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 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-01'.¶

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.
        input_shares = Daf.measurement_to_input_shares(measurement)

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

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

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

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.
        input_shares = Vdaf.measurement_to_input_shares(measurement)

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

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

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

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

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

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 = 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:]

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 "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 input_shares

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

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

    out_share = Prio3.Flp.truncate(input_share)

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

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

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

    if inbound is None:
        return (prep, prep_msg)

    k_joint_rand_check = Prio3.decode_prep_msg(inbound)
    if k_joint_rand_check != k_joint_rand:
        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):
    agg = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
    for agg_share in agg_shares:
        agg = vec_add(agg, Prio3.Flp.Field.decode_vec(agg_share))
    return list(map(lambda x: x.as_unsigned(), agg))

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 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

8. Poplar1

• TODO Update this section in light of removing the public parameter and replacing the verification parameter.¶

• NOTE The spec for Poplar1 is still a work-in-progress. A partial implementation can be found at https://github.com/divviup/libprio-rs/blob/main/src/vdaf/poplar1.rs. The verification logic is nearly complete, however as of this draft the code is missing the IDPF. An implementation of the IDPF can be found at https://github.com/google/distributed_point_functions/.¶

This section specifies Poplar1, a VDAF for the following task. Each Client holds a 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)

9. Security Considerations

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

VDAFs have two essential security goals:¶

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

Note that, to achieve robustness, it is important to ensure that the verification key distributed to the Aggregators (verify_key, see Section 8.2.1) is never revealed to the Clients.¶

It is also possible to consider a stronger form of robustness, where the attacker also controls a subset of Aggregators (see [BBCGGI19], Section 6.3). To satisfy this stronger notion of robustness, it is necessary to prevent the attacker from sharing the verification key with the Clients. It is therefore RECOMMENDED that the Aggregators generate verify_key only after a set of Client inputs has been collected for verification, and 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"
  input_share_0: >-
    ae5483343eb35a52fcb36a62271a7ddb47f09d0ea2c6613807f84ac2e16814c82bca
    bdc9db5080fdf4f4f778734644fc
  input_share_1: >-
    0101010101010101010101010101010101010101010101010101010101010101
  round_0:
    prep_share_0: >-
      22ce013d3aaa7e7574ed01fe1d074cd845dfbbbc5901cabd487d4e2e228274cc
    prep_share_1: >-
      dd31fec1c555818c51ab7ccac14ca5b00aae1c33d835c76dfa9406011a92a8e9
    prep_message: >-
  out_share_0:
    - 12561809521056635474
  out_share_1:
    - 5884934548357948848
agg_share_0: >-
  ae5483343eb35a52
agg_share_1: >-
  51ab7ccac14ca5b0
agg_result: [1]

bits: 8
verify_key: "01010101010101010101010101010101"
upload_0:
  measurement: 100
  nonce: "01010101010101010101010101010101"
  input_share_0: >-
    ae5483353eb35a3371beec8f796e9afd086cb72d05a83a3dbefbe273acb0410787b1
    afba2065df5389011fd8963091e3004fa07fc91018af378da47c89abf1bd85047e40
    874e2cdc5f3bc48f363b89f746770a402a777bed31b5a10c7319b3908d72de0c6512
    15ba78d3cf681e07c564c0b4a9a4508df645bad8fef61e3ddf37fcb36a63271a7dbe
    47f09d0ea2c661396ad006d8915d149ad88f9b1cdb86e1d13d683c359b7ac899a245
    4316051e4e235dfd566f3459c336826555ed7f1baabf241e9a697d458912f3bd3778
    225e832b78cd4f17e57c9b9678cf6043894aff0d0f2e06828982ac3493ae5ded0c98
    86ea13d52bc0f209dc2f4e676c42b95b548a413f67b03ff18e9e6b699338400e9dff
    a800563abb495364acffc17126bf0bf8ff3c5caba82333e91352e03c637d44dc4db1
    59a1b19d8db4d5a3fce356f6f2fca4adc9bcf65bec8d4d962b2b40f7ea413aead099
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    06b171ab61b025b7c19aad8de495a8a97af2baab6d1240d30df417d1cc0fe7a90ada
    ad8115924c0987fe1d16abe0c8a3c297d58a3112b818df72a10a41b34aa6b4ae370b
    1340a6085c8dcd597eead5d2584fdb160f0a086a56ea6a7736666ae34d3012fdb2c2
    4af3d4b2a6ae735edfe837eaab1309eaa2d8273e7dbfe0fd4166d545ce8354e1237b
    48456715d12e38d02cd64c96b9daa01a2281d8a930817088c648b7c115e1550ada14
    b6072ada49be3c7e3f184db2461160d29937caa97db6020a5598063f1dc05653d1d3
    80b34e923bd7170eeeb811bfc3ce12c1df55cf552e986a823743fac4723a48bda6c1
    8ffec653c1f182890197e9fc74631dcbd0283c4258933c03aee9404f010101010101
    01010101010101010101fb0c701f7c07b9407a4a7b77d1ea017e
  input_share_1: >-
    01010101010101010101010101010101010101010101010101010101010101010101
    01010101010101010101010101012d7667bffd0f81b078896503385f6f13
  round_0:
    prep_share_0: >-
      9f7aca77f790b930b46e8cd786ff1a239aa00e7aaaa734cc2bbcb121eb7c5bc00a
      ef22fd95a24cd1a0054bde0dba06062d7667bffd0f81b078896503385f6f13
    prep_share_1: >-
      60853588086f46b34b9173287900e5de2becb1bdb8a8009d2cdc258674f08e8157
      e3a202a38282c20e220e733ab61e4cfb0c701f7c07b9407a4a7b77d1ea017e
    prep_message: >-
      d67a17a0810838f002c31e74e9b56e6d
  out_share_0:
    - 242787699414660215404830418280405596120
  out_share_1:
    - 97494667506278247542035355087495170189
agg_share_0: >-
  b6a735c5636efee29c0c1455e0c0f7d8
agg_share_1: >-
  4958ca3a9c91010163f3ebaa1f3f088d
agg_result: [100]

buckets: [1, 10, 100]
verify_key: "01010101010101010101010101010101"
upload_0:
  measurement: 50
  nonce: "01010101010101010101010101010101"
  input_share_0: >-
    ae5483353eb35a3371beec8f796e9afd086cb72d05a83a3dbefbe273acb0410787b1
    afba2065df5389011fd8963091e3004fa07fc91018af378da47c89abf1bdfcb36a63
    271a7dbe47f09d0ea2c6613956dfe44e1302160dd2ade0205aa0409225caf0f966df
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    c5a9bd78b2e013a50250c8bd44a15ad7d5edac35a58bed81a4088c72430afbd6fe34
    635a737cb7c4d29ffc9947b6b0fb8f3fdede9d8bd495b4d47e8400bded8aa53e4a5a
    2d063c6091c29613e044082b0555ce74c45b823aa8c5804aacdd3dc92a6ac0058755
    7770972dcdc37eefb42eef43a1b401010101010101010101010101010101d5bf864d
    e68bac19204e29697bf9504d
  input_share_1: >-
    01010101010101010101010101010101010101010101010101010101010101010101
    01010101010101010101010101018e2e553b5e45c62e0ced57ec947c8627
  round_0:
    prep_share_0: >-
      93b9dd4a3b46d8941fe7524a5cf1cd47ff8ee9c0c2e9b8230b1b940b665263b7b1
      c4b370652a333ee774ec9cd379b6e78e2e553b5e45c62e0ced57ec947c8627
    prep_share_1: >-
      6c4622b5c4b9274fe018adb5a30e32baa310ddd3e5ed87892dca520d1bed7d0299
      8e38190652ba60a225e19211d77e22d5bf864de68bac19204e29697bf9504d
    prep_message: >-
      5b91d376b8ce6a372ca37e85ef85d66a
  out_share_0:
    - 231724485416847873323492487111470127869
    - 11198307274976669387765744195748249863
    - 180368380143069850478496598824148046307
    - 413446761563421317675646300023681469
  out_share_1:
    - 108557881504090589623373286256430638340
    - 329084059645961793559100029172152516346
    - 159913986777868612468369174543752719903
    - 339868920159375041629190127067877084740
agg_share_0: >-
  ae5483353eb35a3371beec8f796e9afd086cb72d05a83a3dbefbe273acb0410787b1af
  ba2065df5389011fd8963091e3004fa07fc91018af378da47c89abf1bd
agg_share_1: >-
  51ab7ccac14ca5b08e41137086916504f79348d2fa57c5a641041d8c534fbefa784e50
  45df9a209076fee02769cf6e1fffb05f8036efe734c8725b8376540e44
agg_result: [0, 0, 1, 0]

Authors' Addresses