Verifiable Distributed Aggregation Functions

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Table of Contents

1. Introduction

The ubiquity of the Internet makes it an ideal platform for measurement of large-scale 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 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 privacy-preserving measurement system is that of secure multi-party 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 non-colluding 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 multi-party protocols for privacy-preserving measurement proposed in the literature. These protocols vary in their operational and security considerations, sometimes in subtle but consequential ways. This document therefore has two important goals:¶

1. Providing applications like [I-D.draft-gpew-priv-ppm] with a simple, uniform interface for accessing privacy-preserving measurement schemes, and documenting relevant operational and security bounds for that interface:¶

   1. General patterns of communications among the various actors involved in the system (clients, aggregation servers, and measurement collectors);¶
   2. Capabilities of a malicious coalition of servers attempting to divulge information about client inputs; and¶
   3. Conditions that are necessary to ensure that malicious clients cannot corrupt the computation.¶
2. Providing cryptographers with design criteria that allow new constructions to be easily used by applications.¶

This document also specifies two concrete VDAF schemes, each based on a protocol from the literature.¶

• The aforementioned Prio system [CGB17] allows for the privacy-preserving computation of a variety aggregate statistics. The basic idea underlying Prio is fairly simple:¶

  1. Each client shards its input into a sequence of additive shares and distributes the shares among the aggregation servers.¶
  2. Next, each server adds up its shares locally, resulting in an additive share of the aggregate.¶
  3. Finally, the aggregation servers combine their additive shares to obtain the final aggregate.¶

  The difficult part of this system is ensuring that the servers hold shares of a valid input, e.g., the input is an integer in a specific range. Thus Prio specifies a multi-party 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-heavy-hitters problem in a privacy-preserving manner. Here each client holds a bit-string of length n, and the goal of the aggregation servers is to compute the set of inputs that occur at least t times. The core primitive used in their protocol is a generalization of a Distributed Point Function (DPF) [GI14] that allows the servers to "query" their DPF shares on any bit-string 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 multi-party 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 6 describes the Prio3 construction; Section 7 describes the Poplar1 construction; and Section 8 enumerates the security considerations for VDAFs.¶

2. Conventions and Definitions

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶

Algorithms in this document are written in Python 3. Type hints are used to define input and output types. A fatal error in a program (e.g., failure to parse one of the function parameters) is usually handled by raising an exception.¶

Some common functionalities:¶

• zeros(len: Unsigned) -> Bytes returns an array of zero bytes. The length of output MUST be len.¶
• 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 non-negative integer to a byte string and convert a byte string to a non-negative integer.¶
• next_power_of_2(n: Unsigned) -> Unsigned returns the smallest integer greater than or equal to n that is also a power of two.¶

3. Overview

                 +--------------+
           +---->| Aggregator 0 |----+
           |     +--------------+    |
           |             ^           |
           |             |           |
           |             V           |
           |     +--------------+    |
           | +-->| Aggregator 1 |--+ |
           | |   +--------------+  | |
+--------+-+ |           ^         | +->+-----------+
| Client |---+           |         +--->| Collector |--> Aggregate
+--------+-+                         +->+-----------+
           |            ...          |
           |                         |
           |             |           |
           |             V           |
           |    +----------------+   |
           +--->| Aggregator N-1 |---+
                +----------------+

      Input shares           Aggregate shares

In a VDAF-based 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 one-to-one 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 non-collusion 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 non-collusion requirements of a given VDAF instance, it is possible for the same entity to play more than one role. For example, the Collector could also act as an Aggregator, effectively using the other Aggregators to augment a basic client-server protocol.¶

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 Table 1; the methods are defined in the subsections that follow.¶

    Client
    ======

    measurement
      |
      V
    +----------------------------------------------+
    | measurement_to_input_shares                  |
    +----------------------------------------------+
      |              |              ...  |
      V              V                   V
     input_share_0  input_share_1       input_share_[SHARES-1]
      |              |              ...  |
      V              V                   V
    Aggregator 0   Aggregator 1        Aggregator SHARES-1

    Aggregator 0   Aggregator 1        Aggregator SHARES-1
    ============   ============        ===================

    input_share_0  input_share_1       input_share_[SHARES-1]
      |              |              ...  |
      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_[SHARES-1]

    Aggregator 0    Aggregator 1        Aggregator SHARES-1
    ============    ============        ===================

    out_share_0_0   out_share_1_0       out_share_[SHARES-1]_0
    out_share_0_1   out_share_1_1       out_share_[SHARES-1]_1
    out_share_0_2   out_share_1_2       out_share_[SHARES-1]_2
         ...             ...                     ...
    out_share_0_B   out_share_1_B       out_share_[SHARES-1]_B
      |               |                   |
      V               V                   V
    +-----------+   +-----------+       +-----------+
    | out2agg   |   | out2agg   |   ... | out2agg   |
    +-----------+   +-----------+       +-----------+
      |               |                   |
      V               V                   V
    agg_share_0     agg_share_1         agg_share_[SHARES-1]

    Aggregator 0    Aggregator 1        Aggregator SHARES-1
    ============    ============        ===================

    agg_share_0     agg_share_1         agg_share_[SHARES-1]
      |               |                   |
      V               V                   V
    +-----------------------------------------------+
    | agg_shares_to_result                          |
    +-----------------------------------------------+
      |
      V
    agg_result

    Collector
    =========

def run_vdaf(Vdaf,
             agg_param: Vdaf.AggParam,
             nonces: Vec[Bytes],
             measurements: Vec[Vdaf.Measurement]):
    # Distribute long-lived parameters.
    (public_param, verify_params) = Vdaf.setup()

    out_shares = []
    for (nonce, measurement) in zip(nonces, measurements):

        # Each Client shards its input into shares.
        input_shares = Vdaf.measurement_to_input_shares(public_param,
                                                        measurement)

        # Each Aggregator initializes its preparation state.
        prep_states = []
        for j in range(Vdaf.SHARES):
            state = Vdaf.prep_init(verify_params[j],
                                   agg_param,
                                   nonce,
                                   input_shares[j])
            prep_states.append(state)

        # Aggregators recover their output shares.
        inbound = None
        for i in range(Vdaf.ROUNDS+1):
            outbound = []
            for j in range(Vdaf.SHARES):
                out = Vdaf.prep_next(prep_states[j], inbound)
                if i < Vdaf.ROUNDS:
                    (prep_states[j], out) = out
                outbound.append(out)
            # This is where we would send messages over the
            # network in a distributed VDAF computation.
            if i < Vdaf.ROUNDS:
                inbound = Vdaf.prep_shares_to_prep(agg_param,
                                                   outbound)

        # The final outputs of prepare phasre are the output shares.
        out_shares.append(outbound)

    # Each Aggregator aggregates its output shares into an
    # aggregate share.
    agg_shares = []
    for j in range(Vdaf.SHARES):
        out_shares_j = [out[j] for out in out_shares]
        agg_share_j = Vdaf.out_shares_to_agg_share(agg_param,
                                                   out_shares_j)
        agg_shares.append(agg_share_j)

    # Collector unshards the aggregate.
    agg_result = Vdaf.agg_shares_to_result(agg_param, agg_shares)
    return agg_result

5. Preliminaries

This section describes the primitives that are common to the VDAFs specified in this document.¶

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

# 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)))

# 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

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 = AES128-CMAC(seed, info)

    def next(self, length):
        self.length_consumed += length

        # CTR-mode encryption of the all-zero string of the desired
        # length and using a fixed, all-zero IV.
        stream = AES128-CTR(key, zeros(16), zeros(self.length_consumed))
        return stream[-length:]

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 non-zero, 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 multi-party 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".¶

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)

def setup(Prio3):
    k_query_init = gen_rand(Prio3.Prg.SEED_SIZE)
    verify_param = [(j, k_query_init) for j in range(Prio3.SHARES)]
    return (None, verify_param)

def measurement_to_input_shares(Prio3, _public_param, measurement):
    dst = b"vdaf-00 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.SHARES-1):
        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)
        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.SHARES-1):
        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.SHARES-1):
        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,
        k_leader_hint,
    ))
    for j in range(Prio3.SHARES-1):
        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

def prep_init(Prio3, verify_param, _agg_param, nonce, input_share):
    dst = b"vdaf-00 prio3"
    (j, k_query_init) = verify_param

    (input_share, proof_share, k_blind, k_hint) = \
        Prio3.decode_leader_share(input_share) if j == 0 else \
        Prio3.decode_helper_share(dst, j, input_share)

    out_share = Prio3.Flp.truncate(input_share)

    k_query_rand = Prio3.Prg.derive_seed(k_query_init,
                                         byte(255) + nonce)
    query_rand = Prio3.Prg.expand_into_vec(
        Prio3.Flp.Field,
        k_query_rand,
        dst,
        Prio3.Flp.QUERY_RAND_LEN
    )
    joint_rand, k_joint_rand, k_joint_rand_share = [], None, None
    if Prio3.Flp.JOINT_RAND_LEN > 0:
        encoded = Prio3.Flp.Field.encode_vec(input_share)
        k_joint_rand_share = Prio3.Prg.derive_seed(k_blind,
                                                   byte(j) + encoded)
        k_joint_rand = xor(k_hint, k_joint_rand_share)
        joint_rand = Prio3.Prg.expand_into_vec(
            Prio3.Flp.Field,
            k_joint_rand,
            dst,
            Prio3.Flp.JOINT_RAND_LEN
        )
    verifier_share = Prio3.Flp.query(input_share,
                                     proof_share,
                                     query_rand,
                                     joint_rand,
                                     Prio3.SHARES)

    prep_msg = Prio3.encode_prepare_message(verifier_share,
                                            k_joint_rand_share)
    return (out_share, k_joint_rand, prep_msg)

def prep_next(Prio3, prep, inbound):
    (out_share, k_joint_rand, prep_msg) = prep

    if inbound is None:
        return (prep, prep_msg)

    (verifier, k_joint_rand_check) = \
        Prio3.decode_prepare_message(inbound)

    if k_joint_rand_check != k_joint_rand or \
            not Prio3.Flp.decide(verifier):
        raise ERR_VERIFY

    return out_share

def prep_shares_to_prep(Prio3, _agg_param, prep_shares):
    verifier = Prio3.Flp.Field.zeros(Prio3.Flp.VERIFIER_LEN)
    k_joint_rand_check = zeros(Prio3.Prg.SEED_SIZE)
    for encoded in prep_shares:
        (verifier_share, k_joint_rand_share) = \
            Prio3.decode_prepare_message(encoded)

        verifier = vec_add(verifier, verifier_share)

        if Prio3.Flp.JOINT_RAND_LEN > 0:
            k_joint_rand_check = xor(k_joint_rand_check,
                                     k_joint_rand_share)

    return Prio3.encode_prepare_message(verifier,
                                        k_joint_rand_check)

def out_shares_to_agg_share(Prio3, _agg_param, out_shares):
    agg_share = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
    for out_share in out_shares:
        agg_share = vec_add(agg_share, out_share)
    return agg_share

def agg_shares_to_result(Prio3, _agg_param, agg_shares):
    agg = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
    for agg_share in agg_shares:
        agg = vec_add(agg, agg_share)
    return list(map(lambda x: x.as_unsigned(), agg))

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, j, encoded):
    l = Prio3.Prg.SEED_SIZE
    k_input_share, encoded = encoded[:l], encoded[l:]
    input_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field,
                                            k_input_share,
                                            dst + byte(j),
                                            Prio3.Flp.INPUT_LEN)
    k_proof_share, encoded = encoded[:l], encoded[l:]
    proof_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field,
                                            k_proof_share,
                                            dst + byte(j),
                                            Prio3.Flp.PROOF_LEN)
    k_blind, k_hint = None, None
    if Prio3.Flp.JOINT_RAND_LEN > 0:
        k_blind, encoded = encoded[:l], encoded[l:]
        k_hint, encoded = encoded[:l], encoded[l:]
    if len(encoded) != 0:
        raise ERR_DECODE
    return (input_share, proof_share, k_blind, k_hint)

def encode_prepare_message(Prio3, verifier, k_joint_rand):
    encoded = Bytes()
    encoded += Prio3.Flp.Field.encode_vec(verifier)
    if Prio3.Flp.JOINT_RAND_LEN > 0:
        encoded += k_joint_rand
    return encoded

def decode_prepare_message(Prio3, encoded):
    l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.VERIFIER_LEN
    encoded_verifier, encoded = encoded[:l], encoded[l:]
    verifier = Prio3.Flp.Field.decode_vec(encoded_verifier)
    k_joint_rand = None
    if Prio3.Flp.JOINT_RAND_LEN > 0:
        l = Prio3.Prg.SEED_SIZE
        k_joint_rand, encoded = encoded[:l], encoded[l:]
    if len(encoded) != 0:
        raise ERR_DECODE
    return (verifier, k_joint_rand)

C(x_shares[0] + ... + x_shares[SHARES-1]) =
    C(x_shares[0]) + ... + C(x_shares[SHARES-1])

C(x) = (x[0] - 1) * x[0] = x[0]^2 - x[0]

Mul(left, right) = left * right

C(x[0]) = Mul(x[0], x[0]) - x[0]

C(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N-1])

C(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N-1]) + \
            2^0 * inp[0]       + ... + 2^(N-1) * inp[N-1]     - \
            Mul(inp[N], inp[N])

# 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

def Mul(x, y):
    return x * y

def Count(inp: Vec[Field64]):
    return Mul(inp[0], inp[0]) - inp[0]

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]

def Range2(x):
    return x^2 - x

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

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

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

7. Poplar1

• NOTE The spec for Poplar1 is still a work-in-progress. A partial implementation can be found at https://github.com/abetterinternet/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 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.¶

1. Each Clients runs the input-distribution algorithm on its n-bit 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-heavy-hitters. 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.¶

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 non-zero 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 "one-hot" vector, i.e., a vector that is zero everywhere except for one element, which is equal to one.¶

def vdaf_setup():
  k_verify_init = gen_rand(SEED_SIZE)
  return (None, [(0, k_verify_init), (1, k_verify_init)])

    A = -2*a + k
    B = a*a + b - k*a + c

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

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[l-1]

    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
      ]

      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)

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)

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)

8. Security Considerations

• NOTE: This is a brief outline of the security considerations. This section will be filled out more as the draft matures and security analyses are completed.¶

VDAFs have two essential security goals:¶

1. Privacy: An attacker that controls the network, the Collector, and a subset of Clients and Aggregators learns nothing about the measurements of honest Clients beyond what it can deduce from the aggregate result.¶
2. Robustness: An attacker that controls the network and a subset of Clients cannot cause the Collector to compute anything other than the aggregate of the measurements of honest Clients.¶

Note that, to achieve robustness, it is important to ensure that the verification parameters distributed to the Aggregators (verify_params, see Section 6.2.1) is never revealed to the Clients.¶

It is also possible to consider a stronger form of robustness, where the attacker also controls a subset of Aggregators (see [BBCGGI19], Section 6.3). To satisfy this stronger notion of robustness, it is necessary to prevent the attacker from sharing the verification parameters with the Clients. It is therefore RECOMMENDED that the Aggregators generate verify_params only after a set of Client inputs has been collected for verification, and re-generate 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 long-lived parameters.¶

• Establishing secure channels:¶

  • Confidential and authentic channels among Aggregators, and between the Aggregators and the Collector; and¶
  • Confidential and Aggregator-authenticated channels between Clients and Aggregators.¶
• Enforcing the non-collusion properties required of the specific VDAF in use.¶

In such an environment, a VDAF provides the high-level 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 non-collusion 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.¶

Acknowledgments

Thanks to David Cook, Henry Corrigan-Gibbs, Armando Faz-Hernandez, Mariana Raykova, and Christopher Wood for useful feedback on and contributions to the spec.¶

Test Vectors

Test vectors cover the generation of 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.¶

{
    "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
                ]
            ]
        }
    ]
}

{
    "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
                ]
            ]
        }
    ]
}

{
    "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": [
                [
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                ]
            ],
            "out_shares": [
                [
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                ],
                [
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                ]
            ]
        }
    ]
}

Authors' Addresses