InternetDraft  VDAF  March 2022 
Patton, et al.  Expires 6 September 2022  [Page] 
 Workgroup:
 CFRG
 InternetDraft:
 draftpattoncfrgvdaf01
 Published:
 Intended Status:
 Informational
 Expires:
Verifiable Distributed Aggregation Functions
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 Task Force (IETF). Note that other groups may also distribute working documents as InternetDrafts. The list of current InternetDrafts is at https://datatracker.ietf.org/drafts/current/.¶
InternetDrafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use InternetDrafts as reference material or to cite them other than as "work in progress."¶
This InternetDraft will expire on 6 September 2022.¶
Copyright Notice
Copyright (c) 2022 IETF Trust and the persons identified as the document authors. All rights reserved.¶
This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/licenseinfo) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Revised BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Revised BSD License.¶
1. Introduction
The ubiquity of the Internet makes it an ideal platform for measurement of largescale phenomena, whether public health trends or the behavior of computer systems at scale. There is substantial overlap, however, between information that is valuable to measure and information that users consider private.¶
For example, consider an application that provides health information to users. The operator of an application might want to know which parts of their application are used most often, as a way to guide future development of the application. Specific users' patterns of usage, though, could reveal sensitive things about them, such as which users are researching a given health condition.¶
In many situations, the measurement collector is only interested in aggregate statistics, e.g., which portions of an application are most used or what fraction of people have experienced a given disease. Thus systems that provide aggregate statistics while protecting individual measurements can deliver the value of the measurements while protecting users' privacy.¶
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 wellknown distribution and adds it to their input before submitting to the aggregation server. The aggregation server then adds up the noisy inputs, and because it knows the distribution from whence the noise was sampled, it can estimate the true sum with reasonable precision.¶
Differentially private systems like RAPPOR are easy to deploy and provide a useful guarantee. On its own, however, DP falls short of the strongest privacy property one could hope for. Specifically, depending on the "amount" of noise a client adds to its input, it may be possible for a curious aggregator to make a reasonable guess of the input's true value. Indeed, the more noise the clients add, the less reliable will be the server's estimate of the output. Thus systems employing DP techniques alone must strike a delicate balance between privacy and utility.¶
The ideal goal for a privacypreserving measurement system is that of secure multiparty computation: No participant in the protocol should learn anything about an individual input beyond what it can deduce from the aggregate. In this document, we describe Verifiable Distributed Aggregation Functions (VDAFs) as a general class of protocols that achieve this goal.¶
VDAF schemes achieve their privacy goal by distributing the computation of the aggregate among a number of noncolluding aggregation servers. As long as a subset of the servers executes the protocol honestly, VDAFs guarantee that no input is ever accessible to any party besides the client that submitted it. At the same time, VDAFs are "verifiable" in the sense that malformed inputs that would otherwise garble the output of the computation can be detected and removed from the set of inputs.¶
The cost of achieving these security properties is the need for multiple servers to participate in the protocol, and the need to ensure they do not collude to undermine the VDAF's privacy guarantees. Recent implementation experience has shown that practical challenges of coordinating multiple servers can be overcome. The Prio system [CGB17] (essentially a VDAF) has been deployed in systems supporting hundreds of millions of users: The Mozilla Origin Telemetry project [OriginTelemetry] and the Exposure Notification Private Analytics collaboration among the Internet Security Research Group (ISRG), Google, Apple, and others [ENPA].¶
The VDAF abstraction laid out in Section 4 represents a class of multiparty protocols for privacypreserving measurement proposed in the literature. These protocols vary in their operational and security considerations, sometimes in subtle but consequential ways. This document therefore has two important goals:¶

Providing applications like [ID.draftgpewprivppm] with a simple, uniform interface for accessing privacypreserving measurement schemes, and documenting relevant operational and security bounds for that interface:¶
 General patterns of communications among the various actors involved in the system (clients, aggregation servers, and measurement collectors);¶
 Capabilities of a malicious coalition of servers attempting to divulge information about client inputs; and¶
 Conditions that are necessary to ensure that malicious clients cannot corrupt the computation.¶
 Providing cryptographers with design criteria that allow new constructions to be easily used by applications.¶
This document also specifies two concrete VDAF schemes, each based on a protocol from the literature.¶

The aforementioned Prio system [CGB17] allows for the privacypreserving computation of a variety aggregate statistics. The basic idea underlying Prio is fairly simple:¶
 Each client shards its input into a sequence of additive shares and distributes the shares among the aggregation servers.¶
 Next, each server adds up its shares locally, resulting in an additive share of the aggregate.¶
 Finally, the aggregation servers combine their additive shares to obtain the final aggregate.¶
The difficult part of this system is ensuring that the servers hold shares of a valid input, e.g., the input is an integer in a specific range. Thus Prio specifies a multiparty protocol for accomplishing this task.¶
In Section 6 we describe Prio3, a VDAF that follows the same overall framework as the original Prio protocol, but incorporates techniques introduced in [BBCGGI19] that result in significant performance gains.¶

More recently, Boneh et al. [BBCGGI21] described a protocol called Poplar for solving the
t
heavyhitters problem in a privacypreserving manner. Here each client holds a bitstring of lengthn
, and the goal of the aggregation servers is to compute the set of inputs that occur at leastt
times. The core primitive used in their protocol is a generalization of a Distributed Point Function (DPF) [GI14] that allows the servers to "query" their DPF shares on any bitstring of length shorter than or equal ton
. As a result of this query, each of the servers has an additive share of a bit indicating whether the string is a prefix of the client's input. The protocol also specifies a multiparty computation for verifying that at most one string among a set of candidates is a prefix of the client's input.¶In Section 7 we describe a VDAF called Poplar1 that implements this functionality.¶
The remainder of this document is organized as follows: Section 3 gives a brief overview of VDAFs; Section 4 defines the syntax for VDAFs; Section 5 defines various functionalities that are common to our constructions; Section 7 describes the Poplar1 construction; Section 6 describes the Prio3 construction; and Section 8 enumerates the security considerations for VDAFs.¶
2. Conventions and Definitions
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶
Algorithms in this document are written in Python 3. Type hints are used to define input and output types. A fatal error in a program (e.g., failure to parse one of the function parameters) is usually handled by raising an exception.¶
Some common functionalities:¶

zeros(len: Unsigned) > Bytes
returns an array of zero bytes. The length ofoutput
MUST belen
.¶ 
gen_rand(len: Unsigned) > Bytes
returns an array of random bytes. The length ofoutput
MUST belen
.¶ 
byte(int: Unsigned) > Bytes
returns the representation ofint
as a byte string. The value ofint
MUST be in[0,256)
.¶ 
xor(left: Bytes, right: Bytes) > Bytes
returns the bitwise XOR ofleft
andright
. An exception is raised if the inputs are not the same length.¶ 
I2OSP
andOS2IP
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 ton
that is also a power of two.¶
3. Overview
In a VDAFbased private measurement system, we distinguish three types of actors: Clients, Aggregators, and Collectors. The overall flow of the measurement process is as follows:¶
 Clients are configured with public parameters for a set of Aggregators.¶
 To submit an individual measurement, the Client shards the measurement into "input shares" and sends one input share to each Aggregator.¶

The Aggregators verify the validity of the input shares, producing a set of "output shares".¶
 Output shares are in onetoone correspondence with the input shares.¶
 Just as each Aggregator receives one input share of each input, at the end of the validation process, each aggregator holds one output share.¶
 In most VDAFs, Aggregators will need to exchange information among themselves as part of the validation process.¶
 Each Aggregator combines the output shares across inputs in the batch to compute the "aggregate share" for that batch, i.e., its share of the desired aggregate result.¶
 The Aggregators submit their aggregate shares to the Collector, who combines them to obtain the aggregate result over the batch.¶
Aggregators are a new class of actor relative to traditional measurement systems where clients submit measurements to a single server. They are critical for both the privacy properties of the system and the correctness of the measurements obtained. The privacy properties of the system are assured by noncollusion among Aggregators, and Aggregators are the entities that perform validation of client inputs. Thus clients trust Aggregators not to collude (typically it is required that at least one Aggregator is honest), and Collectors trust Aggregators to properly verify Client inputs.¶
Within the bounds of the noncollusion requirements of a given VDAF instance, it is possible for the same entity to play more than one role. For example, the Collector could also act as an Aggregator, effectively using the other Aggregators to augment a basic clientserver protocol.¶
In this document, we describe the computations performed by the actors in this system. It is up to applications to arrange for the required information to be delivered to the proper actors in the proper sequence. In general, we assume that all communications are confidential and mutually authenticated, with the exception that Clients submitting measurements may be anonymous.¶
4. Definition of VDAFs
A concrete VDAF specifies the algorithms involved in evaluating an aggregation function across a batch of inputs. This section specifies the interfaces of these algorithms as they would be exposed to applications.¶
The overall execution of a VDAF comprises the following steps:¶
 Setup  Generating shared parameters for the Aggregators¶
 Sharding  Computing input shares from an individual measurement¶
 Preparation  Conversion and verification of input shares to output shares compatible with the aggregation function being computed¶
 Aggregation  Combining a sequence of output shares into an aggregate share¶
 Unsharding  Combining a sequence of aggregate shares into an aggregate result¶
The setup algorithm is performed once for a given collection of Aggregators. Sharding and preparation are done once per measurement input. Aggregation and unsharding are done over a batch of inputs (more precisely, over the output shares recovered from those inputs).¶
Note that the preparation step performs two functions: Verification and conversion. Conversion translates input shares into output shares that are compatible with the aggregation function. Verification ensures that aggregating the recovered output shares will not lead to a garbled aggregate result.¶
The remainder of this section defines the VDAF interface in terms of an abstract
base class Vdaf
. This class defines the set of methods and attributes a
concrete VDAF must provide. The attributes are listed in [{vdafparam}}; the
methods are defined in the subsections that follow.¶
Parameter  Description 

ROUNDS

Number of rounds of communication during the preparation phase (Section 4.3) 
SHARES

Number of input shares into which each measurement is sharded (Section 4.2) 
Measurement

Type of each measurement 
PublicParam

Type of public parameter used by the Client during the sharding phase (Section 4.2) 
VerifyParam

Type of verification parameter used by each Aggregator during the preparation phase (Section 4.3) 
AggParam

Type of aggregation parameter 
Prep

State of each Aggregator during the preparation phase (Section 4.3) 
OutShare

Type of each output share 
AggShare

Type of each aggregate share 
AggResult

Type of the aggregate result 
4.1. Setup
Before execution of the VDAF can begin, it is necessary to distribute longlived parameters to the Client and Aggregators. The longlived parameters are generated by the following algorithm:¶

Vdaf.setup() > (PublicParam, Vec[VerifyParam])
is the randomized setup algorithm used to generate the public parameter used by the Clients and the verification parameters used by the Aggregators. The length of the latter MUST be equal toSHARES
. In general, an Aggregator's verification parameter is considered secret and MUST NOT be revealed to the Clients, Collector or other Aggregators. The parameters MAY be reused across multiple VDAF evaluations. See Section 8 for a discussion of the security implications this has depending on the threat model.¶
4.2. Sharding
In order to protect the privacy of its measurements, a VDAF Client splits its
measurements into "input shares". The measurement_to_input_shares
method is
used for this purpose.¶

Vdaf.measurement_to_input_shares(public_param: PublicParam, input: Measurement) > Vec[Bytes]
is the randomized inputdistribution algorithm run by each Client. It consumes the public parameter and input measurement and produces a sequence of input shares, one for each Aggregator. The length of the output MUST beSHARES
.¶

CP The
public_param
is intended to allow for protocols that require the Client to use a public key for sharding its measurement. When rotating theverify_param
for such a scheme, it would be necessary to also update thepublic_param
with which the clients are configured. For PPM it would be nice if we could rotate theverify_param
without also having to update the clients. We should consider dropping this at some point. See https://github.com/cjpatton/vdaf/issues/19.¶
4.3. Preparation
To recover and verify output shares, the Aggregators interact with one another
over ROUNDS
rounds. Prior to each round, each Aggregator constructs an
outbound message. Next, the sequence of outbound messages is combined into a
single message, called a "preparation message". (Each of the outbound messages
are called "preparationmessage 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.¶
To facilitate the preparation process, a concrete VDAF implements the following class methods:¶

Vdaf.prep_init(verify_param: VerifyParam, agg_param: AggParam, nonce: Bytes, input_share: Bytes) > Prep
is the deterministic preparationstate initialization algorithm run by each Aggregator to begin processing its input share into an output share. Its inputs are its verification parameter (verify_param
), the aggregation parameter (agg_param
), the nonce provided by the environment (nonce
, see Figure 6), and one of the input shares generated by the client (input_share
). Its output is the Aggregator's initial preparation state.¶ 
Vdaf.prep_next(prep: Prep, inbound: Optional[Bytes]) > Union[Tuple[Prep, Bytes], OutShare]
is the deterministic preparationstate update algorithm run by each Aggregator. It updates the Aggregator's preparation state (prep
) and returns either its next preparation state and its message share for the current round or, if this is the last round, its output share. An exception is raised if a valid output share could not be recovered. The input of this algorithm is the inbound preparation message or, if this is the first round,None
.¶ 
Vdaf.prep_shares_to_prep(agg_param: AggParam, prep_shares: Vec[Bytes]) > Bytes
is the deterministic preparationmessage preprocessing algorithm. It combines the preparationmessage shares generated by the Aggregators in the previous round into the preparation message consumed by each in the next round.¶
In effect, each Aggregator moves through a linear state machine with ROUNDS+1
states. The Aggregator enters the first state on using the initialization
algorithm, and the update algorithm advances the Aggregator to the next state.
Thus, in addition to defining the number of rounds (ROUNDS
), a VDAF instance
defines the state of the Aggregator after each round.¶

TODO Consider how to bake this "linear state machine" condition into the syntax. Given that Python 3 is used as our pseudocode, it's easier to specify the preparation state using a class.¶
The preparationstate update accomplishes two tasks: recovery of output shares from the input shares and ensuring that the recovered output shares are valid. The abstraction boundary is drawn so that an Aggregator only recovers an output share if it is deemed valid (at least, based on the Aggregator's view of the protocol). Another way to draw this boundary would be to have the Aggregators recover output shares first, then verify that they are valid. However, this would allow the possibility of misusing the API by, say, aggregating an invalid output share. Moreover, in protocols like Prio+ [AGJOP21] based on oblivious transfer, it is necessary for the Aggregators to interact in order to recover aggregatable output shares at all.¶
Note that it is possible for a VDAF to specify ROUNDS == 0
, in which case each
Aggregator runs the preparationstate update algorithm once and immediately
recovers its output share without interacting with the other Aggregators.
However, most, if not all, constructions will require some amount of interaction
in order to ensure validity of the output shares (while also maintaining
privacy).¶

OPEN ISSUE Depending on what we do for issue#20, we may end up needing to revise the above paragraph.¶
4.4. Aggregation
Once an Aggregator holds output shares for a batch of measurements (where batches are defined by the application), it combines them into a share of the desired aggregate result. This algorithm is performed locally at each Aggregator, without communication with the other Aggregators.¶

Vdaf.out_shares_to_agg_share(agg_param: AggParam, output_shares: Vec[OutShare]) > agg_share: AggShare
is the deterministic aggregation algorithm. It is run by each Aggregator over the output shares it has computed over a batch of measurement inputs.¶
For simplicity, we have written this algorithm and the unsharding algorithm below in "oneshot" form, where all shares for a batch are provided at the same time. Some VDAFs may also support a "streaming" form, where shares are processed one at a time.¶
4.5. Unsharding
After the Aggregators have aggregated a sufficient number of output shares, each sends its aggregate share to the Collector, who runs the following algorithm to recover the following output:¶

Vdaf.agg_shares_to_result(agg_param: AggParam, agg_shares: Vec[AggShare]) > AggResult
is run by the Collector in order to compute the aggregate result from the Aggregators' shares. The length ofagg_shares
MUST beSHARES
. This algorithm is deterministic.¶

QUESTION Maybe the aggregation algorithms should be randomized in order to allow the Aggregators (or the Collector) to add noise for differential privacy. (See the security considerations of [ID.draftgpewprivppm].) Or is this outofscope of this document?¶
4.6. Execution of a VDAF
Executing a VDAF involves the concurrent evaluation of the VDAF on individual inputs and aggregation of the recovered output shares. This is captured by the following example algorithm:¶
The inputs to this algorithm are the aggregation parameter agg_param
, a list
of nonces nonces
, and a batch of Client inputs input_batch
. The aggregation
parameter is chosen by the Aggregators prior to executing the VDAF. This
document does not specify how the nonces are chosen, but security requires that
the nonces be unique for each measurement. See Section 8 for details.¶
Another important question this document leaves out of scope is how a VDAF is to
be executed by Aggregators distributed over a real network. Algorithm run_vdaf
prescribes the protocol's execution in a "benign" environment in which there is
no adversary and messages are passed among the protocol participants over secure
pointtopoint channels. In reality, these channels need to be instantiated by
some "wrapper protocol", such as [ID.draftgpewprivppm] that implements
suitable cryptographic functionalities. Moreover, some fraction of the
Aggregators (or Clients) may be malicious and diverge from their prescribed
behaviors. Section 8 describes the execution of the VDAF in various
adversarial environments and what properties the wrapper protocol needs to
provide in each.¶
5. Preliminaries
This section describes the primitives that are common to the VDAFs specified in this document.¶
5.1. Finite Fields
Both Prio3 and Poplar1 use finite fields of prime order. Finite field
elements are represented by a class Field
with the following associated
parameters:¶

MODULUS: Unsigned
is the prime modulus that defines the field.¶ 
ENCODED_SIZE: Unsigned
is the number of bytes used to encode a field element as a byte string.¶
A concrete Field
also implements the following class methods:¶

Field.zeros(length: Unsigned) > output: Vec[Field]
returns a vector of zeros. The length ofoutput
MUST belength
.¶ 
Field.rand_vec(length: Unsigned) > output: Vec[Field]
returns a vector of random field elements. The length ofoutput
MUST belength
.¶
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 elementelem
.¶
Likewise, each concrete Field
implements a constructor for converting an
unsigned integer into a field element:¶

Field(integer: Unsigned)
returnsinteger
represented as a field element. Ifinteger >= Field.MODULUS
, theninteger
is first reduced moduloField.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.¶
5.1.1. Auxiliary Functions
The following auxiliary functions on vectors of field elements are used in the remainder of this document. Note that an exception is raised by each function if the operands are not the same length.¶
5.1.2. FFTFriendly Fields
Some VDAFs require fields that are suitable for efficient computation of the discrete Fourier transform. (One example is prio3 (Section 6) when instantiated with the generic FLP of Section 6.3.3.) Specifically, a field is said to be "FFTfriendly" if, in addition to satisfying the interface described in Section 5.1, it implements the following method:¶

Field.gen() > Field
returns the generator of a large subgroup of the multiplicative group.¶
FFTfriendly fields also define the following parameter:¶

GEN_ORDER: Unsigned
is the order of a multiplicative subgroup generated byField.gen()
. This value MUST be a power of 2.¶
5.1.3. Parameters
The tables below define finite fields used in the remainder of this document.¶
Parameter  Value 

MODULUS  2^32 * 4294967295 + 1 
ENCODED_SIZE  8 
Generator  7^4294967295 
GEN_ORDER  2^32 
Parameter  Value 

MODULUS  2^66 * 4611686018427387897 + 1 
ENCODED_SIZE  16 
Generator  7^4611686018427387897 
GEN_ORDER  2^66 
5.2. Pseudorandom Generators
A pseudorandom generator (PRG) is used to expand a short, (pseudo)random seed into a long string of pseudorandom bits. A PRG suitable for this document implements the interface specified in this section. Concrete constructions are described in Section 5.2.1.¶
PRGs are defined by a class Prg
with the following associated parameter:¶

SEED_SIZE: Unsigned
is the size (in bytes) of a seed.¶
A concrete Prg
implements the following class method:¶

Prg(seed: Bytes, info: Bytes)
constructs an instance ofPrg
from the given seed and info string. The seed MUST be of lengthSEED_SIZE
and MUST be generated securely (i.e., it is either the output ofgen_rand
or a previous invocation of the PRG). The info string is used for domain separation.¶ 
prg.next(length: Unsigned)
returns the nextlength
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).¶
5.2.1. Constructions
5.2.1.1. PrgAes128

OPEN ISSUE Phillipp points out that a fixedkey mode of AES may be more performant (https://eprint.iacr.org/2019/074.pdf). See https://github.com/cjpatton/vdaf/issues/32 for details.¶
Our first construction, PrgAes128
, converts a blockcipher, namely AES128,
into a PRG. Seed expansion involves two steps. In the first step, CMAC
[RFC4493] is applied to the seed and info string to get a fresh key. In the
second step, the fresh key is used in CTRmode to produce a key stream for
generating the output. A fixed initialization vector (IV) is used.¶
6. Prio3

NOTE This construction has not undergone significant security analysis.¶
This section describes "Prio3", a VDAF for Prio [CGB17]. Prio is suitable for a wide variety of aggregation functions, including (but not limited to) sum, mean, standard deviation, estimation of quantiles (e.g., median), and linear regression. In fact, the scheme described in this section is compatible with any aggregation function that has the following structure:¶
 Each measurement is encoded as a vector over some finite field.¶
 Input validity is determined by an arithmetic circuit evaluated over the encoded input. (An "arithmetic circuit" is a function comprised of arithmetic operations in the field.) The circuit's output is a single field element: if zero, then the input is said to be "valid"; otherwise, if the output is nonzero, then the input is said to "invalid".¶
 The aggregate result is obtained by summing up the encoded input vectors and computing some function of the sum.¶
At a high level, Prio3 distributes this computation as follows. Each Client first shards its measurement by first encoding it, then splitting the vector into secret shares and sending a share to each Aggregator. Next, in the preparation phase, the Aggregators carry out a multiparty computation to determine if their shares correspond to a valid input (as determined by the arithmetic circuit). This computation involves a "proof" of validity generated by the Client. Next, each Aggregator sums up its input shares locally. Finally, the Collector sums up the aggregate shares and computes the aggregate result.¶
This VDAF does not have an aggregation parameter. Instead, the output share is derived from the input share by applying a fixed map. See Section 7 for an example of a VDAF that makes meaningful use of the aggregation parameter.¶
As the name implies, "Prio3" is a descendant of the original Prio construction. A second iteration was deployed in the [ENPA] system, and like the VDAF described here, the ENPA system was built from techniques introduced in [BBCGGI19] that significantly improve communication cost. That system was specialized for a particular aggregation function; the goal of Prio3 is to provide the same level of generality as the original construction.¶
The core component of Prio3 is a "Fully Linear Proof (FLP)" system. Introduced by [BBCGGI19], the FLP encapsulates the functionality required for encoding and validating inputs. Prio3 can be thought of as a transformation of a particular class of FLPs into a VDAF.¶
The remainder of this section is structured as follows. The syntax for FLPs is described in Section 6.1. The generic transformation of an FLP into Prio3 is specified in Section 6.2. Next, a concrete FLP suitable for any validity circuit is specified in Section 6.3. Finally, instantiations of Prio3 for various types of measurements are specified in Section 6.4. Test vectors can be found in Appendix "Test Vectors".¶
6.1. Fully Linear Proof (FLP) Systems
Conceptually, an FLP is a twoparty protocol executed by a prover and a verifier. In actual use, however, the prover's computation is carried out by the Client, and the verifier's computation is distributed among the Aggregators. The Client generates a "proof" of its input's validity and distributes shares of the proof to the Aggregators. Each Aggregator then performs some a computation on its input share and proof share locally and sends the result to the other Aggregators. Combining the exchanged messages allows each Aggregator to decide if it holds a share of a valid input. (See Section 6.2 for details.)¶
As usual, we will describe the interface implemented by a concrete FLP in terms
of an abstract base class Flp
that specifies the set of methods and parameters
a concrete FLP must provide.¶
The parameters provided by a concrete FLP are listed in Table 4.¶
Parameter  Description 

PROVE_RAND_LEN

Length of the prover randomness, the number of random field elements consumed by the prover when generating a proof 
QUERY_RAND_LEN

Length of the query randomness, the number of random field elements consumed by the verifier 
JOINT_RAND_LEN

Length of the joint randomness, the number of random field elements consumed by both the prover and verifier 
INPUT_LEN

Length of the encoded measurement (Section 6.1.1) 
OUTPUT_LEN

Length of the aggregatable output (Section 6.1.1) 
PROOF_LEN

Length of the proof 
VERIFIER_LEN

Length of the verifier message generated by querying the input and proof 
Measurement

Type of the measurement 
Field

As defined in (Section 5.1) 
An FLP specifies the following algorithms for generating and verifying proofs of validity (encoding is described below in Section 6.1.1):¶

Flp.prove(input: Vec[Field], prove_rand: Vec[Field], joint_rand: Vec[Field]) > Vec[Field]
is the deterministic proofgeneration algorithm run by the prover. Its inputs are the encoded input, the "prover randomness"prove_rand
, and the "joint randomness"joint_rand
. The proof randomness is used only by the prover, but the joint randomness is shared by both the prover and verifier.¶ 
Flp.query(input: Vec[Field], proof: Vec[Field], query_rand: Vec[Field], joint_rand: Vec[Field]) > Vec[Field]
is the 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 randomnessquery_rand
and the joint randomnessjoint_rand
. The former is used only by the verifier.¶ 
Flp.decide(verifier: Vec[Field]) > Bool
is the deterministic decision algorithm run by the verifier. It takes as input the verifier message and outputs a boolean indicating if the input from which it was generated is valid.¶
Our application requires that the FLP is "fully linear" in the sense defined in [BBCGGI19] As a practical matter, what this property implies is that, when run on a share of the input and proof, the querygeneration algorithm outputs a share of the verifier message. Furthermore, the "zeroknowledge" property of the FLP system ensures that the verifier message reveals nothing about the input's validity. Therefore, to decide if an input is valid, the Aggregators will run the querygeneration algorithm locally, exchange verifier shares, combine them to recover the verifier message, and run the decision algorithm.¶
An FLP is executed by the prover and verifier as follows:¶
The proof system is constructed so that, if input
is a valid input, then
run_flp(Flp,. input)
always returns True
. On the other hand, if input
is
invalid, then as long as joint_rand
and query_rand
are generated uniform
randomly, the output is False
with overwhelming probability.¶
We remark that [BBCGGI19] defines a much larger class of fully linear proof systems than we consider here. In particular, what is called an "FLP" here is called a 1.5round, publiccoin, interactive oracle proof system in their paper.¶
6.1.1. Encoding the Input
The type of measurement being aggregated is defined by the FLP. Hence, the FLP also specifies a method of encoding raw measurements as a vector of field elements:¶

Flp.encode(measurement: Measurement) > Vec[Field]
encodes a raw measurement as a vector of field elements. The return value MUST be of lengthINPUT_LEN
.¶
For some FLPs, the encoded input also includes redundant field elements that are
useful for checking the proof, but which are not needed after the proof has been
checked. An example is the "integer sum" data type from [CGB17] in which an
integer in range [0, 2^k)
is encoded as a vector of k
field elements (this
type is also defined in Section 6.4). After consuming this vector,
all that is needed is the integer it represents. Thus the FLP defines an
algorithm for truncating the input to the length of the aggregated output:¶

Flp.truncate(input: Vec[Field]) > Vec[Field]
maps an encoded input to an aggregatable output. The length of the input MUST beINPUT_LEN
and the length of the output MUST beOUTPUT_LEN
.¶
We remark that, taken together, these two functionalities correspond roughly to the notion of "Affineaggregatable encodings (AFEs)" from [CGB17].¶
6.2. Construction
This section specifies Prio3
, an implementation of the Vdaf
interface
(Section 4). It has two generic parameters: an Flp
(Section 6.1) and a Prg
(Section 5.2). The associated constants and types required by the Vdaf
interface
are defined in Table 5. The methods required for sharding, preparation,
aggregation, and unsharding are described in the remaining subsections.¶
Parameter  Value 

ROUNDS

1

SHARES

in [2, 255)

Measurement

Flp.Measurement

PublicParam

None

VerifyParam

Tuple[Unsigned, Bytes]

AggParam

None

Prep

Tuple[Vec[Flp.Field], Optional[Bytes], Bytes]

OutShare

Vec[Flp.Field]

AggShare

Vec[Flp.Field]

AggResult

Vec[Unsigned]

6.2.1. Setup
The setup algorithm generates a symmetric key shared by all of the Aggregators.
The key is used to derive query randomness for the FLP querygeneration
algorithm run by the Aggregators during preparation. An Aggregator's
verification parameter also includes its "ID", a unique integer in [0,
SHARES)
.¶
6.2.3. Preparation
This section describes the process of recovering output shares from the input shares. The highlevel idea is that each Aggregator first queries its input and proof share locally, then exchanges its verifier share with the other Aggregators. The verifier shares are then combined into the verifier message, which is used to decide whether to accept.¶
In addition, the Aggregators must ensure that they have all used the same joint randomness for the querygeneration algorithm. The joint randomness is generated by a PRG seed. Each Aggregator derives an XOR secret share of this seed from its input share and the "blind" generated by the client. Thus, before running the querygeneration algorithm, it must first gather the XOR secret shares derived by the other Aggregators.¶
In order to avoid extra round of communication, the Client sends each Aggregator a "hint" equal to the XOR of the other Aggregators' shares of the joint randomness seed. This leaves open the possibility that the Client cheated by, say, forcing the Aggregators to use joint randomness that biases the proof check procedure some way in its favor. To mitigate this, the Aggregators also check that they have all computed the same joint randomness seed before accepting their output shares. To do so, they exchange their XOR shares of the PRG seed along with their verifier shares.¶

NOTE This optimization somewhat diverges from Section 6.2.3 of [BBCGGI19]. Security analysis is needed.¶
The algorithms required for preparation are defined as follows. These algorithms make use of encoding and decoding methods defined in Section 6.2.6.¶
6.2.4. Aggregation
Aggregating a set of output shares is simply a matter of adding up the vectors elementwise.¶
6.2.5. Unsharding
To unshard a set of aggregate shares, the Collector first adds up the vectors elementwise. It then converts each element of the vector into an integer.¶
6.3. A GeneralPurpose FLP
This section describes an FLP based on the construction from in [BBCGGI19], Section 4.2. We begin in Section 6.3.1 with an overview of their proof system and the extensions to their proof system made here. The construction is specified in Section 6.3.3.¶

OPEN ISSUE We're not yet sure if specifying this generalpurpose FLP is desirable. It might be preferable to specify specialized FLPs for each data type that we want to standardize, for two reasons. First, clear and concise specifications are likely easier to write for specialized FLPs rather than the general one. Second, we may end up tailoring each FLP to the measurement type in a way that improves performance, but breaks compatibility with the generalpurpose FLP.¶
In any case, we can't make this decision until we know which data types to standardize, so for now, we'll stick with the generalpurpose construction. The reference implementation can be found at https://github.com/cjpatton/vdaf/tree/main/poc.¶

OPEN ISSUE Chris Wood points out that the this section reads more like a paper than a standard. Eventually we'll want to work this into something that is readily consumable by the CFRG.¶
6.3.1. Overview
In the proof system of [BBCGGI19], validity is defined via an arithmetic circuit evaluated over the input: If the circuit output is zero, then the input is deemed valid; otherwise, if the circuit output is nonzero, then the input is deemed invalid. Thus the goal of the proof system is merely to allow the verifier to evaluate the validity circuit over the input. For our application (Section 6), this computation is distributed among multiple Aggregators, each of which has only a share of the input.¶
Suppose for a moment that the validity circuit C
is affine, meaning its only
operations are addition and multiplicationbyconstant. In particular, suppose
the circuit does not contain a multiplication gate whose operands are both
nonconstant. Then to decide if an input x
is valid, each Aggregator could
evaluate C
on its share of x
locally, broadcast the output share to its
peers, then combine the output shares locally to recover C(x)
. This is true
because for any SHARES
way 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 nonaffine operation,
x[0]^2
. In order to apply [BBCGGI19], Theorem 4.3, the circuit needs to be
rewritten in terms of a gadget that subsumes this nonaffine operation. For
example, the gadget might be multiplication:¶
Mul(left, right) = left * right¶
The validity circuit can then be rewritten in terms of Mul
like so:¶
C(x[0]) = Mul(x[0], x[0])  x[0]¶
The proof system of [BBCGGI19] allows the verifier to evaluate each instance
of the gadget (i.e., Mul(x[0], x[0])
in our example) using a linear function
of the 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 j
th wire of the k
th
call to the gadget during the evaluation of C(x)
. Suppose there are M
such
calls and fix distinct field elements alpha[0], ..., alpha[M1]
. (We will
require these points to have a special property, as we'll discuss in
Section 6.3.1.1; but for the moment it is only important
that they are distinct.)¶
The prover constructs from wire
and alpha
a polynomial that, when evaluated
at alpha[k1]
, produces the output of the k
th 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 wellformed. This test, and the procedure for constructing
the gadget polynomial, are described in detail in Section 6.3.3.¶
6.3.1.1. Extensions
The FLP described in the next section extends the proof system [BBCGGI19], Section 4.2 in three ways.¶
First, the validity circuit in our construction includes an additional, random
input (this is the "joint randomness" derived from the input shares in Prio3;
see Section 6.2). This allows for circuit optimizations that trade a
small soundness error for a shorter proof. For example, consider a circuit that
recognizes the set of lengthN
vectors for which each element is either one or
zero. A deterministic circuit could be constructed for this language, but it
would involve a large number of multiplications that would result in a large
proof. (See the discussion in [BBCGGI19], Section 5.2 for details). A much
shorter proof can be constructed for the following randomized circuit:¶
C(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N1])¶
(Note that this is a special case of [BBCGGI19], Theorem 5.2.) Here inp
is
the lengthN
input and r
is a random field element. The gadget circuit
Range2
is the "rangecheck" polynomial described above, i.e., Range2(x) = x^2 
x
. The idea is that, if inp
is valid (i.e., each inp[j]
is in [0,2)
),
then the circuit will evaluate to 0 regardless of the value of r
; but if
inp[j]
is not in [0,2)
for some j
, the output will be nonzero with high
probability.¶
The second extension implemented by our FLP allows the validity circuit to
contain multiple gadget types. (This generalization was suggested in
[BBCGGI19], Remark 4.5.) For example, the following circuit is allowed, where
Mul
and Range2
are the gadgets defined above (the input has length N+1
):¶
C(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N1]) + \ 2^0 * inp[0] + ... + 2^(N1) * inp[N1]  \ Mul(inp[N], inp[N])¶
Finally, [BBCGGI19], Theorem 4.3 makes no restrictions on the choice of the
fixed points alpha[0], ..., alpha[M1]
, other than to require that the points
are distinct. In this document, the fixed points are chosen so that the gadget
polynomial can be constructed efficiently using the CooleyTukey FFT ("Fast
Fourier Transform") algorithm. Note that this requires the field to be
"FFTfriendly" as defined in Section 5.1.2.¶
6.3.2. Validity Circuits
The FLP described in Section 6.3.3 is defined in terms of a
validity circuit Valid
that implements the interface described here.¶
A concrete Valid
defines the following parameters:¶
Parameter  Description 

GADGETS

A list of gadgets 
GADGET_CALLS

Number of times each gadget is called 
INPUT_LEN

Length of the input 
OUTPUT_LEN

Length of the aggregatable output 
JOINT_RAND_LEN

Length of the random input 
Measurement

The type of measurement 
Field

An FFTfriendly finite field as defined in Section 5.1.2 
Each gadget G
in GADGETS
defines a constant DEGREE
that specifies the
circuit's "arithmetic degree". This is defined to be the degree of the
polynomial that computes it. For example, the Mul
circuit in
Section 6.3.1 is defined by the polynomial Mul(x) = x * x
, which
has degree 2
. Hence, the arithmetic degree of this gadget is 2
.¶
Each gadget also defines a parameter ARITY
that specifies the circuit's arity
(i.e., the number of input wires).¶
A concrete Valid
provides the following methods for encoding a measurement as
an input vector and truncating an input vector to the length of an aggregatable
output:¶

Valid.encode(measurement: Measurement) > Vec[Field]
returns a vector of lengthINPUT_LEN
representing a measurement.¶ 
Valid.truncate(input: Vec[Field]) > Vec[Field]
returns a vector of lengthOUTPUT_LEN
representing an aggregatable output.¶
Finally, the following class methods are derived for each concrete Valid
:¶
6.3.3. Construction
This section specifies FlpGeneric
, an implementation of the Flp
interface
(Section 6.1). It has as a generic parameter a validity circuit Valid
implementing
the interface defined in Section 6.3.2.¶

NOTE A reference implementation can be found in https://github.com/cjpatton/vdaf/blob/main/poc/flp_generic.sage.¶
The FLP parameters for FlpGeneric
are defined in Table 7. The
required methods for generating the proof, generating the verifier, and deciding
validity are specified in the remaining subsections.¶
In the remainder, we let [n]
denote the set {1, ..., n}
for positive integer
n
. We also define the following constants:¶
Parameter  Value 

PROVE_RAND_LEN

Valid.prove_rand_len() (see Section 6.3.2) 
QUERY_RAND_LEN

Valid.query_rand_len() (see Section 6.3.2) 
JOINT_RAND_LEN

Valid.JOINT_RAND_LEN

INPUT_LEN

Valid.INPUT_LEN

OUTPUT_LEN

Valid.OUTPUT_LEN

PROOF_LEN

Valid.proof_len() (see Section 6.3.2) 
VERIFIER_LEN

Valid.verifier_len() (see Section 6.3.2) 
Measurement

Valid.Measurement

Field

Valid.Field

6.3.3.1. Proof Generation
On input inp
, prove_rand
, and joint_rand
, the proof is computed as
follows:¶
 For each
i
in[H]
create an empty tablewire_i
.¶  Partition the prover randomness
prove_rand
into subvectorsseed_1, ..., seed_H
wherelen(seed_i) == L_i
for alli
in[H]
. Let us call these the "wire seeds" of each gadget.¶  Evaluate
Valid
on input ofinp
andjoint_rand
, recording the inputs of each gadget in the corresponding table. Specifically, for everyi
in[H]
, setwire_i[j1,k1]
to the value on thej
th wire into thek
th call to gadgetG_i
.¶ 
Compute the "wire polynomials". That is, for every
i
in[H]
andj
in[L_i]
, constructpoly_wire_i[j1]
, thej
th wire polynomial for thei
th gadget, as follows:¶ 
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_i1])
. That is, evaluate the circuitG_i
on the wire polynomials for thei
th gadget. (Arithmetic is in the ring of polynomials overField
.)¶
 Let
The proof is the vector proof = seed_1 + coeff_1 + ... + seed_H + coeff_H
,
where coeff_i
is the vector of coefficients of poly_gadget_i
for each i
in
[H]
.¶
6.3.3.2. Query Generation
On input of inp
, proof
, query_rand
, and joint_rand
, the verifier message
is generated as follows:¶
 For every
i
in[H]
create an empty tablewire_i
.¶  Partition
proof
into the subvectorsseed_1
,coeff_1
, ...,seed_H
,coeff_H
defined in Section 6.3.3.1.¶  Evaluate
Valid
on input ofinp
andjoint_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 thek
th call toG_i
by evaluatingpoly_gadget_i(alpha_i^k)
. Letv
denote the output of the circuit evaluation.¶  Compute the wire polynomials just as in the prover's step (4.).¶

Compute the tests for wellformedness of the gadget polynomials. That is, for every
i
in[H]
:¶
The verifier message is the vector verifier = [v] + x_1 + [y_1] + ... + x_H +
[y_H]
.¶
6.4. Instantiations
This section specifies instantiations of Prio3 for various measurement types.
Each uses FlpGeneric
as the FLP (Section 6.3) and is determined by a
validity circuit (Section 6.3.2) and a PRG (Section 5.2). Test vectors for
each can be found in Appendix "Test Vectors".¶

NOTE Reference implementations of each of these VDAFs can be found in https://github.com/cjpatton/vdaf/blob/main/poc/vdaf_prio3.sage.¶
6.4.1. Prio3Aes128Count
Our first instance of Prio3 is for a simple counter: Each measurement is either one or zero and the aggregate result is the sum of the measurements.¶
This instance uses PrgAes128
(see Section 5.2.1) as its PRG. Its
validity circuit, denoted Count
, uses Field64
(Table 2) as its finite
field. Its gadget, denoted Mul
, is the degree2, arity2 gadget defined as¶
def Mul(x, y): return x * y¶
The validity circuit is defined as¶
def Count(inp: Vec[Field64]): return Mul(inp[0], inp[0])  inp[0]¶
The measurement is encoded as a singleton vector in the natural way. The parameters for this circuit are summarized below.¶
Parameter  Value 

GADGETS

[Mul]

GADGET_CALLS

[1]

INPUT_LEN

1

OUTPUT_LEN

1

JOINT_RAND_LEN

0

Measurement

Unsigned , in range [0,2)

Field

Field64 (Table 2) 
6.4.2. Prio3Aes128Sum
The next instance of Prio3 supports summing of integers in a predetermined
range. Each measurement is an integer in range [0, 2^bits)
, where bits
is an
associated parameter.¶
This instance of prio3 uses PrgAes128
(see Section 5.2.1) as its PRG.
Its validity circuit, denoted Sum
, uses Field128
(Table 3) as its
finite field. The measurement is encoded as a lengthbits
vector of field
elements, where the l
th element of the vector represents the l
th 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¶
Parameter  Value 

GADGETS

[Range2]

GADGET_CALLS

[bits]

INPUT_LEN

bits

OUTPUT_LEN

1

JOINT_RAND_LEN

1

Measurement

Unsigned , in range [0, 2^bits)

Field

Field128 (Table 3) 
6.4.3. Prio3Aes128Histogram
This instance of Prio3 allows for estimating the distribution of the measurements by computing a simple histogram. Each measurement is an arbitrary integer and the aggregate result counts the number of measurements that fall in a set of fixed buckets.¶
This instance of prio3 uses PrgAes128
(see Section 5.2.1) as its PRG.
Its validity circuit, denoted Histogram
, uses Field128
(Table 3) as its
finite field. The measurement is encoded as a onehot vector representing the
bucket into which the measurement falls (let bucket
denote a sequence of
monotonically increasing integers):¶
def encode(Histogram, measurement: Integer): boundaries = buckets + [Infinity] encoded = [Field128(0) for _ in range(len(boundaries))] for i in range(len(boundaries)): if measurement <= boundaries[i]: encoded[i] = Field128(1) return encoded def truncate(Histogram, inp: Vec[Field128]): return inp¶
The validity circuit uses Range2
(see Section 6.4.2) as its single gadget. It
checks for onehotness in two steps, as follows:¶
def Histogram(inp: Vec[Field128], joint_rand: Vec[Field128], num_shares: Unsigned): # Check that each bucket is one or zero. range_check = Field128(0) r = joint_rand[0] for x in inp: range_check += r * Range2(x) r *= joint_rand[0] # Check that the buckets sum to 1. sum_check = Field128(1) * Field128(num_shares).inv() for b in inp: sum_check += b out = joint_rand[1] * range_check + \ joint_rand[1]^2 * sum_check return out¶
Note that this circuit depends on the number of shares into which the input is sharded. This is provided to the FLP by Prio3.¶
Parameter  Value 

GADGETS

[Range2]

GADGET_CALLS

[buckets + 1]

INPUT_LEN

buckets + 1

OUTPUT_LEN

buckets + 1

JOINT_RAND_LEN

2

Measurement

Integer

Field

Field128 (Table 3) 
7. Poplar1

NOTE The spec for Poplar1 is still a workinprogress. A partial implementation can be found at https://github.com/abetterinternet/libpriors/blob/main/src/vdaf/poplar1.rs. The verification logic is nearly complete, however as of this draft the code is missing the IDPF. An implementation of the IDPF can be found at https://github.com/google/distributed_point_functions/.¶
This section specifies Poplar1, a VDAF for the following task. Each Client holds
a BITS
bit string and the Aggregators hold a set of l
bit 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.¶
 Each Clients runs the inputdistribution algorithm on its
n
bit string and sends an input share to each Aggregator.¶  The Aggregators agree on an initial set of candidate prefixes, say
0
and1
.¶  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.¶
 The Aggregators send their aggregate shares to the Collector, who combines them to recover the counts of each candidate prefix.¶
 Let
H
denote the set of prefixes that occurred at leastt
times. If the prefixes all have lengthBITS
, thenH
is the set oft
heavyhitters. Otherwise compute the next set of candidate prefixes as follows. For eachp
inH
, add addp  0
andp  1
to the set. Repeat step 3 with the new set of candidate prefixes.¶
Poplar1 is constructed from an "Incremental Distributed Point Function (IDPF)", a primitive described by [BBCGGI21] that generalizes the notion of a Distributed Point Function (DPF) [GI14]. Briefly, a DPF is used to distribute the computation of a "point function", a function that evaluates to zero on every input except at a programmable "point". The computation is distributed in such a way that no one party knows either the point or what it evaluates to.¶
An IDPF generalizes this "point" to a path on a full binary tree from the root to one of the leaves. It is evaluated on an "index" representing a unique node of the tree. If the node is on the path, then function evaluates to to a nonzero value; otherwise it evaluates to zero. This structure allows an IDPF to provide the functionality required for the above protocol, while at the same time ensuring the same degree of privacy as a DPF.¶
Our VDAF composes an IDPF with the "secure sketching" protocol of [BBCGGI21]. This protocol ensures that evaluating a set of input shares on a unique set of candidate prefixes results in shares of a "onehot" vector, i.e., a vector that is zero everywhere except for one element, which is equal to one.¶
7.1. Incremental Distributed Point Functions (IDPFs)
An IDPF is defined over a domain of size 2^BITS
, where BITS
is constant
defined by the IDPF. The Client specifies an index alpha
and values beta
,
one for each "level" 1 <= l <= BITS
. The key generation generates two IDPF
keys, one for each Aggregator. When evaluated at index 0 <= x < 2^l
, each
IDPF share returns an additive share of beta[l]
if x
is the l
bit prefix
of alpha
and shares of zero otherwise.¶

CP What does it mean for
x
to be thel
bit prefix ofalpha
? 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 heightBITS
(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 indexalpha
and the valuesbeta
. The value ofalpha
MUST be in range[0, 2^BITS)
.¶ 
IDPFKey.eval(l: Unsigned, x: Unsigned) > value: Value[l])
is deterministic, stateless keyevaluation algorithm run by each Aggregator. It returns the value corresponding to indexx
. The value ofl
MUST be in[1, BITS]
and the value ofx
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 level1 <= l <= BITS
. Each defines the same methods and associated constants asField
in Section 6.¶
Note that IDPF construction of [BBCGGI21] uses one field for the inner nodes of the tree and a different, larger field for the leaf nodes. See [BBCGGI21], Section 4.3.¶
Finally, an implementation note. The interface for IDPFs specified here is stateless, in the sense that there is no state carried between IDPF evaluations. This is to align the IDPF syntax with the VDAF abstraction boundary, which does not include shared state across across VDAF evaluations. In practice, of course, it will often be beneficial to expose a stateful API for IDPFs and carry the state across evaluations.¶
7.2. Construction
The VDAF involves two rounds of communication (ROUNDS == 2
) and is defined for
two Aggregators (SHARES == 2
).¶
7.2.1. Setup
The verification parameter is a symmetric key shared by both Aggregators. This VDAF has no public parameter.¶
7.2.1.1. Client
The client's input is an IDPF index, denoted alpha
. The values are pairs of
field elements (1, k)
where each k
is chosen at random. This random value is
used as part of the secure sketching protocol of [BBCGGI21]. After evaluating
their IDPF key shares on the set of candidate prefixes, the sketching protocol
is used by the Aggregators to verify that they hold shares of a onehot vector.
In addition, for each level of the tree, the prover generates random elements
a
, b
, and c
and computes¶
A = 2*a + k B = a*a + b  k*a + c¶
and sends additive shares of a
, b
, c
, A
and B
to the Aggregators.
Putting everything together, the inputdistribution algorithm is defined as
follows. Function encode_input_share
is defined in Section 7.2.5.¶
7.2.2. Preparation
The aggregation parameter encodes a sequence of candidate prefixes. When an
Aggregator receives an input share from the Client, it begins by evaluating its
IDPF share on each candidate prefix, recovering a pair of vectors of field
elements data_share
and auth_share
, The Aggregators use auth_share
and the
correlation shares provided by the Client to verify that their data_share
vectors are additive shares of a onehot vector.¶

CP Consider adding aggregation parameter as input to
k_verify_rand
derivation.¶
7.2.5. Helper Functions

TODO Specify the following functionalities:¶

encode_input_share
is used to encode an input share, consisting of an IDPF key share and correlation shares.¶ 
decode_input_share
is used to decode an input share.¶ 
decode_indexes(encoded: Bytes) > (l: Unsigned, indexes: Vec[Unsigned])
decodes a sequence of indexes, i.e., candidate indexes for IDPF evaluation. The value ofl
MUST be in range[1, BITS]
andindexes[i]
MUST be in range[2^(l1), 2^l)
for alli
. An error is raised ifencoded
cannot be decoded.¶
8. Security Considerations

NOTE: This is a brief outline of the security considerations. This section will be filled out more as the draft matures and security analyses are completed.¶
VDAFs have two essential security goals:¶
 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.¶
 Robustness: An attacker that controls the network and a subset of Clients cannot cause the Collector to compute anything other than the aggregate of the measurements of honest Clients.¶
Note that, to achieve robustness, it is important to ensure that the
verification parameters distributed to the Aggregators (verify_params
, see
Section 6.2.1) is never revealed to the Clients.¶
It is also possible to consider a stronger form of robustness, where the
attacker also controls a subset of Aggregators (see [BBCGGI19], Section 6.3).
To satisfy this stronger notion of robustness, it is necessary to prevent the
attacker from sharing the verification parameters with the Clients. It is
therefore RECOMMENDED that the Aggregators generate verify_params
only after a
set of Client inputs has been collected for verification, and regenerate them
for each such set of inputs.¶
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:¶
 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.¶
9. IANA Considerations
This document makes no request of IANA.¶
10. References
10.1. Normative References
 [RFC2119]
 Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, , <https://www.rfceditor.org/rfc/rfc2119>.
 [RFC4493]
 Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The AESCMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, , <https://www.rfceditor.org/rfc/rfc4493>.
 [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, , <https://www.rfceditor.org/rfc/rfc8017>.
 [RFC8174]
 Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, , <https://www.rfceditor.org/rfc/rfc8174>.
10.2. Informative References
 [AGJOP21]
 Addanki, S., Garbe, K., Jaffe, E., Ostrovsky, R., and A. Polychroniadou, "Prio+: Privacy Preserving Aggregate Statistics via Boolean Shares", , <https://ia.cr/2021/576>.
 [BBCGGI19]
 Boneh, D., Boyle, E., CorriganGibbs, H., Gilboa, N., and Y. Ishai, "ZeroKnowledge Proofs on SecretShared Data via Fully Linear PCPs", CRYPTO 2019 , , <https://ia.cr/2019/188>.
 [BBCGGI21]
 Boneh, D., Boyle, E., CorriganGibbs, H., Gilboa, N., and Y. Ishai, "Lightweight Techniques for Private Heavy Hitters", IEEE S&P 2021 , , <https://ia.cr/2021/017>.
 [CGB17]
 CorriganGibbs, H. and D. Boneh, "Prio: Private, Robust, and Scalable Computation of Aggregate Statistics", NSDI 2017 , , <https://dl.acm.org/doi/10.5555/3154630.3154652>.
 [Dou02]
 Douceur, J., "The Sybil Attack", IPTPS 2002 , , <https://doi.org/10.1007/3540457488_24>.
 [Dwo06]
 Dwork, C., "Differential Privacy", ICALP 2006 , , <https://link.springer.com/chapter/10.1007/11787006_1>.
 [ENPA]
 "Exposure Notification Privacypreserving Analytics (ENPA) White Paper", , <https://covid19static.cdnapple.com/applications/covid19/current/static/contacttracing/pdf/ENPA_White_Paper.pdf>.
 [EPK14]
 Erlingsson, Ú., Pihur, V., and A. Korolova, "RAPPOR: Randomized Aggregatable PrivacyPreserving Ordinal Response", CCS 2014 , , <https://dl.acm.org/doi/10.1145/2660267.2660348>.
 [GI14]
 Gilboa, N. and Y. Ishai, "Distributed Point Functions and Their Applications", EUROCRYPT 2014 , , <https://link.springer.com/chapter/10.1007/9783642552205_35>.
 [ID.draftgpewprivppm]
 Geoghegan, T., Patton, C., Rescorla, E., and C. A. Wood, "Privacy Preserving Measurement", Work in Progress, InternetDraft, draftgpewprivppm00, , <https://datatracker.ietf.org/doc/html/draftgpewprivppm00>.
 [OriginTelemetry]
 "Origin Telemetry", , <https://firefoxsourcedocs.mozilla.org/toolkit/components/telemetry/collection/origin.html>.
 [Vad16]
 Vadhan, S., "The Complexity of Differential Privacy", , <https://link.springer.com/chapter/10.1007/9783319570488_7>.
Acknowledgments
Thanks to Henry CorriganGibbs, Armando FazHernandez, Mariana Raykova, and Christopher Wood for useful feedback on and contributions to the spec.¶
Test Vectors
Test vectors cover the generation input shares and the conversion of input shares into output shares. Vectors specify the public and verification parameters, the measurement, the aggregation parameter, the expected input shares, the prepare messages, and the expected output shares.¶
Test vectors are encoded in JSON. Input shares and prepare messages are
represented as hexadecimal streams. To make the tests deterministic,
gen_rand()
was replaced with a function that returns the requested number of
0x01
octets.¶
Prio3Aes128Count
For this test, the value of SHARES
is 2
.¶
{ "public_param": null, "verify_params": [ [ 0, "01010101010101010101010101010101" ], [ 1, "01010101010101010101010101010101" ] ], "agg_param": null, "prep": [ { "measurement": 1, "nonce": "01010101010101010101010101010101", "input_shares": [ "05ac22db75e9b262e9642de9ec1ec37990625f92bf426c52e12c88d7c6e53ed673a3a8a3c7944170e09a52b96573259d", "0101010101010101010101010101010101010101010101010101010101010101" ], "prep_shares": [ [ "48771012eeda70a056cf2fd53022cf7b2edf45090eaa765c2b6cefb7a4abc524", "b788efec11258f61fa53dd238a164da076bfd3f2e4bd966634b24bb64c2fa160" ] ], "out_shares": [ [ 408739992155304546 ], [ 18038004077259279776 ] ] } ] }¶
Prio3Aes128Sum
For this test:¶
{ "public_param": null, "verify_params": [ [ 0, "01010101010101010101010101010101" ], [ 1, "01010101010101010101010101010101" ] ], "agg_param": null, "prep": [ { "measurement": 100, "nonce": "01010101010101010101010101010101", "input_shares": [ "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", "0101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010b9e7430cc7a71a356d6bd09d36d1fdb" ], "prep_shares": [ [ "2bcc0144c56fcf120a7aab22d57cde99fd2ee2301be4c59983d5e68e79f04cd8978b2c4598eafab7b1e8b0af8ba4bda20b9e7430cc7a71a356d6bd09d36d1fdb", "d433febb3a9030d1f58554dd2a8321682eff019a7a26e88471ed622fbd956e5e0af1c04b5573400ed13ae90b325aed4edfed32c071cc6899645ab72c36bd3670" ] ], "out_shares": [ [ 178602842237398423407215704739732627917 ], [ 161679524683540039539650068628168138392 ] ] } ] }¶
Prio3Aes128Histogram
For this test:¶
{ "public_param": null, "verify_params": [ [ 0, "01010101010101010101010101010101" ], [ 1, "01010101010101010101010101010101" ] ], "agg_param": null, "prep": [ { "measurement": 50, "nonce": "01010101010101010101010101010101", "input_shares": [ "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", "0101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101014a3951bdbf7a4564e422499e59351ba7" ], "prep_shares": [ [ "03faf09f79624d05e3a1e17bf1e5117ea6eec816dbd047641506a0c1b1e817fa1ebb27507cea09b00b47e0615ba82ff64a3951bdbf7a4564e422499e59351ba7", "fc050f60869db2de1c5e1e840e1aee830a99091f9a820099b6b9591984046f48b6b6d58ddb1675ef43101aa8773f3025688dfdc50bd6d3a9ecf1613c58b7ced1" ] ], "out_shares": [ [ 7539922107207114695252505926366364067, 198783809130402957557687312006462666532, 261868461448231140209796284667530078285, 19075760356742656327154126012204712008 ], [ 332742444813731348251613267441534402142, 141498557790535505389178461361438099677, 78413905472707322737069488700370687925, 321206606564195806619711647355696054201 ] ] } ] }¶