Verifiable Distributed Aggregation Functions

Status of This Memo

This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79.¶

Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet-Drafts is at https://datatracker.ietf.org/drafts/current/.¶

Internet-Drafts 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 Internet-Drafts as reference material or to cite them other than as "work in progress."¶

This Internet-Draft will expire on 12 January 2023.¶

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/license-info) 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.¶

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 5 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 higher-level protocols like [DAP] 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 the collector of the aggregate result);¶
   2. Capabilities of a malicious coalition of servers attempting to divulge information about client measurements; and¶
   3. Conditions that are necessary to ensure that malicious clients cannot corrupt the computation.¶
2. Providing cryptographers with design criteria that provide a clear deployment roadmap for new constructions.¶

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

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

  1. Each client shards its measurement into a sequence of additive shares and distributes the shares among the aggregation servers.¶
  2. Next, each server adds up its shares locally, resulting in an additive share of the aggregate.¶
  3. Finally, the aggregation servers send their aggregate shares to the data collector, who combines them to obtain the aggregate result.¶

  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 7 we describe Prio3, a VDAF that follows the same overall framework as the original Prio protocol, but incorporates techniques introduced in [BBCGGI19] that result in significant performance gains.¶

• More recently, Boneh et al. [BBCGGI21] described a protocol called Poplar for solving the 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 specialized 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 8 we describe a VDAF called Poplar1 that implements this functionality.¶

Finally, perhaps the most complex aspect of schemes like Prio3 and Poplar1 is the process by which the client-generated measurements are prepared for aggregation. Because these constructions are based on secret sharing, the servers will be required to exchange some amount of information in order to verify the measurement is valid and can be aggregated. Depending on the construction, this process may require multiple round trips over the network.¶

There are applications in which this verification step may not be necessary, e.g., when the client's software is run a trusted execution environment. To support these applications, this document also defines Distributed Aggregation Functions (DAFs) as a simpler class of protocols that aim to provide the same privacy guarantee as VDAFs but fall short of being verifiable.¶

• OPEN ISSUE Decide if we should give one or two example DAFs. There are natural variants of Prio3 and Poplar1 that might be worth describing.¶

The remainder of this document is organized as follows: Section 3 gives a brief overview of DAFs and VDAFs; Section 4 defines the syntax for DAFs; Section 5 defines the syntax for VDAFs; Section 6 defines various functionalities that are common to our constructions; Section 7 describes the Prio3 construction; Section 8 describes the Poplar1 construction; and Section 9 enumerates the security considerations for VDAFs.¶

2. Conventions and Definitions

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

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

A variable with type Bytes is a byte string. This document defines several byte-string constants. When comprised of printable ASCII characters, they are written as Python 3 byte-string literals (e.g., b'some constant string').¶

A global constant VERSION is defined, which algorithms are free to use as desired. Its value SHALL be b'vdaf-02'.¶

This document describes algorithms for multi-party computations in which the parties typically communicate over a network. Wherever a quantity is defined that must be be transmitted from one party to another, this document prescribes a particular encoding of that quantity as a byte string.¶

• OPEN ISSUE It might be better to not be prescriptive about how quantities are encoded on the wire. See issue #58.¶

Some common functionalities:¶

• zeros(len: Unsigned) -> Bytes returns an array of zero bytes. The length of output MUST be len.¶
• gen_rand(len: Unsigned) -> Bytes returns an array of random bytes. The length of output MUST be len.¶
• byte(int: Unsigned) -> Bytes returns the representation of int as a byte string. The value of int MUST be in [0,256).¶
• xor(left: Bytes, right: Bytes) -> Bytes returns the bitwise XOR of left and right. An exception is raised if the inputs are not the same length.¶
• I2OSP and OS2IP from [RFC8017], which are used, respectively, to convert a 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 DAF- or 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:¶

• To submit an individual measurement, the Client shards the measurement into "input shares" and sends one input share to each Aggregator.¶

• The Aggregators convert their input shares into "output shares".¶

  • Output shares are in one-to-one correspondence with the input shares.¶
  • Just as each Aggregator receives one input share of each input, at the end of this process, each aggregator holds one output share.¶
  • In VDAFs, Aggregators will need to exchange information among themselves as part of the validation process.¶
• Each Aggregator combines the output shares across inputs in the batch to compute the "aggregate share" for that batch, i.e., its share of the desired aggregate result.¶
• The Aggregators submit their aggregate shares to the Collector, who combines them to obtain the aggregate result over the batch.¶

Aggregators are a new class of actor relative to traditional measurement systems where clients submit measurements to a single server. They are critical for both the privacy properties of the system and, in the case of VDAFs, the correctness of the measurements obtained. The privacy properties of the system are assured by non-collusion among Aggregators, and Aggregators are the entities that perform validation of Client measurements. Thus clients trust Aggregators not to collude (typically it is required that at least one Aggregator is honest), and Collectors trust Aggregators to correctly run the protocol.¶

Within the bounds of the non-collusion requirements of a given (V)DAF instance, it is possible for the same entity to play more than one role. For example, the Collector could also act as an Aggregator, effectively using the other Aggregator(s) to augment a basic client-server protocol.¶

In this document, we describe the computations performed by the actors in this system. It is up to the higher-level protocol making use of the (V)DAF to arrange for the required information to be delivered to the proper actors in the proper sequence. In general, we assume that all communications are confidential and mutually authenticated, with the exception that Clients submitting measurements may be anonymous.¶

4. Definition of DAFs

By way of a gentle introduction to VDAFs, this section describes a simpler class of schemes called Distributed Aggregation Functions (DAFs). Unlike VDAFs, DAFs do not provide verifiability of the computation. Clients must therefore be trusted to compute their input shares correctly. Because of this fact, the use of a DAF is NOT RECOMMENDED for most applications. See Section 9 for additional discussion.¶

A DAF scheme is used to compute a particular "aggregation function" over a set of measurements generated by Clients. Depending on the aggregation function, the Collector might select an "aggregation parameter" and disseminates it to the Aggregators. The semantics of this parameter is specific to the aggregation function, but in general it is used to represent the set of "queries" that can be made on the measurement set. For example, the aggregation parameter is used to represent the candidate prefixes in Poplar1 Section 8.¶

Execution of a DAF has four distinct stages:¶

• Sharding - Each Client generates input shares from its measurement and distributes them among the Aggregators.¶
• Preparation - Each Aggregator converts each input share into an output share compatible with the aggregation function. This computation involves the aggregation parameter. In general, each aggregation parameter may result in a different an output share.¶
• Aggregation - Each Aggregator combines a sequence of output shares into its aggregate share and sends the aggregate share to the Collector.¶
• Unsharding - The Collector combines the aggregate shares into the aggregate result.¶

Sharding and Preparation are done once per measurement. Aggregation and Unsharding are done over a batch of measurements (more precisely, over the recovered output shares).¶

A concrete DAF specifies an algorithm for the computation needed in each of these stages. The interface of each algorithm is defined in the remainder of this section. In addition, a concrete DAF defines the associated constants and types enumerated in the following table.¶

These types define some of the inputs and outputs of DAF methods at various stages of the computation. Observe that only the measurements, output shares, the aggregate result, and the aggregation parameter have an explicit type. All other values --- in particular, the input shares and the aggregate shares --- have type Bytes and are treated as opaque byte strings. This is because these values must be transmitted between parties over a network.¶

• OPEN ISSUE It might be cleaner to define a type for each value, then have that type implement an encoding where necessary. This way each method parameter has a meaningful type hint. See issue#58.¶

    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
    ============    ============        ===================

    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_daf(Daf,
            agg_param: Daf.AggParam,
            measurements: Vec[Daf.Measurement]):
    out_shares = [ [] for j in range(Daf.SHARES) ]
    for measurement in measurements:
        # Each Client shards its measurement into input shares and
        # distributes them among the Aggregators.
        (public_share, input_shares) = \
            Daf.measurement_to_input_shares(measurement)

        # Each Aggregator prepares its input share for aggregation.
        for j in range(Daf.SHARES):
            out_shares[j].append(
                Daf.prep(j, agg_param, public_share, input_shares[j]))

    # Each Aggregator aggregates its output shares into an aggregate
    # share and sends it to the Collector.
    agg_shares = []
    for j in range(Daf.SHARES):
        agg_share_j = Daf.out_shares_to_agg_share(agg_param,
                                                  out_shares[j])
        agg_shares.append(agg_share_j)

    # Collector unshards the aggregate result.
    num_measurements = len(measurements)
    agg_result = Daf.agg_shares_to_result(agg_param, agg_shares,
                                          num_measurements)
    return agg_result

5. Definition of VDAFs

Like DAFs described in the previous section, a VDAF scheme is used to compute a particular aggregation function over a set of Client-generated measurements. Evaluation of a VDAF involves the same four stages as for DAFs: Sharding, Preparation, Aggregation, and Unsharding. However, the Preparation stage will require interaction among the Aggregators in order to facilitate verifiability of the computation's correctness. Accommodating this interaction will require syntactic changes.¶

Overall execution of a VDAF comprises the following stages:¶

• Sharding - Computing input shares from an individual measurement¶
• Preparation - Conversion and verification of input shares to output shares compatible with the aggregation function being computed¶
• Aggregation - Combining a sequence of output shares into an aggregate share¶
• Unsharding - Combining a sequence of aggregate shares into an aggregate result¶

In contrast to DAFs, the Preparation stage for VDAFs now performs an additional task: Verification of the validity of the recovered output shares. This process ensures that aggregating the output shares will not lead to a garbled aggregate result.¶

The remainder of this section defines the VDAF interface. The attributes are listed in Table 2 are defined by each concrete VDAF.¶

Similarly to DAFs (see {[sec-daf}}), any output of a VDAF method that must be transmitted from one party to another is treated as an opaque byte string. All other quantities are given a concrete type.¶

• OPEN ISSUE It might be cleaner to define a type for each value, then have that type implement an encoding where necessary. See issue#58.¶

    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]

def run_vdaf(Vdaf,
             agg_param: Vdaf.AggParam,
             nonces: Vec[Bytes],
             measurements: Vec[Vdaf.Measurement]):
    # Generate the long-lived verification key.
    verify_key = gen_rand(Vdaf.VERIFY_KEY_SIZE)

    out_shares = []
    for (nonce, measurement) in zip(nonces, measurements):
        # Each Client shards its measurement into input shares.
        (public_share, input_shares) = \
            Vdaf.measurement_to_input_shares(measurement)

        # Each Aggregator initializes its preparation state.
        prep_states = []
        for j in range(Vdaf.SHARES):
            state = Vdaf.prep_init(verify_key, j,
                                   agg_param,
                                   nonce,
                                   public_share,
                                   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 phase are the output shares.
        out_shares.append(outbound)

    # Each Aggregator aggregates its output shares into an
    # aggregate share. In a distributed VDAF computation, the
    # aggregate shares are sent over the network.
    agg_shares = []
    for j in range(Vdaf.SHARES):
        out_shares_j = [out[j] for out in out_shares]
        agg_share_j = Vdaf.out_shares_to_agg_share(agg_param,
                                                   out_shares_j)
        agg_shares.append(agg_share_j)

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

6. 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 = OS2IP(encoded_x)
        if x >= Field.MODULUS:
            raise ERR_DECODE # Integer is larger than modulus
        vec.append(Field(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)

# Output the next `length` pseudorandom elements of `Field`.
def next_vec(self, Field, length: Unsigned):
    m = next_power_of_2(Field.MODULUS) - 1
    vec = []
    while len(vec) < length:
        x = OS2IP(self.next(Field.ENCODED_SIZE))
        x &= m
        if x < Field.MODULUS:
            vec.append(Field(x))
    return vec

# Expand the input `seed` into vector of `length` field elements.
def expand_into_vec(Prg,
                    Field,
                    seed: Bytes,
                    info: Bytes,
                    length: Unsigned):
    prg = Prg(seed, info)
    return prg.next_vec(Field, length)

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:]

7. Prio3

• NOTE This construction has not undergone significant security analysis.¶

This section describes Prio3, a VDAF for Prio [CGB17]. Prio is suitable for a wide variety of aggregation functions, including (but not limited to) sum, mean, standard deviation, estimation of quantiles (e.g., median), and linear regression. In fact, the scheme described in this section is compatible with any aggregation function that has the following structure:¶

• Each measurement is encoded as a vector over some finite field.¶
• Input validity is determined by an arithmetic circuit evaluated over the encoded input. (An "arithmetic circuit" is a function comprised of arithmetic operations in the field.) The circuit's output is a single field element: if zero, then the input is said to be "valid"; otherwise, if the output is non-zero, then the input is said to be "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 8 for an example of a VDAF that makes meaningful use of the aggregation parameter.¶

As the name implies, Prio3 is a descendant of the original Prio construction. A second iteration was deployed in the [ENPA] system, and like the VDAF described here, the ENPA system was built from techniques introduced in [BBCGGI19] that significantly improve communication cost. That system was specialized for a particular aggregation function; the goal of Prio3 is to provide the same level of generality as the original construction.¶

The core component of Prio3 is a "Fully Linear Proof (FLP)" system. Introduced by [BBCGGI19], the FLP encapsulates the functionality required for encoding and validating inputs. Prio3 can be thought of as a transformation of a particular class of FLPs into a VDAF.¶

The remainder of this section is structured as follows. The syntax for FLPs is described in Section 7.1. The generic transformation of an FLP into Prio3 is specified in Section 7.2. Next, a concrete FLP suitable for any validity circuit is specified in Section 7.3. Finally, instantiations of Prio3 for various types of measurements are specified in Section 7.4. Test vectors can be found in Appendix "Test Vectors".¶

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 measurement_to_input_shares(Prio3, measurement):
    # Domain separation tag for PRG info string
    dst = VERSION + b' prio3'
    inp = Prio3.Flp.encode(measurement)
    k_joint_rand = zeros(Prio3.Prg.SEED_SIZE)

    # Generate input shares.
    leader_input_share = inp
    k_helper_input_shares = []
    k_helper_blinds = []
    k_helper_hints = []
    for j in range(Prio3.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 (b'', input_shares)

def prep_init(Prio3, verify_key, agg_id, _agg_param,
              nonce, _public_share, input_share):
    # Domain separation tag for PRG info string
    dst = VERSION + b' prio3'

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

    out_share = Prio3.Flp.truncate(input_share)

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

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

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

    if inbound is None:
        return (prep, prep_msg)

    k_joint_rand_check = Prio3.decode_prep_msg(inbound)
    if k_joint_rand_check != k_joint_rand:
        raise ERR_VERIFY # joint randomness check failed

    return out_share

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

        verifier = vec_add(verifier, verifier_share)

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

    if not Prio3.Flp.decide(verifier):
        raise ERR_VERIFY # proof verifier check failed

    return Prio3.encode_prep_msg(k_joint_rand_check)

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

def agg_shares_to_result(Prio3, _agg_param, agg_shares,
                         num_measurements):
    agg = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
    for agg_share in agg_shares:
        agg = vec_add(agg, Prio3.Flp.Field.decode_vec(agg_share))
    return Prio3.Flp.decode(agg, num_measurements)

def encode_leader_share(Prio3,
                        input_share,
                        proof_share,
                        k_blind,
                        k_hint):
    encoded = Bytes()
    encoded += Prio3.Flp.Field.encode_vec(input_share)
    encoded += Prio3.Flp.Field.encode_vec(proof_share)
    if Prio3.Flp.JOINT_RAND_LEN > 0:
        encoded += k_blind
        encoded += k_hint
    return encoded

def decode_leader_share(Prio3, encoded):
    l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.INPUT_LEN
    encoded_input_share, encoded = encoded[:l], encoded[l:]
    input_share = Prio3.Flp.Field.decode_vec(encoded_input_share)
    l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.PROOF_LEN
    encoded_proof_share, encoded = encoded[:l], encoded[l:]
    proof_share = Prio3.Flp.Field.decode_vec(encoded_proof_share)
    l = Prio3.Prg.SEED_SIZE
    k_blind, k_hint = None, None
    if Prio3.Flp.JOINT_RAND_LEN > 0:
        k_blind, encoded = encoded[:l], encoded[l:]
        k_hint, encoded = encoded[:l], encoded[l:]
    if len(encoded) != 0:
        raise ERR_DECODE
    return (input_share, proof_share, k_blind, k_hint)

def encode_helper_share(Prio3,
                        k_input_share,
                        k_proof_share,
                        k_blind,
                        k_hint):
    encoded = Bytes()
    encoded += k_input_share
    encoded += k_proof_share
    if Prio3.Flp.JOINT_RAND_LEN > 0:
        encoded += k_blind
        encoded += k_hint
    return encoded

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

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

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

def encode_prep_msg(Prio3, k_joint_rand_check):
    encoded = Bytes()
    if Prio3.Flp.JOINT_RAND_LEN > 0:
        encoded += k_joint_rand_check
    return encoded

def decode_prep_msg(Prio3, encoded):
    k_joint_rand_check = None
    if Prio3.Flp.JOINT_RAND_LEN > 0:
        l = Prio3.Prg.SEED_SIZE
        k_joint_rand_check, encoded = encoded[:l], encoded[l:]
    if len(encoded) != 0:
        raise ERR_DECODE
    return k_joint_rand_check

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 decode(Sum, output, _num_measurements):
    return output[0].as_unsigned()

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 decode(Histogram, output: Vec[Field128], _num_measurements):
    return [bucket_count.as_unsigned() for bucket_count in output]

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

8. Poplar1

• NOTE This construction has not undergone significant security analysis.¶

This section specifies Poplar1, a VDAF for the following task. Each Client holds a string of length BITS 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 the Poplar protocol [BBCGGI21]. At a high level, the protocol works as follows.¶

1. Each Client splits its input string into input shares and sends one share to each Aggregator.¶
2. The Aggregators agree on an initial set of candidate prefixes, say 0 and 1.¶
3. The Aggregators evaluate the VDAF on each set of input shares and aggregate the recovered output shares. The aggregation parameter is the set of candidate prefixes.¶
4. The Aggregators send their aggregate shares to the Collector, who combines them to recover the counts of each candidate prefix.¶
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 programmed 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.¶

Poplar1 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.¶

The remainder of this section is structured as follows. IDPFs are defined in Section 8.1; a concrete instantiation is given Section 8.3. The Poplar1 VDAF is defined in Section 8.2 in terms of a generic IDPF. Finally, a concrete instantiation of Poplar1 is specified in Section 8.4; test vectors can be found in Appendix "Test Vectors".¶

def current_field(Idpf, level):
    return Idpf.FieldInner if level < Idpf.BITS-1 \
                else Idpf.FieldLeaf

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

def measurement_to_input_shares(Poplar1, measurement):
    dst = VERSION + b' poplar1'
    prg = Poplar1.Idpf.Prg(
        gen_rand(Poplar1.Idpf.Prg.SEED_SIZE), dst + byte(255))

    # Construct the IDPF values for each level of the IDPF tree.
    # Each "data" value is 1; in addition, the Client generates
    # a random "authenticator" value used by the Aggregators to
    # compute the sketch during preparation. This sketch is used
    # to verify the one-hotness of their output shares.
    beta_inner = [
        [Poplar1.Idpf.FieldInner(1), k] \
            for k in prg.next_vec(Poplar1.Idpf.FieldInner,
                                  Poplar1.Idpf.BITS - 1) ]
    beta_leaf = [Poplar1.Idpf.FieldLeaf(1)] + \
        prg.next_vec(Poplar1.Idpf.FieldLeaf, 1)

    # Generate the IDPF keys.
    (public_share, keys) = \
        Poplar1.Idpf.gen(measurement, beta_inner, beta_leaf)

    # Generate correlated randomness used by the Aggregators to
    # compute a sketch over their output shares. PRG seeds are
    # used to encode shares of the `(a, b, c)` triples.
    # (See [BBCGGI21, Appendix C.4].)
    corr_seed = [
        gen_rand(Poplar1.Idpf.Prg.SEED_SIZE),
        gen_rand(Poplar1.Idpf.Prg.SEED_SIZE),
    ]
    corr_prg = [
        Poplar1.Idpf.Prg(corr_seed[0], dst + byte(0)),
        Poplar1.Idpf.Prg(corr_seed[1], dst + byte(1)),
    ]

    # For each level of the IDPF tree, shares of the `(A, B)`
    # pairs are computed from the corresponding `(a, b, c)`
    # triple and authenticator value `k`.
    corr_inner = [[], []]
    for level in range(Poplar1.Idpf.BITS):
        Field = Poplar1.Idpf.current_field(level)
        k = beta_inner[level][1] if level < Poplar1.Idpf.BITS - 1 \
            else beta_leaf[1]
        (a, b, c) = vec_add(corr_prg[0].next_vec(Field, 3),
                            corr_prg[1].next_vec(Field, 3))
        A = -Field(2) * a + k
        B = a^2 + b - a * k + c
        corr1 = prg.next_vec(Field, 2)
        corr0 = vec_sub([A, B], corr1)
        if level < Poplar1.Idpf.BITS - 1:
            corr_inner[0] += corr0
            corr_inner[1] += corr1
        else:
            corr_leaf = [corr0, corr1]

    # Each input share consists of the Aggregator's IDPF key
    # and a share of the correlated randomness.
    return (public_share,
            Poplar1.encode_input_shares(
                keys, corr_seed, corr_inner, corr_leaf))

def prep_init(Poplar1, verify_key, agg_id, agg_param,
              nonce, public_share, input_share):
    dst = VERSION + b' poplar1'
    (level, prefixes) = agg_param
    (key, corr_seed, corr_inner, corr_leaf) = \
        Poplar1.decode_input_share(input_share)

    # Evaluate the IDPF key at the given set of prefixes.
    value = Poplar1.Idpf.eval(
        agg_id, public_share, key, level, prefixes)

    # Get correlation shares for the given level of the IDPF tree.
    #
    # Implementation note: Computing the shares of `(a, b, c)`
    # requires expanding PRG seeds into a vector of field elements
    # of length proportional to the level of the tree. Typically
    # the IDPF will be evaluated incrementally beginning with
    # `level == 0`. Implementations can save computation by
    # storing the intermediate PRG state between evaluations.
    corr_prg = Poplar1.Idpf.Prg(corr_seed, dst + byte(agg_id))
    for current_level in range(level+1):
        Field = Poplar1.Idpf.current_field(current_level)
        (a_share, b_share, c_share) = corr_prg.next_vec(Field, 3)
    (A_share, B_share) = corr_inner[2*level:2*(level+1)] \
        if level < Poplar1.Idpf.BITS - 1 else corr_leaf

    # Compute the Aggregator's first round of the sketch. These are
    # called the "masked input values" [BBCGGI21, Appendix C.4].
    Field = Poplar1.Idpf.current_field(level)
    verify_rand_prg = Poplar1.Idpf.Prg(verify_key,
        dst + Poplar1.verify_context(nonce, level, prefixes))
    verify_rand = verify_rand_prg.next_vec(Field, len(prefixes))
    sketch_share = [a_share, b_share, c_share]
    out_share = []
    for (i, r) in enumerate(verify_rand):
        (data_share, auth_share) = value[i]
        sketch_share[0] += data_share * r
        sketch_share[1] += data_share * r^2
        sketch_share[2] += auth_share * r
        out_share.append(data_share)

    prep_mem = sketch_share \
                + [A_share, B_share, Field(agg_id)] \
                + out_share
    return (b'ready', level, prep_mem)

def prep_next(Poplar1, prep_state, opt_sketch):
    (step, level, prep_mem) = prep_state
    Field = Poplar1.Idpf.current_field(level)

    # Aggregators exchange masked input values (step (3.)
    # of [BBCGGI21, Appendix C.4]).
    if step == b'ready' and opt_sketch == None:
        sketch_share, prep_mem = prep_mem[:3], prep_mem[3:]
        return ((b'sketch round 1', level, prep_mem),
                Field.encode_vec(sketch_share))

    # Aggregators exchange evaluated shares (step (4.)).
    elif step == b'sketch round 1' and opt_sketch != None:
        prev_sketch = Field.decode_vec(opt_sketch)
        if len(prev_sketch) == 0:
            prev_sketch = Field.zeros(3)
        elif len(prev_sketch) != 3:
            raise ERR_INPUT # prep message malformed
        (A_share, B_share, agg_id), prep_mem = \
            prep_mem[:3], prep_mem[3:]
        sketch_share = [
            agg_id * (prev_sketch[0]^2 \
                        - prev_sketch[1]
                        - prev_sketch[2]) \
                + A_share * prev_sketch[0] \
                + B_share
        ]
        return ((b'sketch round 2', level, prep_mem),
                Field.encode_vec(sketch_share))

    elif step == b'sketch round 2' and opt_sketch != None:
        prev_sketch = Field.decode_vec(opt_sketch)
        if len(prev_sketch) == 0:
            prev_sketch = Field.zeros(1)
        elif len(prev_sketch) != 1:
            raise ERR_INPUT # prep message malformed
        if prev_sketch[0] != Field(0):
            raise ERR_VERIFY
        return prep_mem # Output shares

    raise ERR_INPUT # unexpected input

def prep_shares_to_prep(Poplar1, agg_param, prep_shares):
    if len(prep_shares) != 2:
        raise ERR_INPUT # unexpected number of prep shares
    (level, _) = agg_param
    Field = Poplar1.Idpf.current_field(level)
    sketch = vec_add(Field.decode_vec(prep_shares[0]),
                     Field.decode_vec(prep_shares[1]))
    if sketch == Field.zeros(len(sketch)):
        # In order to reduce communication overhead, let the
        # empty string denote the zero vector of the required
        # length.
        return b''
    return Field.encode_vec(sketch)

def out_shares_to_agg_share(Poplar1, agg_param, out_shares):
    (level, prefixes) = agg_param
    Field = Poplar1.Idpf.current_field(level)
    agg_share = Field.zeros(len(prefixes))
    for out_share in out_shares:
        agg_share = vec_add(agg_share, out_share)
    return Field.encode_vec(agg_share)

def agg_shares_to_result(Poplar1, agg_param,
                         agg_shares, _num_measurements):
    (level, prefixes) = agg_param
    Field = Poplar1.Idpf.current_field(level)
    agg = Field.zeros(len(prefixes))
    for agg_share in agg_shares:
        agg = vec_add(agg, Field.decode_vec(agg_share))
    return list(map(lambda x: x.as_unsigned(), agg))

def encode_input_shares(Poplar1, keys,
                        corr_seed, corr_inner, corr_leaf):
    input_shares = []
    for (key, seed, inner, leaf) in zip(keys,
                                        corr_seed,
                                        corr_inner,
                                        corr_leaf):
        encoded = Bytes()
        encoded += key
        encoded += seed
        encoded += Poplar1.Idpf.FieldInner.encode_vec(inner)
        encoded += Poplar1.Idpf.FieldLeaf.encode_vec(leaf)
        input_shares.append(encoded)
    return input_shares

def decode_input_share(Poplar1, encoded):
    l = Poplar1.Idpf.KEY_SIZE
    key, encoded = encoded[:l], encoded[l:]
    l = Poplar1.Idpf.Prg.SEED_SIZE
    corr_seed, encoded = encoded[:l], encoded[l:]
    l = Poplar1.Idpf.FieldInner.ENCODED_SIZE \
        * 2 * (Poplar1.Idpf.BITS - 1)
    encoded_corr_inner, encoded = encoded[:l], encoded[l:]
    corr_inner = Poplar1.Idpf.FieldInner.decode_vec(
        encoded_corr_inner)
    l = Poplar1.Idpf.FieldLeaf.ENCODED_SIZE * 2
    encoded_corr_leaf, encoded = encoded[:l], encoded[l:]
    corr_leaf = Poplar1.Idpf.FieldLeaf.decode_vec(
        encoded_corr_leaf)
    if len(encoded) != 0:
        raise ERR_INPUT
    return (key, corr_seed, corr_inner, corr_leaf)

def encode_agg_param(Poplar1, level, prefixes):
    if level > 2^16 - 1:
        raise ERR_INPUT # level too deep
    if len(prefixes) > 2^16 - 1:
        raise ERR_INPUT # too many prefixes
    encoded = Bytes()
    encoded += I2OSP(level, 2)
    encoded += I2OSP(len(prefixes), 2)
    packed = 0
    for (i, prefix) in enumerate(prefixes):
        packed |= prefix << ((level+1) * i)
    l = floor(((level+1) * len(prefixes) + 7) / 8)
    encoded += I2OSP(packed, l)
    return encoded

def verify_context(Poplar1, nonce, level, prefixes):
    if len(nonce) > 255:
        raise ERR_INPUT # nonce too long
    context = Bytes()
    context += byte(254)
    context += byte(len(nonce))
    context += nonce
    context += Poplar1.encode_agg_param(level, prefixes)
    return context

def gen(IpdfPoplar, alpha, beta_inner, beta_leaf):
    if alpha >= 2^IdpfPoplar.BITS:
        raise ERR_INPUT # alpha too long
    if len(beta_inner) != IdpfPoplar.BITS - 1:
        raise ERR_INPUT # beta_inner vector is the wrong size

    init_seed = [
        gen_rand(IdpfPoplar.Prg.SEED_SIZE),
        gen_rand(IdpfPoplar.Prg.SEED_SIZE),
    ]

    seed = init_seed.copy()
    ctrl = [Field2(0), Field2(1)]
    correction_words = []
    for level in range(IdpfPoplar.BITS):
        keep = (alpha >> (IdpfPoplar.BITS - level - 1)) & 1
        lose = 1 - keep
        bit = Field2(keep)

        (s0, t0) = IdpfPoplar.extend(seed[0])
        (s1, t1) = IdpfPoplar.extend(seed[1])
        seed_cw = xor(s0[lose], s1[lose])
        ctrl_cw = (
            t0[0] + t1[0] + bit + Field2(1),
            t0[1] + t1[1] + bit,
        )

        x0 = xor(s0[keep], seed_cw) if ctrl[0] == Field2(1) \
                else s0[keep]
        x1 = xor(s1[keep], seed_cw) if ctrl[1] == Field2(1) \
                else s1[keep]
        (seed[0], w0) = IdpfPoplar.convert(level, x0)
        (seed[1], w1) = IdpfPoplar.convert(level, x1)
        ctrl[0] = t0[keep] + ctrl[0] * ctrl_cw[keep]
        ctrl[1] = t1[keep] + ctrl[1] * ctrl_cw[keep]

        b = beta_inner[level] if level < IdpfPoplar.BITS-1 \
                else beta_leaf
        if len(b) != IdpfPoplar.VALUE_LEN:
            raise ERR_INPUT # beta too long or too short

        w_cw = vec_add(vec_sub(b, w0), w1)
        if ctrl[1] == Field2(1):
            w_cw = vec_neg(w_cw)
        correction_words.append((seed_cw, ctrl_cw, w_cw))

    public_share = IdpfPoplar.encode_public_share(correction_words)
    return (public_share, init_seed)

def eval(IdpfPoplar, agg_id, public_share, init_seed,
         level, prefixes):
    if agg_id >= IdpfPoplar.SHARES:
        raise ERR_INPUT # invalid aggregator ID
    if level >= IdpfPoplar.BITS:
        raise ERR_INPUT # level too deep
    if len(set(prefixes)) != len(prefixes):
        raise ERR_INPUT # candidate prefixes are non-unique

    correction_words = IdpfPoplar.decode_public_share(public_share)
    out_share = []
    for prefix in prefixes:
        if prefix >= 2^(level+1):
            raise ERR_INPUT # prefix too long

        # The Aggregator's output share is the value of a node of
        # the IDPF tree at the given `level`. The node's value is
        # computed by traversing the path defined by the candidate
        # `prefix`. Each node in the tree is represented by a seed
        # (`seed`) and a set of control bits (`ctrl`).
        seed = init_seed
        ctrl = Field2(agg_id)
        for current_level in range(level+1):
            bit = (prefix >> (level - current_level)) & 1

            # Implementation note: Typically the current round of
            # candidate prefixes would have been derived from
            # aggregate results computed during previous rounds. For
            # example, when using `IdpfPoplar` to compute heavy
            # hitters, a string whose hit count exceeded the given
            # threshold in the last round would be the prefix of each
            # `prefix` in the current round. (See [BBCGGI21,
            # Section 5.1].) In this case, part of the path would
            # have already been traversed.
            #
            # Re-computing nodes along previously traversed paths is
            # wasteful. Implementations can eliminate this added
            # complexity by caching nodes (i.e., `(seed, ctrl)`
            # pairs) output by previous calls to `eval_next()`.
            (seed, ctrl, y) = IdpfPoplar.eval_next(seed, ctrl,
                correction_words[current_level], current_level, bit)
        out_share.append(y if agg_id == 0 else vec_neg(y))
    return out_share

# Compute the next node in the IDPF tree along the path determined by
# a candidate prefix. The next node is determined by `bit`, the bit
# of the prefix corresponding to the next level of the tree.
#
# TODO Consider implementing some version of the optimization
# discussed at the end of [BBCGGI21, Appendix C.2]. This could on
# average reduce the number of AES calls by a constant factor.
def eval_next(IdpfPoplar, prev_seed, prev_ctrl,
              correction_word, level, bit):
    (seed_cw, ctrl_cw, w_cw) = correction_word
    (s, t) = IdpfPoplar.extend(prev_seed)
    if prev_ctrl == Field2(1):
        s[0] = xor(s[0], seed_cw)
        s[1] = xor(s[1], seed_cw)
        t[0] = t[0] + ctrl_cw[0]
        t[1] = t[1] + ctrl_cw[1]

    next_ctrl = t[bit]
    (next_seed, y) = IdpfPoplar.convert(level, s[bit])
    if next_ctrl == Field2(1):
        y = vec_add(y, w_cw)
    return (next_seed, next_ctrl, y)

def extend(IdpfPoplar, seed):
    dst = VERSION + b' idpf poplar extend'
    prg = IdpfPoplar.Prg(seed, dst)
    s = [
        prg.next(IdpfPoplar.Prg.SEED_SIZE),
        prg.next(IdpfPoplar.Prg.SEED_SIZE),
    ]
    b = OS2IP(prg.next(1))
    t = [Field2(b & 1), Field2((b >> 1) & 1)]
    return (s, t)

def convert(IdpfPoplar, level, seed):
    dst = VERSION + b' idpf poplar convert'
    prg = IdpfPoplar.Prg(seed, dst)
    next_seed = prg.next(IdpfPoplar.Prg.SEED_SIZE)
    Field = IdpfPoplar.current_field(level)
    w = prg.next_vec(Field, IdpfPoplar.VALUE_LEN)
    return (next_seed, w)

def encode_public_share(IdpfPoplar, correction_words):
    encoded = Bytes()
    packed_ctrl = 0
    for (level, (_, ctrl_cw, _)) \
        in enumerate(correction_words):
        packed_ctrl |= ctrl_cw[0].as_unsigned() << (2*level)
        packed_ctrl |= ctrl_cw[1].as_unsigned() << (2*level+1)
    l = floor((2*IdpfPoplar.BITS + 7) / 8)
    encoded += I2OSP(packed_ctrl, l)
    for (level, (seed_cw, _, w_cw)) \
        in enumerate(correction_words):
        Field = IdpfPoplar.current_field(level)
        encoded += seed_cw
        encoded += Field.encode_vec(w_cw)
    return encoded

def decode_public_share(IdpfPoplar, encoded):
    l = floor((2*IdpfPoplar.BITS + 7) / 8)
    encoded_ctrl, encoded = encoded[:l], encoded[l:]
    packed_ctrl = OS2IP(encoded_ctrl)
    correction_words = []
    for level in range(IdpfPoplar.BITS):
        Field = IdpfPoplar.current_field(level)
        ctrl_cw = (Field2(packed_ctrl & 1),
                   Field2((packed_ctrl >> 1) & 1))
        packed_ctrl >>= 2
        l = IdpfPoplar.Prg.SEED_SIZE
        seed_cw, encoded = encoded[:l], encoded[l:]
        l = Field.ENCODED_SIZE * IdpfPoplar.VALUE_LEN
        encoded_w_cw, encoded = encoded[:l], encoded[l:]
        w_cw = Field.decode_vec(encoded_w_cw)
        correction_words.append((seed_cw, ctrl_cw, w_cw))
    if len(encoded) != 0:
        raise ERR_DECODE
    return correction_words

9. Security Considerations

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

VDAFs have two essential security goals:¶

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

Note that, to achieve robustness, it is important to ensure that the verification key distributed to the Aggregators (verify_key, see Section 5.2) is never revealed to the Clients.¶

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

10. IANA Considerations

This document makes no request of IANA.¶

Acknowledgments

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

Test Vectors

• NOTE Machine-readable test vectors can be found at https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/test_vec.¶

Test vectors cover the generation of input shares and the conversion of input shares into output shares. Vectors specify the verification key, measurements, aggregation parameter, and any parameters needed to construct the VDAF. (For example, for Prio3AesSum, the user specifies the number of bits for representing each summand.)¶

Byte strings are encoded in hexadecimal To make the tests deterministic, gen_rand() was replaced with a function that returns the requested number of 0x01 octets.¶

verify_key: "01010101010101010101010101010101"
upload_0:
  measurement: 1
  nonce: "01010101010101010101010101010101"
  public_share: >-
  input_share_0: >-
    a46091d07a930ace79bb11d1d03596bf4f921f7e70b853cf6b8d4969287dfb504a1d
    ea6b0eb01a5f50ad24225186c104
  input_share_1: >-
    0101010101010101010101010101010101010101010101010101010101010101
  round_0:
    prep_share_0: >-
      cdbbf14ef0c19728578ce1e50411e1e26eb49eea4dadfada9ed3d13c71d846b8
    prep_share_1: >-
      32440eb00f3e68d95b9f6e2e856cf534948881bf63c24d315e4954159611ebb5
    prep_message: >-
  out_share_0:
    - 11844627344580086478
  out_share_1:
    - 6602116724834497844
agg_share_0: >-
  a46091d07a930ace
agg_share_1: >-
  5b9f6e2e856cf534
agg_result: 1

bits: 8
verify_key: "01010101010101010101010101010101"
upload_0:
  measurement: 100
  nonce: "01010101010101010101010101010101"
  public_share: >-
  input_share_0: >-
    a46091d17a930aaf4d02cff822f817a3d10d4d74d526daf00112da46b9b33deaa222
    0ff24fe104b34007f47c4531dc7471b6832358f5d6859fb3b2112e79ec6eb940251d
    14855b16689ab3cfd9100b568729aadfb5558a3bd0277d3ae7c5d967e3525839f26f
    5cb210e4fb0b0aeb0394ce0791de9c4c07f8840dcdd1ce09bb2179bb11d1d03596be
    4f921f7f70b853ce99c2bd80daf08029b7bb7820b7a7632e3b9200ff98adde77a833
    6469ba7fa61ebf135e7f303568ddf63e39978954c40a3fcfa08412f5f03f61699643
    85f167c68c3ae5205e1c3dd075fc76224bf9407ef3fd54334bec36d453460381ee54
    4027bb520f55394873f5dff846794007cc7d77e903b1474095c61140d1e488a98e00
    f433c22491825191261e982affc44e20756db024e0dcdda9746aba73d2ec9a083953
    4a1d0178802fc0ee9393160b34dca94999c91fa788b8308b15880ea3146aabf38e2f
    a4d715c0be7e183fe7f158f47438eaa5a55406da5be56d724afd86e579bc076c669c
    334c93fb724891e0b72c8737a9f65c2d3d56bae20dce052f43293712a603f8a8b9e8
    bcdafad8420ae14201f9a8b4891e9b60dd418d0f71ee2395b9026cb4978f901765e0
    155b127a4fa95a0e0751b95d9ada245c78df820e4bc7f0c9329a5bbad2ee85688cea
    47336da7a798705b6c465678363574e844d1974f6f369e9ef13e920d383f580aa382
    8b7641a77f352b73134886c58e822040747311a9b094137c5f46d6e9ed05bb17d9c7
    0f01b5491805ac83c447035560b00b8e7049c334fcfa003941fa0e307cf52884e461
    8f794848936bfcd8d7f895781d27b006673a6349bdfa04c8c3506b2b64efff5335f7
    6723bce0f1092571cae3d125dcabbb6da4867963ffd29172ef699ecf010101010101
    0101010101010101010132dd9fa5604fadc81ff1636918a6db69
  input_share_1: >-
    01010101010101010101010101010101010101010101010101010101010101010101
    0101010101010101010101010101646430cb3212da0c23bff5618647609c
  round_0:
    prep_share_0: >-
      ede97ab0d3d2dc256a1f82261078cc06b5a423c5f221e9fa029ba023bc71e70fd1
      9e2db541507b4171409bc06c3ac292646430cb3212da0c23bff5618647609c
    prep_share_1: >-
      1216854f2c2d23be95e07dd9ef8733fb779dce023a0e4f7147b74c75c5fd1723a1
      1b7a116c0a89c6b97471597eb2a83c32dd9fa5604fadc81ff1636918a6db69
    prep_message: >-
      56b9af6e525d77c43c4e96089ee1bbf5
  out_share_0:
    - 231691668211413349799981210015670190003
  out_share_1:
    - 108590698709525113146884563352230576306
agg_share_0: >-
  ae4e3131e0f5bea01ba67704684fbfb3
agg_share_1: >-
  51b1cece1f0a4143e45988fb97b040b2
agg_result: 100

buckets: [1, 10, 100]
verify_key: "01010101010101010101010101010101"
upload_0:
  measurement: 50
  nonce: "01010101010101010101010101010101"
  public_share: >-
  input_share_0: >-
    a46091d17a930aaf4d02cff822f817a3d10d4d74d526daf00112da46b9b33deaa222
    0ff24fe104b34007f47c4531dc7471b6832358f5d6859fb3b2112e79ec6e79bb11d1
    d03596be4f921f7f70b853cec138b5a4f1a798810bde46a7265fd6bed9bb8cc8d058
    bb224cc75b17f2ebc7f7b6c7a708b11022329fab97ed7d23309116a6fb53d1d9ff45
    9a67eafcddbd190e69069ef82f361a6d4c9616ec3396776834a7d3d5848409cdd89f
    ec7920a858074db9c2e7adfc4b3397339532f80e327f25f31975e3f5d149ba8c8d5b
    f101488f4882ab2b38ddedc84f814435c2b227ffeee7affc5076d78819d7ce68f247
    355af67c52fc7ef9bbcbf8bc910af7c047a28dfeb4f92158934ce93dac747c1b55a8
    f5efb18e944e0579602c918e670f8d543b530d8ba3730d0a43e8bcd2d54f736c4db1
    ac64c9c49dc5ab20c0ac0cf6530b01010101010101010101010101010101d451ca17
    ca18344ac4925510c71e5ac8
  input_share_1: >-
    01010101010101010101010101010101010101010101010101010101010101010101
    0101010101010101010101010101ef377f05b86ffa85ed011213230bb1d5
  round_0:
    prep_share_0: >-
      a1a7093768a9f8a3c01e31f7e9c6dc7925407d03831c510799fcf30a03c61bcd35
      78329d502e06cd01e994c227d3d1afef377f05b86ffa85ed011213230bb1d5
    prep_share_1: >-
      5e58f6c8975607403fe1ce081639238809c0227ef8c371e897e7950d763d8a37f7
      ee6acb826a97f3ef37fd59e6a95511d451ca17ca18344ac4925510c71e5ac8
    prep_message: >-
      3b66b5127277cecf29934703e415eb1d
  out_share_0:
    - 218494809353158975333103031759951894435
    - 277877721980181983205633439801202130410
    - 215511796844427285058159435311010143348
    - 151150421348124479376036084370309246062
  out_share_1:
    - 121787557567779487613762741607948871774
    - 62404644940756479741232333566698635799
    - 124770570076511177888706338056890622862
    - 189131945572813983570829688997591520147
agg_share_0: >-
  a46091d17a930aaf4d02cff822f817a3d10d4d74d526daf00112da46b9b33deaa2220f
  f24fe104b34007f47c4531dc7471b6832358f5d6859fb3b2112e79ec6e
agg_share_1: >-
  5b9f6e2e856cf534b2fd3007dd07e85e2ef2b28b2ad924f3feed25b9464cc2175dddf0
  0db01efb30bff80b83bace238e8e497cdca70a295e604c4deed1861393
agg_result: [0, 0, 1, 0]

bits: 4
upload_0:
  measurement: 13
  nonce: "01010101010101010101010101010101"
  public_share: >-
    9a00000000000000000000000000000000ffffffff00000000bd5fef66a8181fb000
    000000000000000000000000000000ffffffff00000000dbf324f2b4351711000000
    00000000000000000000000000ffffffff00000000984542b43f4008f40000000000
    00000000000000000000000000000000000000000000000000000000000000000000
    00000000000000000113aec130616497e38eaafb95841bad26f773979c71d5422d1c
    875c6c007575ce
  input_share_0: >-
    01010101010101010101010101010101010101010101010101010101010101011926
    0bd7b833c3b313d7ba6d15a9f16fec202c82ddf2848909acc634a461a6b8b7758b10
    e0adb885083645e2bbe0c575137f41a015e6cc2ce8fed760ca08589f957eef3d1667
    07d9044d22411142e73568bba4e2f7150dab3316e30966a1d73f660ec47316af0198
    e8682372a232bcef
  input_share_1: >-
    01010101010101010101010101010101010101010101010101010101010101013422
    59973b9807dbbb662f6a81067220a977e09ad54ba5048edb2118c95bb8f3ce11a0dd
    6f0bda66dc2dc274f26ec69e091747a0b26b57ad629d905fae98204c5171179db567
    12e5dd66f792eddc6da955476f1471eb5c12aac55b35c1d80a2022d30da1ac00dfc5
    8c581e8304ab59f1

verify_key: "01010101010101010101010101010101"
agg_param: (0, [0, 1])
upload_0:
  round_0:
    prep_share_0: >-
      48c3581799a109398b1943c10957b8c2536f292900de6436
    prep_share_1: >-
      8377f3fcd4d2d1f33233948d19fe2c385f97c52d8e007f76
    prep_message: >-
      cc3b4c146e73db2cbd4cd84e2355e4fab306ee568edee3ac
  round_1:
    prep_share_0: >-
      64ce1f88d77b2a65
    prep_share_1: >-
      9b31e0762884d59c
    prep_message: >-
  out_share_0:
    - 18352387916526274078
    - 14559030685141423577
  out_share_1:
    - 94356152888310243
    - 3887713384273160745
agg_share_0: >-
  feb0c79330b8461eca0c114565ddd5d9
agg_share_1: >-
  014f386bcf47b9e335f3eeb99a222a29
agg_result: [0, 1]

verify_key: "01010101010101010101010101010101"
agg_param: (1, [0, 1, 2, 3])
upload_0:
  round_0:
    prep_share_0: >-
      37c72ca578a032eaeffbd9e63d29e6eaa205d9d67628d055
    prep_share_1: >-
      ab956611d15cfcef0773a4e435fdcb83909785aa50c17513
    prep_message: >-
      e35c92b749fd2fd9f76f7eca7327b26d329d5f81c6ea4567
  round_1:
    prep_share_0: >-
      ee63dfb515814411
    prep_share_1: >-
      119c2049ea7ebbf0
    prep_message: >-
  out_share_0:
    - 7623875273889432259
    - 4753532626118505607
    - 9999595461072379379
    - 1320535832873937471
  out_share_1:
    - 10822868795525152062
    - 13693211443296078714
    - 8447148608342204942
    - 17126208236540646851
agg_share_0: >-
  69cd722b281042c341f7f0bcf805f4878ac5b3177ad3ddf312537c57efa02a3f
agg_share_1: >-
  96328dd3d7efbd3ebe080f4207fa0b7a753a4ce7852c220eedac83a7105fd5c3
agg_result: [0, 0, 0, 1]

verify_key: "01010101010101010101010101010101"
agg_param: (2, [0, 2, 4, 6])
upload_0:
  round_0:
    prep_share_0: >-
      7eb8dbda5ea6807e53f22d3f74a43f59ed83ef578b070654
    prep_share_1: >-
      567632dc63ec2294c6f030077726f87ec11e2fc53a38f640
    prep_message: >-
      d52f0eb6c292a3121ae25d47ebcb37d6aea21f1dc53ffc93
  round_1:
    prep_share_0: >-
      9e92c793980229fd
    prep_share_1: >-
      616d386b67fdd604
    prep_message: >-
  out_share_0:
    - 1646250657834468215
    - 16923505979406688621
    - 18330814903122515861
    - 12155602082232726549
  out_share_1:
    - 16800493411580116106
    - 1523238090007895700
    - 115929166292068460
    - 6291141987181857773
agg_share_0: >-
  16d8a840477a0377eadc5f0610bc216dfe642308980dd795a8b15fb0ce7af415
agg_share_1: >-
  e92757beb885fc8a1523a0f8ef43de94019bdcf667f2286c574ea04e31850bed
agg_result: [0, 0, 0, 1]

verify_key: "01010101010101010101010101010101"
agg_param: (3, [1, 3, 5, 7, 9, 13, 15])
upload_0:
  round_0:
    prep_share_0: >-
      246777985de3e9fd0ea9e8ea429ae3c7255c0b2aa84cff6bd5c0079a122be38b53
      4e75cbfe8abb2c54b5b277089bb2708b274a17ce26df0ea9881487ac852eee30c1
      8ba6dc40beb7f33e5003fdf065eec844e320d0b9ff49b0b29f5bbbcbcc8d
    prep_share_1: >-
      3dea0b9e72389ea0d62521e91214c654ebf9b2a3d855a32219755216f4cb8eca0d
      864838efc99f1a5d2adecf87e4ff0b839fb7191d4c58c5fd6753c0c6a2fb155071
      efff9719f0e9645d70b398ca24ecb2fa8fbb56a6a90ce6d95b675e1e28f2
    prep_message: >-
      62518336d01c889de4cf0ad354afaa1c1155bdce80a2a28def3559b106f7725560
      d4be04ee545a46b1e091469080b17c0ec70130eb7337d4a6ef684873282a030133
      7ba6735aafa1579bc0b796ba8adb7b3f72dc2760a856978bfac319e9f592
  round_1:
    prep_share_0: >-
      40521213baba54c83b4ddc0cb1e3e4c16897bb8831cc12ea91a82927dd30ae58
    prep_share_1: >-
      3fadedec4545ab37c4b223f34e1c1b3e97684477ce33ed156e57d6d822cf5195
    prep_message: >-
  out_share_0:
    - 27022242238524926041754024638978288940369802983157338113702795433392718442891
    - 8433446103891669428510589487794410319530091467635619512122120272876905203470
    - 31927171851194534364500488903327032909992419490289770138187660233081601268509
    - 13674389066590417186009193737398514883303633790292716630453797389670854762435
    - 49245655988555337792760994399532100867010365510706353464574012401106060986059
    - 38418164104898327452350816690091778787588836972103923142413897241001677348897
    - 17473652484574293745985786122460988541722879368489070298827855759184581179349
  out_share_1:
    - 30873802380133171670031467865365664986265189349662943906025996570563846377058
    - 49462598514766428283274903016549543607104900865184662507606671731079659616479
    - 25968872767463563347285003601016921016642572842530511881541131770874963551440
    - 44221655552067680525776298766945439043331358542527565389274994614285710057514
    - 8650388630102759919024498104811853059624626822113928555154779602850503833890
    - 19477880513759770259434675814252175139046155360716358877314894762954887471053
    - 40422392134083803965799706381882965384912112964331211720900936244771983640600
agg_share_0: >-
  3bbe0c0f2a3ecf406419e5a8636f13c1102df2df86b4dec82daa4669b24f718b12a529
  4d99b7a604a6944dbe1b81cfcc98e4a5a1fa0d08ac727f620ee0b4870e4696238e0b88
  71bb6d93ab6c8fc1680da244dfc59bd0b6f70b8dae1f4c1a1b1d1e3b6e1053afb851cc
  ae5d9ddabb8bff59afea5f6f905e0b2dd1b8a9cc911fc36ce00db4dfb57d404c6bddc9
  0a93a35fa5a476634aab5c2c46b467e2a6ce46cb54efe909f5c78891ac58d782afb5a4
  4dbc5b2f0c190d4ebe8ca7eb1e4af8e02126a1bca0b523a4c67c7176bebb04afed7b50
  6fc5c343e3e7727d3027d68937d5
agg_share_1: >-
  4441f3f0d5c130bf9be61a579c90ec3eefd20d20794b2137d255b9964db08e626d5ad6
  b2664859fb596bb241e47e3033671b5a5e05f2f7538d809df11f4b78df3969dc71f477
  8e44926c5493703e97f25dbb203a642f4908f47251e0b3e5e4d061c491efac5047ae33
  51a26225447400a65015a0906fa1f4d22e4756336ee02a131ff24b204a82bfb3942236
  f56c5ca05a5b899cb554a3d3b94b981d5931b9222b1016f60a38776e53a7287d504a5b
  b243a4d0f3e6f2b141735814e1b5071fcd595e435f4adc5b39838e894144fb501284af
  903a3cbc1c188d82cfd82976c818
agg_result: [0, 0, 0, 0, 0, 1, 0]

Authors' Addresses