG R. L. Barnes
InternetDraft Cisco
Intended status: Informational C. Patton
Expires: 27 November 2022 Cloudflare
P. Schoppmann
Google
26 May 2022
Verifiable Distributed Aggregation Functions
draftirtfcfrgvdaf01
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
This InternetDraft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
InternetDrafts are working documents of the Internet Engineering
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InternetDrafts are draft documents valid for a maximum of six months
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material or to cite them other than as "work in progress."
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Copyright Notice
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document authors. All rights reserved.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Change Log . . . . . . . . . . . . . . . . . . . . . . . 7
2. Conventions and Definitions . . . . . . . . . . . . . . . . . 7
3. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4. Definition of DAFs . . . . . . . . . . . . . . . . . . . . . 10
4.1. Sharding . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2. Preparation . . . . . . . . . . . . . . . . . . . . . . . 13
4.3. Aggregation . . . . . . . . . . . . . . . . . . . . . . . 13
4.4. Unsharding . . . . . . . . . . . . . . . . . . . . . . . 14
4.5. Execution of a DAF . . . . . . . . . . . . . . . . . . . 15
5. Definition of VDAFs . . . . . . . . . . . . . . . . . . . . . 16
5.1. Sharding . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2. Preparation . . . . . . . . . . . . . . . . . . . . . . . 18
5.3. Aggregation . . . . . . . . . . . . . . . . . . . . . . . 21
5.4. Unsharding . . . . . . . . . . . . . . . . . . . . . . . 21
5.5. Execution of a VDAF . . . . . . . . . . . . . . . . . . . 22
6. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 23
6.1. Finite Fields . . . . . . . . . . . . . . . . . . . . . . 23
6.1.1. Auxiliary Functions . . . . . . . . . . . . . . . . . 24
6.1.2. FFTFriendly Fields . . . . . . . . . . . . . . . . . 25
6.1.3. Parameters . . . . . . . . . . . . . . . . . . . . . 25
6.2. Pseudorandom Generators . . . . . . . . . . . . . . . . . 26
6.2.1. PrgAes128 . . . . . . . . . . . . . . . . . . . . . . 27
7. Prio3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.1. Fully Linear Proof (FLP) Systems . . . . . . . . . . . . 29
7.1.1. Encoding the Input . . . . . . . . . . . . . . . . . 32
7.2. Construction . . . . . . . . . . . . . . . . . . . . . . 33
7.2.1. Sharding . . . . . . . . . . . . . . . . . . . . . . 33
7.2.2. Preparation . . . . . . . . . . . . . . . . . . . . . 36
7.2.3. Aggregation . . . . . . . . . . . . . . . . . . . . . 38
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7.2.4. Unsharding . . . . . . . . . . . . . . . . . . . . . 38
7.2.5. Auxiliary Functions . . . . . . . . . . . . . . . . . 39
7.3. A GeneralPurpose FLP . . . . . . . . . . . . . . . . . . 41
7.3.1. Overview . . . . . . . . . . . . . . . . . . . . . . 41
7.3.2. Validity Circuits . . . . . . . . . . . . . . . . . . 44
7.3.3. Construction . . . . . . . . . . . . . . . . . . . . 46
7.4. Instantiations . . . . . . . . . . . . . . . . . . . . . 49
7.4.1. Prio3Aes128Count . . . . . . . . . . . . . . . . . . 50
7.4.2. Prio3Aes128Sum . . . . . . . . . . . . . . . . . . . 51
7.4.3. Prio3Aes128Histogram . . . . . . . . . . . . . . . . 52
8. Poplar1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8.1. Incremental Distributed Point Functions (IDPFs) . . . . . 55
8.2. Construction . . . . . . . . . . . . . . . . . . . . . . 56
8.2.1. Setup . . . . . . . . . . . . . . . . . . . . . . . . 56
8.2.2. Preparation . . . . . . . . . . . . . . . . . . . . . 58
8.2.3. Aggregation . . . . . . . . . . . . . . . . . . . . . 60
8.2.4. Unsharding . . . . . . . . . . . . . . . . . . . . . 60
8.2.5. Helper Functions . . . . . . . . . . . . . . . . . . 61
9. Security Considerations . . . . . . . . . . . . . . . . . . . 61
10. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 63
11. References . . . . . . . . . . . . . . . . . . . . . . . . . 63
11.1. Normative References . . . . . . . . . . . . . . . . . . 63
11.2. Informative References . . . . . . . . . . . . . . . . . 63
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 64
Test Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Prio3Aes128Count . . . . . . . . . . . . . . . . . . . . . . . 65
Prio3Aes128Sum . . . . . . . . . . . . . . . . . . . . . . . . 65
Prio3Aes128Histogram . . . . . . . . . . . . . . . . . . . . . 67
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 68
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.
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Thus systems that provide aggregate statistics while protecting
individual measurements can deliver the value of the measurements
while protecting users' privacy.
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
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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].
The VDAF abstraction laid out in Section 5 represents a class of
multiparty protocols for privacypreserving measurement proposed in
the literature. These protocols vary in their operational and
security considerations, sometimes in subtle but consequential ways.
This document therefore has two important goals:
1. Providing higherlevel protocols like [DAP] with a simple,
uniform interface for accessing privacypreserving measurement
schemes, 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 the
collector of the aggregate result);
2. Capabilities of a malicious coalition of servers attempting
to divulge information about client measurements; and
3. Conditions that are necessary to ensure that malicious
clients cannot corrupt the computation.
2. Providing cryptographers with design criteria that provide a
clear deployment roadmap for new constructions.
This document also specifies two concrete VDAF schemes, each based on
a protocol from the literature.
* The aforementioned Prio system [CGB17] allows for the privacy
preserving computation of a variety aggregate statistics. The
basic idea underlying Prio is fairly simple:
1. Each client shards its measurement into a sequence of additive
shares and distributes the shares among the aggregation
servers.
2. Next, each server adds up its shares locally, resulting in an
additive share of the aggregate.
3. Finally, the aggregation servers send their aggregate shares
to the data collector, who combines them to obtain the
aggregate result.
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The difficult part of this system is ensuring that the servers
hold shares of a valid input, e.g., the input is an integer in a
specific range. Thus Prio specifies a multiparty protocol for
accomplishing this task.
In Section 7 we describe Prio3, a VDAF that follows the same
overall framework as the original Prio protocol, but incorporates
techniques introduced in [BBCGGI19] that result in significant
performance gains.
* More recently, Boneh et al. [BBCGGI21] described a protocol
called Poplar for solving the theavyhitters problem in a
privacypreserving manner. Here each client holds a bitstring of
length n, and the goal of the aggregation servers is to compute
the set of inputs that occur at least t times. The core primitive
used in their protocol is a specialized Distributed Point Function
(DPF) [GI14] that allows the servers to "query" their DPF shares
on any bitstring of length shorter than or equal to n. As a
result of this query, each of the servers has an additive share of
a bit indicating whether the string is a prefix of the client's
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 8 we describe a VDAF called Poplar1 that implements
this functionality.
Finally, perhaps the most complex aspect of schemes like Prio3 and
Poplar1 is the process by which the clientgenerated measurements are
prepared for aggregation. Because these constructions are based on
secret sharing, the servers will be required to exchange some amount
of information in order to verify the measurement is valid and can be
aggregated. Depending on the construction, this process may require
multiple round trips over the network.
There are applications in which this verification step may not be
necessary, e.g., when the client's software is run a trusted
execution environment. To support these applications, this document
also defines Distributed Aggregation Functions (DAFs) as a simpler
class of protocols that aim to provide the same privacy guarantee as
VDAFs but fall short of being verifiable.
OPEN ISSUE Decide if we should give one or two example DAFs.
There are natural variants of Prio3 and Poplar1 that might be
worth describing.
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The remainder of this document is organized as follows: Section 3
gives a brief overview of DAFs and VDAFs; Section 4 defines the
syntax for DAFs; Section 5 defines the syntax for VDAFs; Section 6
defines various functionalities that are common to our constructions;
Section 7 describes the Prio3 construction; Section 8 describes the
Poplar1 construction; and Section 9 enumerates the security
considerations for VDAFs.
1.1. Change Log
(*) Indicates a change that breaks compatibility with the previous
draft.
01:
* Require that VDAFs specify serialization of aggregate shares.
* Define Distributed Aggregation Functions (DAFs).
* Prio3: Move proof verifier check from prep_next() to
prep_shares_to_prep(). (*)
* Remove public parameter and replace verification parameter with a
"verification key" and "Aggregator ID".
2. Conventions and Definitions
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in
BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
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.
A variable with type Bytes is a byte string. This document defines
several bytestring constants. When comprised of printable ASCII
characters, they are written as Python 3 bytestring literals (e.g.,
b'some constant string').
A global constant VERSION is defined, which algorithms are free to
use as desired. Its value SHALL be b'vdaf01'.
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This document describes algorithms for multiparty computations in
which the parties typically communicate over a network. Wherever a
quantity is defined that must be be transmitted from one party to
another, this document prescribes a particular encoding of that
quantity as a byte string.
OPEN ISSUE It might be better to not be prescriptive about how
quantities are encoded on the wire. See issue #58.
Some common functionalities:
* zeros(len: Unsigned) > Bytes returns an array of zero bytes. The
length of output MUST be len.
* 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
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++
+> Aggregator 0 +
 ++ 
 ^ 
  
 V 
 ++ 
 +> Aggregator 1 + 
  ++  
+++  ^  +>++
 Client +  +> Collector > Aggregate
+++ +>++
 ... 
 
  
 V 
 ++ 
+> Aggregator N1 +
++
Input shares Aggregate shares
Figure 1: Overall data flow of a (V)DAF
In a DAF or VDAFbased private measurement system, we distinguish
three types of actors: Clients, Aggregators, and Collectors. The
overall flow of the measurement process is as follows:
* To submit an individual measurement, the Client shards the
measurement into "input shares" and sends one input share to each
Aggregator.
* The Aggregators convert their input shares into "output shares".
 Output shares are in onetoone correspondence with the input
shares.
 Just as each Aggregator receives one input share of each input,
at the end of this process, each aggregator holds one output
share.
 In VDAFs, Aggregators will need to exchange information among
themselves as part of the validation process.
* Each Aggregator combines the output shares across inputs in the
batch to compute the "aggregate share" for that batch, i.e., its
share of the desired aggregate result.
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* The Aggregators submit their aggregate shares to the Collector,
who combines them to obtain the aggregate result over the batch.
Aggregators are a new class of actor relative to traditional
measurement systems where clients submit measurements to a single
server. They are critical for both the privacy properties of the
system and, in the case of VDAFs, the correctness of the measurements
obtained. The privacy properties of the system are assured by non
collusion among Aggregators, and Aggregators are the entities that
perform validation of Client measurements. Thus clients trust
Aggregators not to collude (typically it is required that at least
one Aggregator is honest), and Collectors trust Aggregators to
correctly run the protocol.
Within the bounds of the noncollusion requirements of a given (V)DAF
instance, it is possible for the same entity to play more than one
role. For example, the Collector could also act as an Aggregator,
effectively using the other Aggregator(s) to augment a basic client
server protocol.
In this document, we describe the computations performed by the
actors in this system. It is up to the higherlevel protocol making
use of the (V)DAF to arrange for the required information to be
delivered to the proper actors in the proper sequence. In general,
we assume that all communications are confidential and mutually
authenticated, with the exception that Clients submitting
measurements may be anonymous.
4. Definition of DAFs
By way of a gentle introduction to VDAFs, this section describes a
simpler class of schemes called Distributed Aggregation Functions
(DAFs). Unlike VDAFs, DAFs do not provide verifiability of the
computation. Clients must therefore be trusted to compute their
input shares correctly. Because of this fact, the use of a DAF is
NOT RECOMMENDED for most applications. See Section 9 for additional
discussion.
A DAF scheme is used to compute a particular "aggregation function"
over a set of measurements generated by Clients. Depending on the
aggregation function, the Collector might select an "aggregation
parameter" and disseminates it to the Aggregators. The semantics of
this parameter is specific to the aggregation function, but in
general it is used to represent the set of "queries" that can be made
on the measurement set. For example, the aggregation parameter is
used to represent the candidate prefixes in Poplar1 Section 8.
Execution of a DAF has four distinct stages:
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* Sharding  Each Client generates input shares from its measurement
and distributes them among the Aggregators.
* Preparation  Each Aggregator converts each input share into an
output share compatible with the aggregation function. This
computation involves the aggregation parameter. In general, each
aggregation parameter may result in a different an output share.
* Aggregation  Each Aggregator combines a sequence of output shares
into its aggregate share and sends the aggregate share to the
Collector.
* Unsharding  The Collector combines the aggregate shares into the
aggregate result.
Sharding and Preparation are done once per measurement. Aggregation
and Unsharding are done over a batch of measurements (more precisely,
over the recovered output shares).
A concrete DAF specifies an algorithm for the computation needed in
each of these stages. The interface of each algorithm is defined in
the remainder of this section. In addition, a concrete DAF defines
the associated constants and types enumerated in the following table.
+=============+===================================+
 Parameter  Description 
+=============+===================================+
 SHARES  Number of input shares into which 
  each measurement is sharded 
+++
 Measurement  Type of each measurement 
+++
 AggParam  Type of aggregation parameter 
+++
 OutShare  Type of each output share 
+++
 AggResult  Type of the aggregate result 
+++
Table 1: Constants and types defined by each
concrete DAF.
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These types define some of the inputs and outputs of DAF methods at
various stages of the computation. Observe that only the
measurements, output shares, the aggregate result, and the
aggregation parameter have an explicit type. All other values  in
particular, the input shares and the aggregate shares  have type
Bytes and are treated as opaque byte strings. This is because these
values must be transmitted between parties over a network.
OPEN ISSUE It might be cleaner to define a type for each value,
then have that type implement an encoding where necessary. This
way each method parameter has a meaningful type hint. See
issue#58.
4.1. Sharding
In order to protect the privacy of its measurements, a DAF Client
shards its measurements into a sequence of input shares. The
measurement_to_input_shares method is used for this purpose.
* Daf.measurement_to_input_shares(input: Measurement) > Vec[Bytes]
is the randomized inputdistribution algorithm run by each Client.
It consumes the measurement and produces a sequence of input
shares, one for each Aggregator. The length of the output vector
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 into input shares
and distributes them to the Aggregators.
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4.2. Preparation
Once an Aggregator has received an input share form a Client, the
next step is to prepare the input share for aggregation. This is
accomplished using the following algorithm:
* Daf.prep(agg_id: Unsigned, agg_param: AggParam, input_share:
Bytes) > OutShare is the deterministic preparation algorithm. It
takes as input an input share generated by a Client, the
Aggregator's unique identifier, and the aggregation parameter
selected by the Collector and returns an output share.
The protocol in which the DAF is used MUST ensure that the
Aggregator's identifier is equal to the integer in range [0, SHARES)
that matches the index of input_share in the sequence of input shares
output by the Client.
4.3. Aggregation
Once an Aggregator holds output shares for a batch of measurements
(where batches are defined by the application), it combines them into
a share of the desired aggregate result:
* Daf.out_shares_to_agg_share(agg_param: AggParam, out_shares:
Vec[OutShare]) > agg_share: Bytes is the deterministic
aggregation algorithm. It is run by each Aggregator a set of
recovered output shares.
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 3: Aggregation of output shares. `B` indicates the number of
measurements in the batch.
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For simplicity, we have written this algorithm in a "oneshot" form,
where all output shares for a batch are provided at the same time.
Many DAFs may also support a "streaming" form, where shares are
processed one at a time.
OPEN ISSUE It may be worthwhile to explicitly define the
"streaming" API. See issue#47.
4.4. Unsharding
After the Aggregators have aggregated a sufficient number of output
shares, each sends its aggregate share to the Collector, who runs the
following algorithm to recover the following output:
* Daf.agg_shares_to_result(agg_param: AggParam, agg_shares:
Vec[Bytes]) > 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 4: 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 [DAP].)
Or is this outofscope of this document? See https://github.com/
ietfwgppm/ppmspecification/issues/19.
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4.5. Execution of a DAF
Securely executing a DAF involves emulating the following procedure.
def run_daf(Daf,
agg_param: Daf.AggParam,
measurements: Vec[Daf.Measurement]):
out_shares = [ [] for j in range(Daf.SHARES) ]
for measurement in measurements:
# Each Client shards its measurement into input shares and
# distributes them among the Aggregators.
input_shares = Daf.measurement_to_input_shares(measurement)
# Each Aggregator prepares its input share for aggregation.
for j in range(Daf.SHARES):
out_shares[j].append(
Daf.prep(j, agg_param, input_shares[j]))
# Each Aggregator aggregates its output shares into an aggregate
# share and it to the Collector.
agg_shares = []
for j in range(Daf.SHARES):
agg_share_j = Daf.out_shares_to_agg_share(agg_param,
out_shares[j])
agg_shares.append(agg_share_j)
# Collector unshards the aggregate result.
agg_result = Daf.agg_shares_to_result(agg_param, agg_shares)
return agg_result
Figure 5: Execution of a DAF.
The inputs to this procedure are the same as the aggregation function
computed by the DAF: An aggregation parameter and a sequence of
measurements. The procedure prescribes how a DAF is executed in a
"benign" environment in which there is no adversary and the messages
are passed among the protocol participants over secure pointtopoint
channels. In reality, these channels need to be instantiated by some
"wrapper protocol", such as [DAP], that realizes these channels using
suitable cryptographic mechanisms. Moreover, some fraction of the
Aggregators (or Clients) may be malicious and diverge from their
prescribed behaviors. Section 9 describes the execution of the DAF
in various adversarial environments and what properties the wrapper
protocol needs to provide in each.
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5. Definition of VDAFs
Like DAFs described in the previous section, a VDAF scheme is used to
compute a particular aggregation function over a set of Client
generated measurements. Evaluation of a VDAF involves the same four
stages as for DAFs: Sharding, Preparation, Aggregation, and
Unsharding. However, the Preparation stage will require interaction
among the Aggregators in order to facilitate verifiability of the
computation's correctness. Accommodating this interaction will
require syntactic changes.
Overall execution of a VDAF comprises the following stages:
* Sharding  Computing input shares from an individual measurement
* Preparation  Conversion and verification of input shares to
output shares compatible with the aggregation function being
computed
* Aggregation  Combining a sequence of output shares into an
aggregate share
* Unsharding  Combining a sequence of aggregate shares into an
aggregate result
In contrast to DAFs, the Preparation stage for VDAFs now performs an
additional task: Verification of the validity of the recovered output
shares. This process ensures that aggregating the output shares will
not lead to a garbled aggregate result.
The remainder of this section defines the VDAF interface. The
attributes are listed in Table 2 are defined by each concrete VDAF.
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+=================+==========================================+
 Parameter  Description 
+=================+==========================================+
 VERIFY_KEY_SIZE  Size (in bytes) of the verification key 
  (Section 5.2) 
+++
 ROUNDS  Number of rounds of communication during 
  the Preparation stage (Section 5.2) 
+++
 SHARES  Number of input shares into which each 
  measurement is sharded (Section 5.1) 
+++
 Measurement  Type of each measurement 
+++
 AggParam  Type of aggregation parameter 
+++
 Prep  State of each Aggregator during 
  Preparation (Section 5.2) 
+++
 OutShare  Type of each output share 
+++
 AggResult  Type of the aggregate result 
+++
Table 2: Constants and types defined by each concrete VDAF.
Similarly to DAFs (see {[secdaf}}), any output of a VDAF method that
must be transmitted from one party to another is treated as an opaque
byte string. All other quantities are given a concrete type.
OPEN ISSUE It might be cleaner to define a type for each value,
then have that type implement an encoding where necessary. See
issue#58.
5.1. Sharding
Sharding is syntactically identical to DAFs (cf. Section 4.1):
* Vdaf.measurement_to_input_shares(measurement: Measurement) >
Vec[Bytes] is the randomized inputdistribution algorithm run by
each Client. It consumes the measurement and produces a sequence
of input shares, one for each Aggregator. Depending on the VDAF,
the input shares may encode additional information used to verify
the recovered output shares (e.g., the "proof shares" in Prio3
Section 7). The length of the output vector MUST be SHARES.
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5.2. Preparation
To recover and verify output shares, the Aggregators interact with
one another over ROUNDS rounds. Prior to each round, each Aggregator
constructs an outbound message. Next, the sequence of outbound
messages is combined into a single message, called a "preparation
message". (Each of the outbound messages are called "preparation
message shares".) Finally, the preparation message is distributed to
the Aggregators to begin the next round.
An Aggregator begins the first round with its input share and it
begins each subsequent round with the previous 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.
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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 6: VDAF preparation process on the input shares for a single
measurement. At the end of the computation, each Aggregator holds an
output share or an error.
To facilitate the preparation process, a concrete VDAF implements the
following class methods:
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* Vdaf.prep_init(verify_key: Bytes, agg_id: Unsigned, 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 the shared verification key (verify_key),
the Aggregator's unique identifier (agg_id), the aggregation
parameter (agg_param), the nonce provided by the environment
(nonce, see Figure 7), and one of the input shares generated by
the client (input_share). Its output is the Aggregator's initial
preparation state.
The length of verify_key MUST be VERIFY_KEY_SIZE. It is up to the
high level protocol in which the VDAF is used to arrange for the
distribution of the verification key among the Aggregators prior
to the start of this phase of VDAF evaluation.
OPEN ISSUE What security properties do we need for this key
exchange? See issue#18.
Protocols using the VDAF MUST ensure that the Aggregator's
identifier is equal to the integer in range [0, SHARES) that
matches the index of input_share in the sequence of input shares
output by the Client.
* 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.
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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
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 accommodating 0round VDAFs may require syntax changes
if, for example, public keys are required. On the other hand, we
could consider defining this class of schemes as a different
primitive. See issue#77.
5.3. Aggregation
VDAF Aggregation is identical to DAF Aggregation (cf. Section 4.3):
* Vdaf.out_shares_to_agg_share(agg_param: AggParam, out_shares:
Vec[OutShare]) > agg_share: Bytes is the deterministic
aggregation algorithm. It is run by each Aggregator over the
output shares it has computed over a batch of measurement inputs.
The data flow for this stage is illustrated in Figure 3. Here again,
we have the aggregation algorithm in a "oneshot" form, where all
shares for a batch are provided at the same time. VDAFs typically
also support a "streaming" form, where shares are processed one at a
time.
5.4. Unsharding
VDAF Unsharding is identical to DAF Unsharding (cf. Section 4.4):
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* Vdaf.agg_shares_to_result(agg_param: AggParam, agg_shares:
Vec[Bytes]) > 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.
The data flow for this stage is illustrated in Figure 4.
5.5. Execution of a VDAF
Secure execution of a VDAF involves simulating the following
procedure.
def run_vdaf(Vdaf,
agg_param: Vdaf.AggParam,
nonces: Vec[Bytes],
measurements: Vec[Vdaf.Measurement]):
# Generate the longlived verification key.
verify_key = gen_rand(Vdaf.VERIFY_KEY_SIZE)
out_shares = []
for (nonce, measurement) in zip(nonces, measurements):
# Each Client shards its measurement into input shares.
input_shares = Vdaf.measurement_to_input_shares(measurement)
# Each Aggregator initializes its preparation state.
prep_states = []
for j in range(Vdaf.SHARES):
state = Vdaf.prep_init(verify_key, 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)
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# The final outputs of prepare phase are the output shares.
out_shares.append(outbound)
# Each Aggregator aggregates its output shares into an
# aggregate share. In a distributed VDAF computation, the
# aggregate shares are sent over the network.
agg_shares = []
for j in range(Vdaf.SHARES):
out_shares_j = [out[j] for out in out_shares]
agg_share_j = Vdaf.out_shares_to_agg_share(agg_param,
out_shares_j)
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 7: Execution of a VDAF.
The inputs to this algorithm are the aggregation parameter, a list of
measurements, and a nonce for each measurement. This document does
not specify how the nonces are chosen, but security requires that the
nonces be unique. See Section 9 for details. As explained in
Section 4.5, the secure execution of a VDAF requires the application
to instantiate secure channels between each of the protocol
participants.
6. Preliminaries
This section describes the primitives that are common to the VDAFs
specified in this document.
6.1. Finite Fields
Both Prio3 and Poplar1 use finite fields of prime order. Finite
field elements are represented by a class Field with the following
associated parameters:
* MODULUS: 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 8: Derived class methods for finite fields.
6.1.1. Auxiliary Functions
The following auxiliary functions on vectors of field elements are
used in the remainder of this document. Note that an exception is
raised by each function if the operands are not the same length.
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# 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 9: Common functions for finite fields.
6.1.2. FFTFriendly Fields
Some VDAFs require fields that are suitable for efficient computation
of the discrete Fourier transform. (One example is Prio3 (Section 7)
when instantiated with the generic FLP of Section 7.3.3.)
Specifically, a field is said to be "FFTfriendly" if, in addition to
satisfying the interface described in Section 6.1, it implements the
following method:
* Field.gen() > Field returns the generator of a large subgroup of
the multiplicative group.
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.
6.1.3. 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 3: Field64, an FFTfriendly field.
+==============+================================+
 Parameter  Value 
+==============+================================+
 MODULUS  2^66 * 4611686018427387897 + 1 
+++
 ENCODED_SIZE  16 
+++
 Generator  7^4611686018427387897 
+++
 GEN_ORDER  2^66 
+++
Table 4: Field128, an FFTfriendly field.
6.2. 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 the subsections
that follow.
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 10: Derived class methods for PRGs.
6.2.1. PrgAes128
OPEN ISSUE Phillipp points out that a fixedkey mode of AES may be
more performant (https://eprint.iacr.org/2019/074.pdf). See
issue#32.
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 11: Definition of PRG PrgAes128.
7. 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 8 for an example of a VDAF that makes meaningful use of
the aggregation parameter.
As the name implies, Prio3 is a descendant of the original Prio
construction. A second iteration was deployed in the [ENPA] system,
and like the VDAF described here, the ENPA system was built from
techniques introduced in [BBCGGI19] that significantly improve
communication cost. That system was specialized for a particular
aggregation function; the goal of Prio3 is to provide the same level
of generality as the original construction.
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 7.1. The generic transformation of
an FLP into Prio3 is specified in Section 7.2. Next, a concrete FLP
suitable for any validity circuit is specified in Section 7.3.
Finally, instantiations of Prio3 for various types of measurements
are specified in Section 7.4. Test vectors can be found in
Appendix "Test Vectors".
7.1. Fully Linear Proof (FLP) Systems
Conceptually, an FLP is a twoparty protocol executed by a prover and
a verifier. In actual use, however, the prover's computation is
carried out by the Client, and the verifier's computation is
distributed among the Aggregators. The Client generates a "proof" of
its 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 7.2 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 5.
+================+==========================================+
 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 7.1.1) 
+++
 OUTPUT_LEN  Length of the aggregatable output 
  (Section 7.1.1) 
+++
 PROOF_LEN  Length of the proof 
+++
 VERIFIER_LEN  Length of the verifier message generated 
  by querying the input and proof 
+++
 Measurement  Type of the measurement 
+++
 Field  As defined in (Section 6.1) 
+++
Table 5: Constants and types defined by a concrete FLP.
An FLP specifies the following algorithms for generating and
verifying proofs of validity (encoding is described below in
Section 7.1.1):
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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], num_shares: Unsigned) >
Vec[Field] is the querygeneration 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.
The semantics of num_shares is discussed below.
* 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 "strong 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.
The querygeneration algorithm includes a parameter num_shares that
specifies the number of shares of the input and proof that were
generated. If these data are not secret shared, then num_shares ==
1. This parameter is useful for certain FLP constructions. For
example, the FLP in Section 7.3 is defined in terms of an arithmetic
circuit; when the circuit contains constants, it is sometimes
necessary to normalize those constants to ensure that the circuit's
output, when run on a valid input, is the same regardless of the
number of shares.
An FLP is executed by the prover and verifier as follows:
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def run_flp(Flp, 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)
Figure 12: Execution of an FLP.
The proof system is constructed so that, if input is a valid input,
then run_flp(Flp, input, 1) 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.
7.1.1. Encoding the Input
The type of measurement being aggregated is defined by the FLP.
Hence, the FLP also specifies a method of encoding raw measurements
as a vector of field elements:
* Flp.encode(measurement: Measurement) > 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 7.4). 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:
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* 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].
7.2. Construction
This section specifies Prio3, an implementation of the Vdaf interface
(Section 5). It has two generic parameters: an Flp (Section 7.1) and
a Prg (Section 6.2). The associated constants and types required by
the Vdaf interface are defined in Table 6. The methods required for
sharding, preparation, aggregation, and unsharding are described in
the remaining subsections.
+=================+===============================================+
 Parameter  Value 
+=================+===============================================+
 VERIFY_KEY_SIZE  Prg.SEED_SIZE 
+++
 ROUNDS  1 
+++
 SHARES  in [2, 255) 
+++
 Measurement  Flp.Measurement 
+++
 AggParam  None 
+++
 Prep  Tuple[Vec[Flp.Field], Optional[Bytes], Bytes] 
+++
 OutShare  Vec[Flp.Field] 
+++
 AggResult  Vec[Unsigned] 
+++
Table 6: Associated parameters for the Prio3 VDAF.
7.2.1. Sharding
Recall from Section 7.1 that the FLP syntax calls for "joint
randomness" shared by the prover (i.e., the Client) and the verifier
(i.e., the Aggregators). VDAFs have no such notion. Instead, the
Client derives the joint randomness from its 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].)
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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 7.2.5.
def measurement_to_input_shares(Prio3, measurement):
# Domain separation tag for PRG info string
dst = VERSION + b' 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,
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)
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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
)
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,
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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.
7.2.2. Preparation
This section describes the process of recovering output shares from
the input shares. The highlevel idea is that each Aggregator first
queries its 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.
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 7.2.5.
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def prep_init(Prio3,
verify_key, agg_id, _agg_param, nonce, input_share):
# Domain separation tag for PRG info string
dst = VERSION + b'prio3'
(input_share, proof_share, k_blind, k_hint) = \
Prio3.decode_leader_share(input_share) if agg_id == 0 else \
Prio3.decode_helper_share(dst, agg_id, input_share)
out_share = Prio3.Flp.truncate(input_share)
k_query_rand = Prio3.Prg.derive_seed(verify_key, 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(agg_id) + 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_prep_share(verifier_share,
k_joint_rand_share)
return (out_share, k_joint_rand, prep_msg)
def prep_next(Prio3, prep, inbound):
(out_share, k_joint_rand, prep_msg) = prep
if inbound is None:
return (prep, prep_msg)
k_joint_rand_check = Prio3.decode_prep_msg(inbound)
if k_joint_rand_check != k_joint_rand:
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raise ERR_VERIFY # joint randomness check failed
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_prep_share(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)
if not Prio3.Flp.decide(verifier):
raise ERR_VERIFY # proof verifier check failed
return Prio3.encode_prep_msg(k_joint_rand_check)
Figure 14: Preparation state for Prio3.
7.2.3. 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 Prio3.Flp.Field.encode_vec(agg_share)
Figure 15: Aggregation algorithm for Prio3.
7.2.4. 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, Prio3.Flp.Field.decode_vec(agg_share))
return list(map(lambda x: x.as_unsigned(), agg))
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Figure 16: Computation of the aggregate result for Prio3.
7.2.5. 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,
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, agg_id, encoded):
l = Prio3.Prg.SEED_SIZE
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k_input_share, encoded = encoded[:l], encoded[l:]
input_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field,
k_input_share,
dst + byte(agg_id),
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(agg_id),
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_prep_share(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_prep_share(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)
def encode_prep_msg(Prio3, k_joint_rand_check):
encoded = Bytes()
if Prio3.Flp.JOINT_RAND_LEN > 0:
encoded += k_joint_rand_check
return encoded
def decode_prep_msg(Prio3, encoded):
k_joint_rand_check = None
if Prio3.Flp.JOINT_RAND_LEN > 0:
l = Prio3.Prg.SEED_SIZE
k_joint_rand_check, encoded = encoded[:l], encoded[l:]
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if len(encoded) != 0:
raise ERR_DECODE
return k_joint_rand_check
Figure 17: Helper functions required for Prio3.
7.3. A GeneralPurpose FLP
This section describes an FLP based on the construction from in
[BBCGGI19], Section 4.2. We begin in Section 7.3.1 with an overview
of their proof system and the extensions to their proof system made
here. The construction is specified in Section 7.3.3.
OPEN ISSUE 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/cfrg/draftirtfcfrgvdaf/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.
7.3.1. 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 7),
this computation is distributed among multiple Aggregators, each of
which has only a share of the input.
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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
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 7.3.1.1; but for the moment it is only important that they
are distinct.)
The prover constructs from wire and alpha a polynomial that, when
evaluated at alpha[k1], produces the output of the kth call to the
gadget. Let us call this the "gadget polynomial". Polynomial
evaluation is linear, which means that, in the distributed setting,
the Client can disseminate additive shares of the gadget polynomial
that the Aggregators then use to compute additive shares of each
gadget output, allowing each Aggregator to compute its share of C(x)
locally.
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 7.3.3.
7.3.1.1. 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 7.2). This allows for circuit
optimizations that trade a small soundness error for a shorter proof.
For example, consider a circuit that recognizes the set of lengthN
vectors for which each element is either one or zero. A
deterministic circuit could be constructed for this language, but it
would involve a large number of multiplications that would result in
a large proof. (See the discussion in [BBCGGI19], Section 5.2 for
details). A much shorter proof can be constructed for the following
randomized circuit:
C(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 6.1.2.
7.3.2. Validity Circuits
The FLP described in Section 7.3.3 is defined in terms of a validity
circuit Valid that implements the interface described here.
A concrete Valid defines the following parameters:
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+================+=======================================+
 Parameter  Description 
+================+=======================================+
 GADGETS  A list of gadgets 
+++
 GADGET_CALLS  Number of times each gadget is called 
+++
 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 6.1.2 
+++
Table 7: Validity circuit parameters.
Each gadget G in GADGETS defines a constant DEGREE that specifies the
circuit's "arithmetic degree". This is defined to be the degree of
the polynomial that computes it. For example, the Mul circuit in
Section 7.3.1 is defined by the polynomial Mul(x) = x * x, which has
degree 2. Hence, the arithmetic degree of this gadget is 2.
Each gadget also defines a parameter ARITY that specifies the
circuit's arity (i.e., the number of input wires).
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.
7.3.3. Construction
This section specifies FlpGeneric, an implementation of the Flp
interface (Section 7.1). It has as a generic parameter a validity
circuit Valid implementing the interface defined in Section 7.3.2.
NOTE A reference implementation can be found in
https://github.com/cfrg/draftirtfcfrgvdaf/blob/main/poc/
flp_generic.sage.
The FLP parameters for FlpGeneric are defined in Table 8. The
required methods for generating the proof, generating the verifier,
and deciding validity are specified in the remaining subsections.
In the remainder, we let [n] denote the set {1, ..., n} for positive
integer n. We also define the following constants:
* Let H = len(Valid.GADGETS)
* For each i in [H]:
 Let G_i = Valid.GADGETS[i]
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 Let L_i = Valid.GADGETS[i].ARITY
 Let M_i = Valid.GADGET_CALLS[i]
 Let P_i = next_power_of_2(M_i+1)
 Let alpha_i = Field.gen()^(Field.GEN_ORDER / P_i)
+================+============================================+
 Parameter  Value 
+================+============================================+
 PROVE_RAND_LEN  Valid.prove_rand_len() (see Section 7.3.2) 
+++
 QUERY_RAND_LEN  Valid.query_rand_len() (see Section 7.3.2) 
+++
 JOINT_RAND_LEN  Valid.JOINT_RAND_LEN 
+++
 INPUT_LEN  Valid.INPUT_LEN 
+++
 OUTPUT_LEN  Valid.OUTPUT_LEN 
+++
 PROOF_LEN  Valid.proof_len() (see Section 7.3.2) 
+++
 VERIFIER_LEN  Valid.verifier_len() (see Section 7.3.2) 
+++
 Measurement  Valid.Measurement 
+++
 Field  Valid.Field 
+++
Table 8: FLP Parameters of FlpGeneric.
7.3.3.1. 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.
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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.
4. Compute the "wire polynomials". That is, for every i in [H] and
j in [L_i], construct poly_wire_i[j1], the jth wire polynomial
for the ith gadget, as follows:
* Let w = [seed_i[j1], wire_i[j1,0], ..., wire_i[j1,M_i1]].
* Let padded_w = w + Field.zeros(P_i  len(w)).
NOTE We pad w to the nearest power of 2 so that we can use FFT
for interpolating the wire polynomials. Perhaps there is some
clever math for picking wire_inp in a way that avoids having
to pad.
* Let poly_wire_i[j1] be the lowest degree polynomial for which
poly_wire_i[j1](alpha_i^k) == padded_w[k] for all k in [P_i].
5. Compute the "gadget polynomials". That is, for every i in [H]:
* Let poly_gadget_i = G_i(poly_wire_i[0], ..., poly_wire_i[L_i
1]). That is, evaluate the circuit G_i on the wire
polynomials for the ith gadget. (Arithmetic is in the ring of
polynomials over Field.)
The proof is the vector proof = seed_1 + coeff_1 + ... + seed_H +
coeff_H, where coeff_i is the vector of coefficients of poly_gadget_i
for each i in [H].
7.3.3.2. Query Generation
On input of 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 7.3.3.1.
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.
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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.)
* Let y_i = poly_gadget_i(t).
* For each j in [0,L_i) let x_i[j1] = poly_wire_i[j1](t).
The verifier message is the vector verifier = [v] + x_1 + [y_1] + ...
+ x_H + [y_H].
7.3.3.3. Decision
On input of vector verifier, the verifier decides if the input is
valid as follows:
1. Parse verifier into v, x_1, y_1, ..., x_H, y_H as defined in
Section 7.3.3.2.
2. Check for wellformedness of the gadget polynomials. For every i
in [H]:
* Let z = G_i(x_i). That is, evaluate the circuit G_i on x_i
and set z to the output.
* If z != y_i, then return False and halt.
3. Return True if v == 0 and False otherwise.
7.3.3.4. Encoding
The FLP encoding and truncation methods invoke Valid.encode and
Valid.truncate in the natural way.
7.4. Instantiations
This section specifies instantiations of Prio3 for various
measurement types. Each uses FlpGeneric as the FLP (Section 7.3) and
is determined by a validity circuit (Section 7.3.2) and a PRG
(Section 6.2). Test vectors for each can be found in Appendix "Test
Vectors".
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NOTE Reference implementations of each of these VDAFs can be found
in https://github.com/cfrg/draftirtfcfrgvdaf/blob/main/poc/
vdaf_prio3.sage.
7.4.1. 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.
This instance uses PrgAes128 (Section 6.2.1) as its PRG. Its
validity circuit, denoted Count, uses Field64 (Table 3) 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 3) 
+++
Table 9: Parameters of validity circuit
Count.
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7.4.2. 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.
This instance of Prio3 uses PrgAes128 (Section 6.2.1) as its PRG.
Its validity circuit, denoted Sum, uses Field128 (Table 4) 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 4) 
+++
Table 10: Parameters of validity circuit Sum.
7.4.3. 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 (Section 6.2.1) as its PRG.
Its validity circuit, denoted Histogram, uses Field128 (Table 4) 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 7.4.2) 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 4) 
+++
Table 11: Parameters of validity
circuit Histogram.
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8. Poplar1
TODO Update this section in light of removing the public parameter
and replacing the verification parameter.
NOTE The spec for Poplar1 is still a workinprogress. A partial
implementation can be found at https://github.com/divviup/libprio
rs/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.
8.1. 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 a pair of 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 7.
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.
8.2. Construction
The VDAF involves two rounds of communication (ROUNDS == 2) and is
defined for two Aggregators (SHARES == 2).
8.2.1. 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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8.2.1.1. 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 8.2.5.
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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(alpha, 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 shares of a, b,
and c using PRG seeds as suggested in [BBCGGI21].
8.2.2. Preparation
The aggregation parameter encodes a sequence of candidate prefixes.
When an Aggregator receives an input share from the Client, it begins
by evaluating its IDPF share on each candidate prefix, recovering a
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 = self.idpf_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
]
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self.step = 'sketch round 2'
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.
8.2.3. 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.
8.2.4. 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.
8.2.5. 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.
9. 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 key distributed to the Aggregators (verify_key, see
Section 8.2.1) 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 key
with the Clients. It is therefore RECOMMENDED that the Aggregators
generate verify_key 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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10. IANA Considerations
This document makes no request of IANA.
11. References
11.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
.
[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, .
11.2. Informative References
[AGJOP21] Addanki, S., Garbe, K., Jaffe, E., Ostrovsky, R., and A.
Polychroniadou, "Prio+: Privacy Preserving Aggregate
Statistics via Boolean Shares", 2021,
.
[BBCGGI19] Boneh, D., Boyle, E., CorriganGibbs, H., Gilboa, N., and
Y. Ishai, "ZeroKnowledge Proofs on SecretShared Data via
Fully Linear PCPs", CRYPTO 2019 , 2019,
.
[BBCGGI21] Boneh, D., Boyle, E., CorriganGibbs, H., Gilboa, N., and
Y. Ishai, "Lightweight Techniques for Private Heavy
Hitters", IEEE S&P 2021 , 2021, .
[CGB17] CorriganGibbs, H. and D. Boneh, "Prio: Private, Robust,
and Scalable Computation of Aggregate Statistics", NSDI
2017 , 2017,
.
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[DAP] Geoghegan, T., Patton, C., Rescorla, E., and C. A. Wood,
"Distributed Aggregation Protocol for Privacy Preserving
Measurement", Work in Progress, InternetDraft, draft
ietfppmdap00, 4 May 2022,
.
[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,
.
[OriginTelemetry]
"Origin Telemetry", 2020, .
Acknowledgments
Thanks to David Cook, Henry CorriganGibbs, Armando FazHernandez,
Simon Friedberger, Tim Geoghegan, Mariana Raykova, Jacob Rothstein,
and Christopher Wood for useful feedback on and contributions to the
spec.
Test Vectors
NOTE Machinereadable test vectors can be found at
https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc/
test_vec.
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Test vectors cover the generation of input shares and the conversion
of input shares into output shares. Vectors specify the verification
key, measurements, aggregation parameter, and any parameters needed
to construct the VDAF. (For example, for Prio3AesSum, the user
specifies the number of bits for representing each summand.)
Byte strings are encoded in hexadecimal To make the tests
deterministic, gen_rand() was replaced with a function that returns
the requested number of 0x01 octets.
Prio3Aes128Count
verify_key: "01010101010101010101010101010101"
upload_0:
measurement: 1
nonce: "01010101010101010101010101010101"
input_share_0: >
ae5483343eb35a52fcb36a62271a7ddb47f09d0ea2c6613807f84ac2e16814c82bca
bdc9db5080fdf4f4f778734644fc
input_share_1: >
0101010101010101010101010101010101010101010101010101010101010101
round_0:
prep_share_0: >
22ce013d3aaa7e7574ed01fe1d074cd845dfbbbc5901cabd487d4e2e228274cc
prep_share_1: >
dd31fec1c555818c51ab7ccac14ca5b00aae1c33d835c76dfa9406011a92a8e9
prep_message: >
out_share_0:
 12561809521056635474
out_share_1:
 5884934548357948848
agg_share_0: >
ae5483343eb35a52
agg_share_1: >
51ab7ccac14ca5b0
agg_result: [1]
Prio3Aes128Sum
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bits: 8
verify_key: "01010101010101010101010101010101"
upload_0:
measurement: 100
nonce: "01010101010101010101010101010101"
input_share_0: >
ae5483353eb35a3371beec8f796e9afd086cb72d05a83a3dbefbe273acb0410787b1
afba2065df5389011fd8963091e3004fa07fc91018af378da47c89abf1bd85047e40
874e2cdc5f3bc48f363b89f746770a402a777bed31b5a10c7319b3908d72de0c6512
15ba78d3cf681e07c564c0b4a9a4508df645bad8fef61e3ddf37fcb36a63271a7dbe
47f09d0ea2c661396ad006d8915d149ad88f9b1cdb86e1d13d683c359b7ac899a245
4316051e4e235dfd566f3459c336826555ed7f1baabf241e9a697d458912f3bd3778
225e832b78cd4f17e57c9b9678cf6043894aff0d0f2e06828982ac3493ae5ded0c98
86ea13d52bc0f209dc2f4e676c42b95b548a413f67b03ff18e9e6b699338400e9dff
a800563abb495364acffc17126bf0bf8ff3c5caba82333e91352e03c637d44dc4db1
59a1b19d8db4d5a3fce356f6f2fca4adc9bcf65bec8d4d962b2b40f7ea413aead099
79d4958707bf4098bb28829b79e381aaad8f69b7f2e6c159bbcb342ef7df2d9c56a9
06b171ab61b025b7c19aad8de495a8a97af2baab6d1240d30df417d1cc0fe7a90ada
ad8115924c0987fe1d16abe0c8a3c297d58a3112b818df72a10a41b34aa6b4ae370b
1340a6085c8dcd597eead5d2584fdb160f0a086a56ea6a7736666ae34d3012fdb2c2
4af3d4b2a6ae735edfe837eaab1309eaa2d8273e7dbfe0fd4166d545ce8354e1237b
48456715d12e38d02cd64c96b9daa01a2281d8a930817088c648b7c115e1550ada14
b6072ada49be3c7e3f184db2461160d29937caa97db6020a5598063f1dc05653d1d3
80b34e923bd7170eeeb811bfc3ce12c1df55cf552e986a823743fac4723a48bda6c1
8ffec653c1f182890197e9fc74631dcbd0283c4258933c03aee9404f010101010101
01010101010101010101fb0c701f7c07b9407a4a7b77d1ea017e
input_share_1: >
01010101010101010101010101010101010101010101010101010101010101010101
01010101010101010101010101012d7667bffd0f81b078896503385f6f13
round_0:
prep_share_0: >
9f7aca77f790b930b46e8cd786ff1a239aa00e7aaaa734cc2bbcb121eb7c5bc00a
ef22fd95a24cd1a0054bde0dba06062d7667bffd0f81b078896503385f6f13
prep_share_1: >
60853588086f46b34b9173287900e5de2becb1bdb8a8009d2cdc258674f08e8157
e3a202a38282c20e220e733ab61e4cfb0c701f7c07b9407a4a7b77d1ea017e
prep_message: >
d67a17a0810838f002c31e74e9b56e6d
out_share_0:
 242787699414660215404830418280405596120
out_share_1:
 97494667506278247542035355087495170189
agg_share_0: >
b6a735c5636efee29c0c1455e0c0f7d8
agg_share_1: >
4958ca3a9c91010163f3ebaa1f3f088d
agg_result: [100]
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Prio3Aes128Histogram
buckets: [1, 10, 100]
verify_key: "01010101010101010101010101010101"
upload_0:
measurement: 50
nonce: "01010101010101010101010101010101"
input_share_0: >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input_share_1: >
01010101010101010101010101010101010101010101010101010101010101010101
01010101010101010101010101018e2e553b5e45c62e0ced57ec947c8627
round_0:
prep_share_0: >
93b9dd4a3b46d8941fe7524a5cf1cd47ff8ee9c0c2e9b8230b1b940b665263b7b1
c4b370652a333ee774ec9cd379b6e78e2e553b5e45c62e0ced57ec947c8627
prep_share_1: >
6c4622b5c4b9274fe018adb5a30e32baa310ddd3e5ed87892dca520d1bed7d0299
8e38190652ba60a225e19211d77e22d5bf864de68bac19204e29697bf9504d
prep_message: >
5b91d376b8ce6a372ca37e85ef85d66a
out_share_0:
 231724485416847873323492487111470127869
 11198307274976669387765744195748249863
 180368380143069850478496598824148046307
 413446761563421317675646300023681469
out_share_1:
 108557881504090589623373286256430638340
 329084059645961793559100029172152516346
 159913986777868612468369174543752719903
 339868920159375041629190127067877084740
agg_share_0: >
ae5483353eb35a3371beec8f796e9afd086cb72d05a83a3dbefbe273acb0410787b1af
ba2065df5389011fd8963091e3004fa07fc91018af378da47c89abf1bd
agg_share_1: >
51ab7ccac14ca5b08e41137086916504f79348d2fa57c5a641041d8c534fbefa784e50
45df9a209076fee02769cf6e1fffb05f8036efe734c8725b8376540e44
agg_result: [0, 0, 1, 0]
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Authors' Addresses
Richard L. Barnes
Cisco
Email: rlb@ipv.sx
Christopher Patton
Cloudflare
Email: chrispatton+ietf@gmail.com
Phillipp Schoppmann
Google
Email: schoppmann@google.com
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