| Internet-Draft | VDAF | February 2023 |
| Barnes, et al. | Expires 28 August 2023 | [Page] |
This document describes Verifiable Distributed Aggregation Functions (VDAFs), a family of multi-party protocols for computing aggregate statistics over user measurements. These protocols are designed to ensure that, as long as at least one aggregation server executes the protocol honestly, individual measurements are never seen by any server in the clear. At the same time, VDAFs allow the servers to detect if a malicious or misconfigured client submitted an input that would result in an incorrect aggregate result.¶
This note is to be removed before publishing as an RFC.¶
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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 (MPC): 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 input measurements.¶
In addition to these MPC-style security goals, VDAFs can be composed with various mechanisms for differential privacy, thereby providing the added assurance that the aggregate result itself does not leak too much information about any one measurement.¶
TODO(issue #94) Provide guidance for local and central DP and point to it here.¶
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:¶
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:¶
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:¶
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.¶
(*) Indicates a change that breaks wire compatibility with the previous draft.¶
04:¶
03:¶
02:¶
01:¶
prep_next() to
prep_shares_to_prep(). (*)¶
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 of type Unsigned is defined, which algorithms are
free to use as desired. Its value SHALL be 4.¶
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).¶
concat(parts: Vec[Bytes]) -> Bytes returns the concatenation of the input
byte strings, i.e., parts[0] || ... || parts[len(parts)-1].¶
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.¶
+--------------+
+---->| 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:¶
The Aggregators convert their input shares into "output shares".¶
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.¶
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 and Preparation are done once per measurement. Aggregation and Unsharding are done over a batch of measurements (more precisely, over the recovered output shares).¶
A concrete DAF specifies an algorithm for the computation needed in each of these stages. The interface of each algorithm is defined in the remainder of this section. In addition, a concrete DAF defines the associated constants and types enumerated in the following table.¶
| Parameter | Description |
|---|---|
ID
|
Algorithm identifier for this DAF. |
SHARES
|
Number of input shares into which each measurement is sharded |
Measurement
|
Type of each measurement |
AggParam
|
Type of aggregation parameter |
OutShare
|
Type of each output share |
AggResult
|
Type of the aggregate result |
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.¶
Each DAF is identified by a unique, 32-bit integer ID. Identifiers for each
(V)DAF specified in this document are defined in Table 18.¶
In order to protect the privacy of its measurements, a DAF Client shards its
measurements into a sequence of input shares. The measurement_to_input_shares
method is used for this purpose.¶
Daf.measurement_to_input_shares(input: Measurement) -> (Bytes, Vec[Bytes])
is the randomized input-distribution algorithm run by each Client. It consumes
the measurement and produces a "public share", distributed to each of the
Aggregators, and a corresponding sequence of input shares, one for each
Aggregator. The length of the output vector MUST be SHARES.¶
Once an Aggregator has received the public share and one of the input shares, the next step is to prepare the input share for aggregation. This is accomplished using the following algorithm:¶
Daf.prep(agg_id: Unsigned, agg_param: AggParam, public_share: Bytes,
input_share: Bytes) -> OutShare is the deterministic preparation algorithm.
It takes as input the public share and one of the input shares generated by a
Client, the Aggregator's unique identifier, and the aggregation parameter
selected by the Collector and returns an output share.¶
The protocol in which the DAF is used MUST ensure that the Aggregator's
identifier is equal to the integer in range [0, SHARES) that matches the index
of input_share in the sequence of input shares output by the Client.¶
Concrete DAFs implementations MAY impose certain restrictions for input shares
and aggregation parameters. Protocols using a DAF MUST ensure that for each
input share and aggregation parameter agg_param, Daf.prep is only called if
Daf.is_valid(agg_param, previous_agg_params) returns True, where
previous_agg_params contains all aggregation parameters that have previously
been used with the same input share.¶
DAFs MUST implement the following function:¶
Daf.is_valid(agg_param: AggParam, previous_agg_params: Vec[AggParam]) ->
Bool: Checks if the agg_param is compatible with all elements of
previous_agg_params.¶
Once an Aggregator holds output shares for a batch of measurements (where batches are defined by the application), it combines them into a share of the desired aggregate result:¶
Daf.out_shares_to_agg_share(agg_param: AggParam, out_shares: Vec[OutShare])
-> agg_share: Bytes is the deterministic aggregation algorithm. It is run by
each Aggregator a set of recovered output shares.¶
Aggregator 0 Aggregator 1 Aggregator 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]
For simplicity, we have written this algorithm in a "one-shot" form, where all output shares for a batch are provided at the same time. Many DAFs may also support a "streaming" form, where shares are processed one at a time.¶
OPEN ISSUE It may be worthwhile to explicitly define the "streaming" API. See issue#47.¶
After the Aggregators have aggregated a sufficient number of output shares, each sends its aggregate share to the Collector, who runs the following algorithm to recover the following output:¶
Daf.agg_shares_to_result(agg_param: AggParam,
agg_shares: Vec[Bytes], num_measurements: Unsigned) -> AggResult is
run by the Collector in order to compute the aggregate result from
the Aggregators' shares. The length of agg_shares MUST be SHARES.
num_measurements is the number of measurements that contributed to
each of the aggregate shares. This algorithm is deterministic.¶
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
=========
QUESTION Maybe the aggregation algorithms should be randomized in order to allow the Aggregators (or the Collector) to add noise for differential privacy. (See the security considerations of [DAP].) Or is this out-of-scope of this document? See https://github.com/ietf-wg-ppm/ppm-specification/issues/19.¶
Securely executing a DAF involves emulating the following procedure.¶
def run_daf(Daf,
agg_param: Daf.AggParam,
measurements: Vec[Daf.Measurement]):
out_shares = [ [] for j in range(Daf.SHARES) ]
for measurement in measurements:
# Each Client shards its measurement into input shares and
# distributes them among the Aggregators.
(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
The inputs to this procedure are the same as the aggregation function computed by the DAF: An aggregation parameter and a sequence of measurements. The procedure prescribes how a DAF is executed in a "benign" environment in which there is no adversary and the messages are passed among the protocol participants over secure point-to-point channels. In reality, these channels need to be instantiated by some "wrapper protocol", such as [DAP], that realizes these channels using suitable cryptographic mechanisms. Moreover, some fraction of the Aggregators (or Clients) may be malicious and diverge from their prescribed behaviors. Section 9 describes the execution of the DAF in various adversarial environments and what properties the wrapper protocol needs to provide in each.¶
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:¶
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.¶
| Parameter | Description |
|---|---|
ID
|
Algorithm identifier for this VDAF |
VERIFY_KEY_SIZE
|
Size (in bytes) of the verification key (Section 5.2) |
NONCE_SIZE ` |
Size (in bytes) of the nonce |
ROUNDS
|
Number of rounds of communication during the Preparation stage (Section 5.2) |
SHARES
|
Number of input shares into which each measurement is sharded (Section 5.1) |
Measurement
|
Type of each measurement |
AggParam
|
Type of aggregation parameter |
Prep
|
State of each Aggregator during Preparation (Section 5.2) |
OutShare
|
Type of each output share |
AggResult
|
Type of the aggregate result |
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.¶
Each VDAF is identified by a unique, 32-bit integer ID. Identifiers for each
(V)DAF specified in this document are defined in Table 18. The following
method is defined for every VDAF:¶
def custom(Vdaf, usage: Unsigned) -> Bytes:
return format_custom(0, Vdaf.ID, usage)
¶
It is used to construct a customization string for an instance of Prg used by
the VDAF. (See Section 6.2.)¶
Sharding transforms a measurement into input shares as it does in DAFs (cf. Section 4.1); in addition, it takes a nonce as input and produces a public share:¶
Vdaf.measurement_to_input_shares(measurement: Measurement, nonce:
Bytes[Vdaf.NONCE_SIZE]) -> (Bytes, Vec[Bytes]) is the randomized
input-distribution algorithm run by each Client. It consumes the measurement
and the nonce and produces a public share, distributed to each of Aggregators,
and the corresponding sequence of input shares, one for each Aggregator.
Depending on the VDAF, the input shares may encode additional information used
to verify the recovered output shares (e.g., the "proof shares" in Prio3
Section 7). The length of the output vector MUST be SHARES.¶
In order to ensure privacy of the measurement, the Client MUST generate the nonce using a cryptographically secure pseudorandom number generator (CSPRNG). (See Section 9 for details.)¶
To recover and verify output shares, the Aggregators interact with one another
over ROUNDS rounds. Prior to each round, each Aggregator constructs an
outbound message. Next, the sequence of outbound messages is combined into a
single message, called a "preparation message". (Each of the outbound messages
are called "preparation-message shares".) Finally, the preparation message is
distributed to the Aggregators to begin the next round.¶
An Aggregator begins the first round with its input share and it begins each subsequent round with the previous preparation message. Its output in the last round is its output share and its output in each of the preceding rounds is a preparation-message share.¶
This process involves a value called the "aggregation parameter" used to map the input shares to output shares. The Aggregators need to agree on this parameter before they can begin preparing inputs for aggregation.¶
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]
To facilitate the preparation process, a concrete VDAF implements the following class methods:¶
Vdaf.prep_init(verify_key: Bytes[Vdaf.VERIFY_KEY_SIZE], agg_id: Unsigned,
agg_param: AggParam, nonce: Bytes[Vdaf.NONCE_SIZE], public_share: Bytes,
input_share: Bytes) -> Prep is the deterministic preparation-state
initialization algorithm run by each Aggregator to begin processing its input
share into an output share. Its inputs are the shared verification key
(verify_key), the Aggregator's unique identifier (agg_id), the aggregation
parameter (agg_param), the nonce provided by the environment (nonce, see
Figure 7), the public share (public_share), and one of the input
shares generated by the client (input_share). Its output is the Aggregator's
initial preparation state.¶
It is up to the high level protocol in which the VDAF is used to arrange for the distribution of the verification key prior to generating and processing reports. (See Section 9 for details.)¶
Protocols using the VDAF MUST ensure that the Aggregator's identifier is equal
to the integer in range [0, SHARES) that matches the index of input_share
in the sequence of input shares output by the Client.¶
Protocols MUST ensure that public share consumed by each of the Aggregators is identical. This is security critical for VDAFs such as Poplar1 that require an extractable distributed point function. (See Section 8 for details.)¶
Vdaf.prep_next(prep: Prep, inbound: Optional[Bytes]) -> Union[Tuple[Prep,
Bytes], OutShare] is the deterministic preparation-state update algorithm run
by each Aggregator. It updates the Aggregator's preparation state (prep) and
returns either its next preparation state and its message share for the
current round or, if this is the last round, its output share. An exception is
raised if a valid output share could not be recovered. The input of this
algorithm is the inbound preparation message or, if this is the first round,
None.¶
Vdaf.prep_shares_to_prep(agg_param: AggParam, prep_shares: Vec[Bytes]) ->
Bytes is the deterministic preparation-message pre-processing algorithm. It
combines the preparation-message shares generated by the Aggregators in the
previous round into the preparation message consumed by each in the next
round.¶
In effect, each Aggregator moves through a linear state machine with ROUNDS+1
states. The Aggregator enters the first state on using the initialization
algorithm, and the update algorithm advances the Aggregator to the next state.
Thus, in addition to defining the number of rounds (ROUNDS), a VDAF instance
defines the state of the Aggregator after each round.¶
TODO Consider how to bake this "linear state machine" condition into the syntax. Given that Python 3 is used as our pseudocode, it's easier to specify the preparation state using a class.¶
The preparation-state update accomplishes two tasks: recovery of output shares from the input shares and ensuring that the recovered output shares are valid. The abstraction boundary is drawn so that an Aggregator only recovers an output share if it is deemed valid (at least, based on the Aggregator's view of the protocol). Another way to draw this boundary would be to have the Aggregators recover output shares first, then verify that they are valid. However, this would allow the possibility of misusing the API by, say, aggregating an invalid output share. Moreover, in protocols like Prio+ [AGJOP21] based on oblivious transfer, it is necessary for the Aggregators to interact in order to recover aggregatable output shares at all.¶
Note that it is possible for a VDAF to specify ROUNDS == 0, in which case each
Aggregator runs the preparation-state update algorithm once and immediately
recovers its output share without interacting with the other Aggregators.
However, most, if not all, constructions will require some amount of interaction
in order to ensure validity of the output shares (while also maintaining
privacy).¶
OPEN ISSUE accommodating 0-round VDAFs may require syntax changes if, for example, public keys are required. On the other hand, we could consider defining this class of schemes as a different primitive. See issue#77.¶
Similar to DAFs (see Section 4.3), VDAFs MAY impose
restrictions for input shares and aggregation parameters. Protocols using a VDAF
MUST ensure that for each input share and aggregation parameter agg_param, the
preparation phase (including Vdaf.prep_init, Vdaf.prep_next, and
Vdaf.prep_shares_to_prep; see Section 5.2) is only called if
Vdaf.is_valid(agg_param, previous_agg_params) returns True, where
previous_agg_params contains all aggregation parameters that have previously
been used with the same input share.¶
VDAFs MUST implement the following function:¶
Vdaf.is_valid(agg_param: AggParam, previous_agg_params: Vec[AggParam]) ->
Bool: Checks if the agg_param is compatible with all elements of
previous_agg_params.¶
VDAF Aggregation is identical to DAF Aggregation (cf. Section 4.4):¶
Vdaf.out_shares_to_agg_share(agg_param: AggParam, out_shares: Vec[OutShare])
-> agg_share: Bytes is the deterministic aggregation algorithm. It is run by
each Aggregator over the output shares it has computed over a batch of
measurement inputs.¶
The data flow for this stage is illustrated in Figure 3. Here again, we have the aggregation algorithm in a "one-shot" form, where all shares for a batch are provided at the same time. VDAFs typically also support a "streaming" form, where shares are processed one at a time.¶
VDAF Unsharding is identical to DAF Unsharding (cf. Section 4.5):¶
Vdaf.agg_shares_to_result(agg_param: AggParam,
agg_shares: Vec[Bytes], num_measurements: Unsigned) -> AggResult is
run by the Collector in order to compute the aggregate result from
the Aggregators' shares. The length of agg_shares MUST be SHARES.
num_measurements is the number of measurements that contributed to
each of the aggregate shares. This algorithm is deterministic.¶
Secure execution of a VDAF involves simulating the following procedure.¶
def run_vdaf(Vdaf,
verify_key: Bytes[Vdaf.VERIFY_KEY_SIZE],
agg_param: Vdaf.AggParam,
nonces: Vec[Bytes[Vdaf.NONCE_SIZE]],
measurements: Vec[Vdaf.Measurement]):
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, nonce)
# 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
The inputs to this algorithm are the aggregation parameter, a list of measurements, and a nonce for each measurement. This document does not specify how the nonces are chosen, but security requires that the nonces be unique. See Section 9 for details. As explained in Section 4.6, the secure execution of a VDAF requires the application to instantiate secure channels between each of the protocol participants.¶
This section describes the primitives that are common to the VDAFs specified in this document.¶
Both Prio3 and Poplar1 use finite fields of prime order. Finite field
elements are represented by a class Field with the following associated
parameters:¶
MODULUS: Unsigned is the prime modulus that defines the field.¶
ENCODED_SIZE: Unsigned is the number of bytes used to encode a field element
as a byte string.¶
A concrete Field also implements the following class methods:¶
Field.zeros(length: Unsigned) -> output: Vec[Field] returns a vector of
zeros. The length of output MUST be length.¶
Field.rand_vec(length: Unsigned) -> output: Vec[Field] returns a vector of
random field elements. The length of output MUST be length.¶
A field element is an instance of a concrete Field. The concrete class defines
the usual arithmetic operations on field elements. In addition, it defines the
following instance method for converting a field element to an unsigned integer:¶
elem.as_unsigned() -> Unsigned returns the integer representation of
field element elem.¶
Likewise, each concrete Field implements a constructor for converting an
unsigned integer into a field element:¶
Field(integer: Unsigned) returns integer represented as a field element.
The value of integer MUST be less than Field.MODULUS.¶
Finally, each concrete Field has two derived class methods, one for encoding
a vector of field elements as a byte string and another for decoding a vector of
field elements.¶
def encode_vec(Field, data: Vec[Field]) -> Bytes:
encoded = Bytes()
for x in data:
encoded += I2OSP(x.as_unsigned(), Field.ENCODED_SIZE)
return encoded
def decode_vec(Field, encoded: Bytes) -> Vec[Field]:
L = Field.ENCODED_SIZE
if len(encoded) % L != 0:
raise ERR_DECODE
vec = []
for i in range(0, len(encoded), L):
encoded_x = encoded[i:i+L]
x = OS2IP(encoded_x)
if x >= Field.MODULUS:
raise ERR_DECODE # Integer is larger than modulus
vec.append(Field(x))
return vec
The following auxiliary functions on vectors of field elements are used in the remainder of this document. Note that an exception is raised by each function if the operands are not the same length.¶
# 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)))
Some VDAFs require fields that are suitable for efficient computation of the discrete Fourier transform, as this allows for fast polynomial interpolation. (One example is Prio3 (Section 7) when instantiated with the generic FLP of Section 7.3.3.) Specifically, a field is said to be "FFT-friendly" if, in addition to satisfying the interface described in Section 6.1, it implements the following method:¶
Field.gen() -> Field returns the generator of a large subgroup of the
multiplicative group. To be FFT-friendly, the order of this subgroup MUST be a
power of 2. In addition, the size of the subgroup dictates how large
interpolated polynomials can be. It is RECOMMENDED that a generator is chosen
with order at least 2^20.¶
FFT-friendly fields also define the following parameter:¶
GEN_ORDER: Unsigned is the order of a multiplicative subgroup generated by
Field.gen().¶
The tables below define finite fields used in the remainder of this document.¶
| Parameter | Value |
|---|---|
| MODULUS | 2^32 * 4294967295 + 1 |
| ENCODED_SIZE | 8 |
| Generator | 7^4294967295 |
| GEN_ORDER | 2^32 |
| Parameter | Value |
|---|---|
| MODULUS | 2^66 * 4611686018427387897 + 1 |
| ENCODED_SIZE | 16 |
| Generator | 7^4611686018427387897 |
| GEN_ORDER | 2^66 |
OPEN ISSUE We currently use big-endian for encoding field elements. However,
for implementations of GF(2^255-19), little endian is more common. See
issue#90.¶
A pseudorandom generator (PRG) is used to expand a short, (pseudo)random seed into a long string of pseudorandom bits. A PRG suitable for this document implements the interface specified in this section.¶
PRGs are defined by a class Prg with the following associated parameter:¶
SEED_SIZE: Unsigned is the size (in bytes) of a seed.¶
A concrete Prg implements the following class method:¶
Prg(seed: Bytes[Prg.SEED_SIZE], custom: Bytes, binder: Bytes) constructs an
instance of Prg from the given seed and customization and binder strings.
(See below for definitions of these.) The seed MUST be of length SEED_SIZE
and MUST be generated securely (i.e., it is either the output of gen_rand or
a previous invocation of the PRG).¶
prg.next(length: Unsigned) returns the next length bytes of output of PRG.
If the seed was securely generated, the output can be treated as pseudorandom.¶
Each Prg has two derived class methods. The first is used to derive a fresh
seed from an existing one. The second is used to compute a sequence of
pseudorandom field elements. For each method, the seed MUST be of length
SEED_SIZE and MUST be generated securely (i.e., it is either the output of
gen_rand or a previous invocation of the PRG).¶
# Derive a new seed.
def derive_seed(Prg, seed: Bytes[Prg.SEED_SIZE], custom: Bytes, binder: Bytes):
prg = Prg(seed, custom, binder)
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[Prg.SEED_SIZE],
custom: Bytes,
binder: Bytes,
length: Unsigned):
prg = Prg(seed, custom, binder)
return prg.next_vec(Field, length)
This section describes PrgSha3, a PRG based on the Keccak permutation of SHA-3 [FIPS202]. Keccak is used in the cSHAKE128 mode of operation [SP800-185].¶
class PrgSha3(Prg):
# Associated parameters
SEED_SIZE = 16
def __init__(self, seed, custom, binder):
self.l = 0
self.x = seed + binder
self.s = custom
def next(self, length: Unsigned) -> Bytes:
self.l += length
# Function `cSHAKE128(x, l, n, s)` is as defined in
# [SP800-185, Section 3.3].
#
# Implementation note: Rather than re-generate the output
# stream each time `next()` is invoked, most implementations
# of SHA-3 will expose an "absorb-then-squeeze" API that
# allows stateful handling of the stream.
stream = cSHAKE128(self.x, self.l, b'', self.s)
return stream[-length:]
PRGs are used to map a seed to a finite domain, e.g., a fresh seed or a vector of field elements. To ensure domain separation, the derivation is needs to be bound to some distinguished "customization string". The customization string encodes the following values:¶
VERSION);¶
The following algorithm is used in the remainder of this document in order to format the customization string:¶
def format_custom(algo_class: Unsigned, algo: Unsigned, usage: Unsigned):
return I2OSP(VERSION, 1) + \
I2OSP(algo_class, 1) + \
I2OSP(algo, 4) + \
I2OSP(usage, 2)
¶
It is also sometimes necessary to bind the output to some ephemeral value that multiple parties need to agree on. We call this input the "binder string".¶
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:¶
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".¶
Conceptually, an FLP is a two-party protocol executed by a prover and a verifier. In actual use, however, the prover's computation is carried out by the Client, and the verifier's computation is distributed among the Aggregators. The Client generates a "proof" of its input's validity and distributes shares of the proof to the Aggregators. Each Aggregator then performs some a computation on its input share and proof share locally and sends the result to the other Aggregators. Combining the exchanged messages allows each Aggregator to decide if it holds a share of a valid input. (See Section 7.2 for details.)¶
As usual, we will describe the interface implemented by a concrete FLP in terms
of an abstract base class Flp that specifies the set of methods and parameters
a concrete FLP must provide.¶
The parameters provided by a concrete FLP are listed in Table 6.¶
| Parameter | Description |
|---|---|
PROVE_RAND_LEN
|
Length of the prover randomness, the number of random field elements consumed by the prover when generating a proof |
QUERY_RAND_LEN
|
Length of the query randomness, the number of random field elements consumed by the verifier |
JOINT_RAND_LEN
|
Length of the joint randomness, the number of random field elements consumed by both the prover and verifier |
INPUT_LEN
|
Length of the encoded measurement (Section 7.1.1) |
OUTPUT_LEN
|
Length of the aggregatable output (Section 7.1.1) |
PROOF_LEN
|
Length of the proof |
VERIFIER_LEN
|
Length of the verifier message generated by querying the input and proof |
Measurement
|
Type of the measurement |
AggResult
|
Type of the aggregate result |
Field
|
As defined in (Section 6.1) |
An FLP specifies the following algorithms for generating and verifying proofs of validity (encoding is described below in Section 7.1.1):¶
Flp.prove(input: Vec[Field], prove_rand: Vec[Field], joint_rand: Vec[Field])
-> Vec[Field] is the deterministic proof-generation algorithm run by the
prover. Its inputs are the encoded input, the "prover randomness"
prove_rand, and the "joint randomness" joint_rand. The prover randomness is
used only by the prover, but the joint randomness is shared by both the prover
and verifier.¶
Flp.query(input: Vec[Field], proof: Vec[Field], query_rand: Vec[Field],
joint_rand: Vec[Field], num_shares: Unsigned) -> Vec[Field] is the
query-generation algorithm run by the verifier. This is used to "query" the
input and proof. The result of the query (i.e., the output of this function)
is called the "verifier message". In addition to the input and proof, this
algorithm takes as input the query randomness query_rand and the joint
randomness joint_rand. The former is used only by the verifier. num_shares
specifies how many input and proof shares were generated.¶
Flp.decide(verifier: Vec[Field]) -> Bool is the deterministic decision
algorithm run by the verifier. It takes as input the verifier message and
outputs a boolean indicating if the input from which it was generated is
valid.¶
Our application requires that the FLP is "fully linear" in the sense defined in [BBCGGI19]. As a practical matter, what this property implies is that, when run on a share of the input and proof, the query-generation algorithm outputs a share of the verifier message. Furthermore, the "strong zero-knowledge" property of the FLP system ensures that the verifier message reveals nothing about the input's validity. Therefore, to decide if an input is valid, the Aggregators will run the query-generation algorithm locally, exchange verifier shares, combine them to recover the verifier message, and run the decision algorithm.¶
The query-generation algorithm includes a parameter num_shares that specifies
the number of shares of the input and proof that were generated. If these data
are not secret shared, then num_shares == 1. This parameter is useful for
certain FLP constructions. For example, the FLP in Section 7.3 is defined in
terms of an arithmetic circuit; when the circuit contains constants, it is
sometimes necessary to normalize those constants to ensure that the circuit's
output, when run on a valid input, is the same regardless of the number of
shares.¶
An FLP is executed by the prover and verifier as follows:¶
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)
The proof system is constructed so that, if input is a valid input, then
run_flp(Flp, input, 1) always returns True. On the other hand, if input is
invalid, then as long as joint_rand and query_rand are generated uniform
randomly, the output is False with overwhelming probability.¶
We remark that [BBCGGI19] defines a much larger class of fully linear proof systems than we consider here. In particular, what is called an "FLP" here is called a 1.5-round, public-coin, interactive oracle proof system in their paper.¶
The type of measurement being aggregated is defined by the FLP. Hence, the FLP also specifies a method of encoding raw measurements as a vector of field elements:¶
Flp.encode(measurement: Measurement) -> Vec[Field] encodes a raw measurement
as a vector of field elements. The return value MUST be of length INPUT_LEN.¶
For some FLPs, the encoded input also includes redundant field elements that are
useful for checking the proof, but which are not needed after the proof has been
checked. An example is the "integer sum" data type from [CGB17] in which an
integer in range [0, 2^k) is encoded as a vector of k field elements (this
type is also defined in Section 7.4.2). After consuming this vector, all that is
needed is the integer it represents. Thus the FLP defines an algorithm for
truncating the input to the length of the aggregated output:¶
Flp.truncate(input: Vec[Field]) -> Vec[Field] maps an encoded input to an
aggregatable output. The length of the input MUST be INPUT_LEN and the length
of the output MUST be OUTPUT_LEN.¶
Once the aggregate shares have been computed and combined together, their sum can be converted into the aggregate result. This could be a projection from the FLP's field to the integers, or it could include additional post-processing.¶
Flp.decode(output: Vec[Field], num_measurements: Unsigned) -> AggResult
maps a sum of aggregate shares to an aggregate result. The length of the
input MUST be OUTPUT_LEN. num_measurements is the number of measurements
that contributed to the aggregated output.¶
We remark that, taken together, these three functionalities correspond roughly to the notion of "Affine-aggregatable encodings (AFEs)" from [CGB17].¶
This section specifies Prio3, an implementation of the Vdaf interface
(Section 5). It has two generic parameters: an Flp (Section 7.1) and a Prg
(Section 6.2). The associated constants and types required by the Vdaf interface
are defined in Table 7. The methods required for sharding, preparation,
aggregation, and unsharding are described in the remaining subsections. These
methods refer to constants enumerated in Table 8.¶
| Parameter | Value |
|---|---|
VERIFY_KEY_SIZE
|
Prg.SEED_SIZE
|
ROUNDS
|
1
|
SHARES
|
in [2, 256)
|
Measurement
|
Flp.Measurement
|
AggParam
|
None
|
Prep
|
Tuple[Vec[Flp.Field], Optional[Bytes], Bytes]
|
OutShare
|
Vec[Flp.Field]
|
AggResult
|
Flp.AggResult
|
| Variable | Value |
|---|---|
DST_MEASUREMENT_SHARE: Unsigned
|
1 |
DST_PROOF_SHARE: Unsigned
|
2 |
DST_JOINT_RANDOMNESS: Unsigned
|
3 |
DST_PROVE_RANDOMNESS: Unsigned
|
4 |
DST_QUERY_RANDOMNESS: Unsigned
|
5 |
DST_JOINT_RAND_SEED: Unsigned
|
6 |
DST_JOINT_RAND_PART: Unsigned
|
7 |
This section describes the process of recovering output shares from the input shares. The high-level idea is that each Aggregator first queries its input and proof share locally, then exchanges its verifier share with the other Aggregators. The verifier shares are then combined into the verifier message, which is used to decide whether to accept.¶
In addition, the Aggregators must ensure that they have all used the same joint randomness for the query-generation algorithm. The joint randomness is generated by a PRG seed. Each Aggregator derives a "part" of this seed from its input share and the "blind" generated by the client. The seed is derived by hashing together the parts, so before running the query-generation algorithm, it must first gather the parts derived by the other Aggregators.¶
In order to avoid extra round of communication, the Client sends each Aggregator a "hint" consisting of the other Aggregators' parts of the joint randomness seed. This leaves open the possibility that the Client cheated by, say, forcing the Aggregators to use joint randomness that biases the proof check procedure some way in its favor. To mitigate this, the Aggregators also check that they have all computed the same joint randomness seed before accepting their output shares. To do so, they exchange their parts of the joint randomness along with their verifier shares.¶
The definitions of constants and a few auxiliary functions are defined in Section 7.2.6.¶
def prep_init(Prio3, verify_key, agg_id, _agg_param,
nonce, public_share, input_share):
k_joint_rand_parts = Prio3.decode_public_share(public_share)
(meas_share, proof_share, k_blind) = \
Prio3.decode_leader_share(input_share) if agg_id == 0 else \
Prio3.decode_helper_share(agg_id, input_share)
out_share = Prio3.Flp.truncate(meas_share)
# Compute joint randomness.
joint_rand = []
k_corrected_joint_rand, k_joint_rand_part = None, None
if Prio3.Flp.JOINT_RAND_LEN > 0:
encoded = Prio3.Flp.Field.encode_vec(meas_share)
k_joint_rand_part = Prio3.Prg.derive_seed(k_blind,
Prio3.custom(DST_JOINT_RAND_PART),
byte(agg_id) + nonce + encoded)
k_joint_rand_parts[agg_id] = k_joint_rand_part
k_corrected_joint_rand = Prio3.joint_rand(k_joint_rand_parts)
joint_rand = Prio3.Prg.expand_into_vec(
Prio3.Flp.Field,
k_corrected_joint_rand,
Prio3.custom(DST_JOINT_RANDOMNESS),
b'',
Prio3.Flp.JOINT_RAND_LEN,
)
# Query the measurement and proof share.
query_rand = Prio3.Prg.expand_into_vec(
Prio3.Flp.Field,
verify_key,
Prio3.custom(DST_QUERY_RANDOMNESS),
nonce,
Prio3.Flp.QUERY_RAND_LEN,
)
verifier_share = Prio3.Flp.query(meas_share,
proof_share,
query_rand,
joint_rand,
Prio3.SHARES)
prep_msg = Prio3.encode_prep_share(verifier_share,
k_joint_rand_part)
return (out_share, k_corrected_joint_rand, prep_msg)
def prep_next(Prio3, prep, inbound):
(out_share, k_corrected_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_corrected_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_parts = []
for encoded in prep_shares:
(verifier_share, k_joint_rand_part) = \
Prio3.decode_prep_share(encoded)
verifier = vec_add(verifier, verifier_share)
if Prio3.Flp.JOINT_RAND_LEN > 0:
k_joint_rand_parts.append(k_joint_rand_part)
if not Prio3.Flp.decide(verifier):
raise ERR_VERIFY # proof verifier check failed
k_joint_rand_check = None
if Prio3.Flp.JOINT_RAND_LEN > 0:
k_joint_rand_check = Prio3.joint_rand(k_joint_rand_parts)
return Prio3.encode_prep_msg(k_joint_rand_check)
Every input share MUST only be used once, regardless of the aggregation parameters used.¶
def is_valid(agg_param, previous_agg_params):
return len(previous_agg_params) == 0
Aggregating a set of output shares is simply a matter of adding up the vectors element-wise.¶
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)
To unshard a set of aggregate shares, the Collector first adds up the vectors element-wise. It then converts each element of the vector into an integer.¶
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 joint_rand(Prio3, k_joint_rand_parts):
return Prio3.Prg.derive_seed(
zeros(Prio3.Prg.SEED_SIZE),
Prio3.custom(DST_JOINT_RAND_SEED),
concat(k_joint_rand_parts),
)
¶
def encode_leader_share(Prio3,
meas_share,
proof_share,
k_blind):
encoded = Bytes()
encoded += Prio3.Flp.Field.encode_vec(meas_share)
encoded += Prio3.Flp.Field.encode_vec(proof_share)
if Prio3.Flp.JOINT_RAND_LEN > 0:
encoded += k_blind
return encoded
def decode_leader_share(Prio3, encoded):
l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.INPUT_LEN
encoded_meas_share, encoded = encoded[:l], encoded[l:]
meas_share = Prio3.Flp.Field.decode_vec(encoded_meas_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
if Prio3.Flp.JOINT_RAND_LEN == 0:
if len(encoded) != 0:
raise ERR_DECODE
return (meas_share, proof_share, None)
k_blind, encoded = encoded[:l], encoded[l:]
if len(encoded) != 0:
raise ERR_DECODE
return (meas_share, proof_share, k_blind)
def encode_helper_share(Prio3,
k_meas_share,
k_proof_share,
k_blind):
encoded = Bytes()
encoded += k_meas_share
encoded += k_proof_share
if Prio3.Flp.JOINT_RAND_LEN > 0:
encoded += k_blind
return encoded
def decode_helper_share(Prio3, agg_id, encoded):
c_meas_share = Prio3.custom(DST_MEASUREMENT_SHARE)
c_proof_share = Prio3.custom(DST_PROOF_SHARE)
l = Prio3.Prg.SEED_SIZE
k_meas_share, encoded = encoded[:l], encoded[l:]
meas_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field,
k_meas_share,
c_meas_share,
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,
c_proof_share,
byte(agg_id),
Prio3.Flp.PROOF_LEN)
if Prio3.Flp.JOINT_RAND_LEN == 0:
if len(encoded) != 0:
raise ERR_DECODE
return (meas_share, proof_share, None)
k_blind, encoded = encoded[:l], encoded[l:]
if len(encoded) != 0:
raise ERR_DECODE
return (meas_share, proof_share, k_blind)
def encode_public_share(Prio3,
k_joint_rand_parts):
encoded = Bytes()
if Prio3.Flp.JOINT_RAND_LEN > 0:
encoded += concat(k_joint_rand_parts)
return encoded
def decode_public_share(Prio3, encoded):
l = Prio3.Prg.SEED_SIZE
if Prio3.Flp.JOINT_RAND_LEN == 0:
if len(encoded) != 0:
raise ERR_DECODE
return None
k_joint_rand_parts = []
for i in range(Prio3.SHARES):
k_joint_rand_part, encoded = encoded[:l], encoded[l:]
k_joint_rand_parts.append(k_joint_rand_part)
if len(encoded) != 0:
raise ERR_DECODE
return k_joint_rand_parts
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)
if Prio3.Flp.JOINT_RAND_LEN == 0:
if len(encoded) != 0:
raise ERR_DECODE
return (verifier, None)
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):
if Prio3.Flp.JOINT_RAND_LEN == 0:
if len(encoded) != 0:
raise ERR_DECODE
return None
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
¶
This section describes an FLP based on the construction from in [BBCGGI19], Section 4.2. We begin in Section 7.3.1 with an overview of their proof system and the extensions to their proof system made here. The construction is specified in Section 7.3.3.¶
OPEN ISSUE We're not yet sure if specifying this general-purpose FLP is desirable. It might be preferable to specify specialized FLPs for each data type that we want to standardize, for two reasons. First, clear and concise specifications are likely easier to write for specialized FLPs rather than the general one. Second, we may end up tailoring each FLP to the measurement type in a way that improves performance, but breaks compatibility with the general-purpose FLP.¶
In any case, we can't make this decision until we know which data types to standardize, so for now, we'll stick with the general-purpose construction. The reference implementation can be found at https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc.¶
OPEN ISSUE Chris Wood points out that the this section reads more like a paper than a standard. Eventually we'll want to work this into something that is readily consumable by the CFRG.¶
In the proof system of [BBCGGI19], validity is defined via an arithmetic circuit evaluated over the input: If the circuit output is zero, then the input is deemed valid; otherwise, if the circuit output is non-zero, then the input is deemed invalid. Thus the goal of the proof system is merely to allow the verifier to evaluate the validity circuit over the input. For our application (Section 7), this computation is distributed among multiple Aggregators, each of which has only a share of the input.¶
Suppose for a moment that the validity circuit C is affine, meaning its only
operations are addition and multiplication-by-constant. In particular, suppose
the circuit does not contain a multiplication gate whose operands are both
non-constant. Then to decide if an input x is valid, each Aggregator could
evaluate C on its share of x locally, broadcast the output share to its
peers, then combine the output shares locally to recover C(x). This is true
because for any SHARES-way secret sharing of x it holds that¶
C(x_shares[0] + ... + x_shares[SHARES-1]) =
C(x_shares[0]) + ... + C(x_shares[SHARES-1])
¶
(Note that, for this equality to hold, it may be necessary to scale any
constants in the circuit by SHARES.) However this is not the case if C is
not-affine (i.e., it contains at least one multiplication gate whose operands
are non-constant). In the proof system of [BBCGGI19], the proof is designed to
allow the (distributed) verifier to compute the non-affine operations using only
linear operations on (its share of) the input and proof.¶
To make this work, the proof system is restricted to validity circuits that
exhibit a special structure. Specifically, an arithmetic circuit with "G-gates"
(see [BBCGGI19], Definition 5.2) is composed of affine gates and any number of
instances of a distinguished gate G, which may be non-affine. We will refer to
this class of circuits as 'gadget circuits' and to G as the "gadget".¶
As an illustrative example, consider a validity circuit C that recognizes the
set L = set([0], [1]). That is, C takes as input a length-1 vector x and
returns 0 if x[0] is in [0,2) and outputs something else otherwise. This
circuit can be expressed as the following degree-2 polynomial:¶
C(x) = (x[0] - 1) * x[0] = x[0]^2 - x[0]¶
This polynomial recognizes L because x[0]^2 = x[0] is only true if x[0] ==
0 or x[0] == 1. Notice that the polynomial involves a non-affine operation,
x[0]^2. In order to apply [BBCGGI19], Theorem 4.3, the circuit needs to be
rewritten in terms of a gadget that subsumes this non-affine operation. For
example, the gadget might be multiplication:¶
Mul(left, right) = left * right¶
The validity circuit can then be rewritten in terms of Mul like so:¶
C(x[0]) = Mul(x[0], x[0]) - x[0]¶
The proof system of [BBCGGI19] allows the verifier to evaluate each instance
of the gadget (i.e., Mul(x[0], x[0]) in our example) using a linear function
of the input and proof. The proof is constructed roughly as follows. Let C be
the validity circuit and suppose the gadget is arity-L (i.e., it has L input
wires.). Let wire[j-1,k-1] denote the value of the jth wire of the kth
call to the gadget during the evaluation of C(x). Suppose there are M such
calls and fix distinct field elements alpha[0], ..., alpha[M-1]. (We will
require these points to have a special property, as we'll discuss in
Section 7.3.1.1; but for the moment it is only important
that they are distinct.)¶
The prover constructs from wire and alpha a polynomial that, when evaluated
at alpha[k-1], produces the output of the kth call to the gadget. Let us
call this the "gadget polynomial". Polynomial evaluation is linear, which means
that, in the distributed setting, the Client can disseminate additive shares of
the gadget polynomial that the Aggregators then use to compute additive shares
of each gadget output, allowing each Aggregator to compute its share of C(x)
locally.¶
There is one more wrinkle, however: It is still possible for a malicious prover
to produce a gadget polynomial that would result in C(x) being computed
incorrectly, potentially resulting in an invalid input being accepted. To
prevent this, the verifier performs a probabilistic test to check that the
gadget polynomial is well-formed. This test, and the procedure for constructing
the gadget polynomial, are described in detail in Section 7.3.3.¶
The FLP described in the next section extends the proof system [BBCGGI19], Section 4.2 in three ways.¶
First, the validity circuit in our construction includes an additional, random
input (this is the "joint randomness" derived from the input shares in Prio3;
see Section 7.2). This allows for circuit optimizations that trade a
small soundness error for a shorter proof. For example, consider a circuit that
recognizes the set of length-N vectors for which each element is either one or
zero. A deterministic circuit could be constructed for this language, but it
would involve a large number of multiplications that would result in a large
proof. (See the discussion in [BBCGGI19], Section 5.2 for details). A much
shorter proof can be constructed for the following randomized circuit:¶
C(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N-1])¶
(Note that this is a special case of [BBCGGI19], Theorem 5.2.) Here inp is
the length-N input and r is a random field element. The gadget circuit
Range2 is the "range-check" polynomial described above, i.e., Range2(x) = x^2 -
x. The idea is that, if inp is valid (i.e., each inp[j] is in [0,2)),
then the circuit will evaluate to 0 regardless of the value of r; but if
inp[j] is not in [0,2) for some j, the output will be non-zero with high
probability.¶
The second extension implemented by our FLP allows the validity circuit to
contain multiple gadget types. (This generalization was suggested in
[BBCGGI19], Remark 4.5.) For example, the following circuit is allowed, where
Mul and Range2 are the gadgets defined above (the input has length N+1):¶
C(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N-1]) + \
2^0 * inp[0] + ... + 2^(N-1) * inp[N-1] - \
Mul(inp[N], inp[N])
¶
Finally, [BBCGGI19], Theorem 4.3 makes no restrictions on the choice of the
fixed points alpha[0], ..., alpha[M-1], other than to require that the points
are distinct. In this document, the fixed points are chosen so that the gadget
polynomial can be constructed efficiently using the Cooley-Tukey FFT ("Fast
Fourier Transform") algorithm. Note that this requires the field to be
"FFT-friendly" as defined in Section 6.1.2.¶
The FLP described in Section 7.3.3 is defined in terms of a
validity circuit Valid that implements the interface described here.¶
A concrete Valid defines the following parameters:¶
| Parameter | Description |
|---|---|
GADGETS
|
A list of gadgets |
GADGET_CALLS
|
Number of times each gadget is called |
INPUT_LEN
|
Length of the input |
OUTPUT_LEN
|
Length of the aggregatable output |
JOINT_RAND_LEN
|
Length of the random input |
Measurement
|
The type of measurement |
AggResult
|
Type of the aggregate result |
Field
|
An FFT-friendly finite field as defined in Section 6.1.2 |
Each gadget G in GADGETS defines a constant DEGREE that specifies the
circuit's "arithmetic degree". This is defined to be the degree of the
polynomial that computes it. For example, the Mul circuit in
Section 7.3.1 is defined by the polynomial Mul(x) = x * x, which
has degree 2. Hence, the arithmetic degree of this gadget is 2.¶
Each gadget also defines a parameter ARITY that specifies the circuit's arity
(i.e., the number of input wires).¶
A concrete Valid provides the following methods for encoding a measurement as
an input vector, truncating an input vector to the length of an aggregatable
output, and converting an aggregated output to an aggregate result:¶
Valid.encode(measurement: Measurement) -> Vec[Field] returns a vector of
length INPUT_LEN representing a measurement.¶
Valid.truncate(input: Vec[Field]) -> Vec[Field] returns a vector of length
OUTPUT_LEN representing an aggregatable output.¶
Valid.decode(output: Vec[Field], num_measurements: Unsigned) -> AggResult
returns an aggregate result.¶
Finally, the following class methods are derived for each concrete Valid:¶
# 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
This section specifies FlpGeneric, an implementation of the Flp interface
(Section 7.1). It has as a generic parameter a validity circuit Valid implementing
the interface defined in Section 7.3.2.¶
NOTE A reference implementation can be found in https://github.com/cfrg/draft-irtf-cfrg-vdaf/blob/main/poc/flp_generic.sage.¶
The FLP parameters for FlpGeneric are defined in Table 10. The
required methods for generating the proof, generating the verifier, and deciding
validity are specified in the remaining subsections.¶
In the remainder, we let [n] denote the set {1, ..., n} for positive integer
n. We also define the following constants:¶
| Parameter | Value |
|---|---|
PROVE_RAND_LEN
|
Valid.prove_rand_len() (see Section 7.3.2) |
QUERY_RAND_LEN
|
Valid.query_rand_len() (see Section 7.3.2) |
JOINT_RAND_LEN
|
Valid.JOINT_RAND_LEN
|
INPUT_LEN
|
Valid.INPUT_LEN
|
OUTPUT_LEN
|
Valid.OUTPUT_LEN
|
PROOF_LEN
|
Valid.proof_len() (see Section 7.3.2) |
VERIFIER_LEN
|
Valid.verifier_len() (see Section 7.3.2) |
Measurement
|
Valid.Measurement
|
Field
|
Valid.Field
|
On input inp, prove_rand, and joint_rand, the proof is computed as
follows:¶
i in [H] create an empty table wire_i.¶
prove_rand into sub-vectors seed_1, ...,
seed_H where len(seed_i) == L_i for all i in [H]. Let us call these
the "wire seeds" of each gadget.¶
Valid on input of inp and joint_rand, recording the inputs of
each gadget in the corresponding table. Specifically, for every i in [H],
set wire_i[j-1,k-1] to the value on the jth wire into the kth call to
gadget G_i.¶
Compute the "wire polynomials". That is, for every i in [H] and j in
[L_i], construct poly_wire_i[j-1], the jth wire polynomial for the
ith gadget, as follows:¶
w = [seed_i[j-1], wire_i[j-1,0], ..., wire_i[j-1,M_i-1]].¶
padded_w = w + Field.zeros(P_i - len(w)).¶
NOTE We pad w to the nearest power of 2 so that we can use FFT for
interpolating the wire polynomials. Perhaps there is some clever math for
picking wire_inp in a way that avoids having to pad.¶
poly_wire_i[j-1] be the lowest degree polynomial for which
poly_wire_i[j-1](alpha_i^k) == padded_w[k] for all k in [P_i].¶
Compute the "gadget polynomials". That is, for every i in [H]:¶
poly_gadget_i = G_i(poly_wire_i[0], ..., poly_wire_i[L_i-1]). That
is, evaluate the circuit G_i on the wire polynomials for the ith
gadget. (Arithmetic is in the ring of polynomials over Field.)¶
The proof is the vector proof = seed_1 + coeff_1 + ... + seed_H + coeff_H,
where coeff_i is the vector of coefficients of poly_gadget_i for each i in
[H].¶
On input of inp, proof, query_rand, and joint_rand, the verifier message
is generated as follows:¶
i in [H] create an empty table wire_i.¶
proof into the sub-vectors seed_1, coeff_1, ..., seed_H,
coeff_H defined in Section 7.3.3.1.¶
Valid on input of inp and joint_rand, recording the inputs of
each gadget in the corresponding table. This step is similar to the prover's
step (3.) except the verifier does not evaluate the gadgets. Instead, it
computes the output of the kth call to G_i by evaluating
poly_gadget_i(alpha_i^k). Let v denote the output of the circuit
evaluation.¶
Compute the tests for well-formedness of the gadget polynomials. That is, for
every i in [H]:¶
The verifier message is the vector verifier = [v] + x_1 + [y_1] + ... + x_H +
[y_H].¶
This section specifies instantiations of Prio3 for various measurement types.
Each uses FlpGeneric as the FLP (Section 7.3) and is determined by a
validity circuit (Section 7.3.2) and a PRG (Section 6.2). Test vectors for
each can be found in Appendix "Test Vectors".¶
NOTE Reference implementations of each of these VDAFs can be found in https://github.com/cfrg/draft-irtf-cfrg-vdaf/blob/main/poc/vdaf_prio3.sage.¶
Our first instance of Prio3 is for a simple counter: Each measurement is either one or zero and the aggregate result is the sum of the measurements.¶
This instance uses PrgSha3 (Section 6.2.1) as its PRG. Its validity
circuit, denoted Count, uses Field64 (Table 3) as its finite field. Its
gadget, denoted Mul, is the degree-2, arity-2 gadget defined as¶
def Mul(x, y):
return x * y
¶
The validity circuit is defined as¶
def Count(inp: Vec[Field64]):
return Mul(inp[0], inp[0]) - inp[0]
¶
The measurement is encoded and decoded as a singleton vector in the natural way. The parameters for this circuit are summarized below.¶
| Parameter | Value |
|---|---|
GADGETS
|
[Mul]
|
GADGET_CALLS
|
[1]
|
INPUT_LEN
|
1
|
OUTPUT_LEN
|
1
|
JOINT_RAND_LEN
|
0
|
Measurement
|
Unsigned, in range [0,2)
|
AggResult
|
Unsigned
|
Field
|
Field64 (Table 3) |
The next instance of Prio3 supports summing of integers in a pre-determined
range. Each measurement is an integer in range [0, 2^bits), where bits is an
associated parameter.¶
This instance of Prio3 uses PrgSha3 (Section 6.2.1) as its PRG. Its validity
circuit, denoted Sum, uses Field128 (Table 4) as its finite field. The
measurement is encoded as a length-bits vector of field elements, where the
lth element of the vector represents the lth bit of the summand:¶
def encode(Sum, measurement: Integer):
if 0 > measurement or measurement >= 2^Sum.INPUT_LEN:
raise ERR_INPUT
encoded = []
for l in range(Sum.INPUT_LEN):
encoded.append(Sum.Field((measurement >> l) & 1))
return encoded
def truncate(Sum, inp):
decoded = Sum.Field(0)
for (l, b) in enumerate(inp):
w = Sum.Field(1 << l)
decoded += w * b
return [decoded]
def decode(Sum, output, _num_measurements):
return output[0].as_unsigned()
¶
The validity circuit checks that the input consists of ones and zeros. Its
gadget, denoted Range2, is the degree-2, arity-1 gadget defined as¶
def Range2(x):
return x^2 - x
¶
The validity circuit is defined as¶
def Sum(inp: Vec[Field128], joint_rand: Vec[Field128]):
out = Field128(0)
r = joint_rand[0]
for x in inp:
out += r * Range2(x)
r *= joint_rand[0]
return out
¶
| Parameter | Value |
|---|---|
GADGETS
|
[Range2]
|
GADGET_CALLS
|
[bits]
|
INPUT_LEN
|
bits
|
OUTPUT_LEN
|
1
|
JOINT_RAND_LEN
|
1
|
Measurement
|
Unsigned, in range [0, 2^bits)
|
AggResult
|
Unsigned
|
Field
|
Field128 (Table 4) |
This instance of Prio3 allows for estimating the distribution of the measurements by computing a simple histogram. Each measurement is an arbitrary integer and the aggregate result counts the number of measurements that fall in a set of fixed buckets.¶
This instance of Prio3 uses PrgSha3 (Section 6.2.1) as its PRG. Its validity
circuit, denoted Histogram, uses Field128 (Table 4) as its finite
field. The measurement is encoded as a one-hot vector representing the bucket
into which the measurement falls (let bucket denote a sequence of
monotonically increasing integers):¶
def encode(Histogram, measurement: Integer):
boundaries = buckets + [Infinity]
encoded = [Field128(0) for _ in range(len(boundaries))]
for i in range(len(boundaries)):
if measurement <= boundaries[i]:
encoded[i] = Field128(1)
return encoded
def truncate(Histogram, inp: Vec[Field128]):
return inp
def decode(Histogram, output: Vec[Field128], _num_measurements):
return [bucket_count.as_unsigned() for bucket_count in output]
¶
The validity circuit uses Range2 (see Section 7.4.2) as its single gadget. It
checks for one-hotness in two steps, as follows:¶
def Histogram(inp: Vec[Field128],
joint_rand: Vec[Field128],
num_shares: Unsigned):
# Check that each bucket is one or zero.
range_check = Field128(0)
r = joint_rand[0]
for x in inp:
range_check += r * Range2(x)
r *= joint_rand[0]
# Check that the buckets sum to 1.
sum_check = -Field128(1) * Field128(num_shares).inv()
for b in inp:
sum_check += b
out = joint_rand[1] * range_check + \
joint_rand[1]^2 * sum_check
return out
¶
Note that this circuit depends on the number of shares into which the input is sharded. This is provided to the FLP by Prio3.¶
| Parameter | Value |
|---|---|
GADGETS
|
[Range2]
|
GADGET_CALLS
|
[buckets + 1]
|
INPUT_LEN
|
buckets + 1
|
OUTPUT_LEN
|
buckets + 1
|
JOINT_RAND_LEN
|
2
|
Measurement
|
Integer
|
AggResult
|
Vec[Unsigned]
|
Field
|
Field128 (Table 4) |
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.¶
0 and
1.¶
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".¶
An IDPF is defined over a domain of size 2^BITS, where BITS is constant
defined by the IDPF. Indexes into the IDPF tree are encoded as integers in range
[0, 2^BITS). The Client specifies an index alpha and a vector of
values beta, one for each "level" L in range [0, BITS). The key generation
algorithm generates one IDPF "key" for each Aggregator. When evaluated at level
L and index 0 <= prefix < 2^L, each IDPF key returns an additive share of
beta[L] if prefix is the L-bit prefix of alpha and shares of zero
otherwise.¶
An index x is defined to be a prefix of another index y as follows. Let
LSB(x, N) denote the least significant N bits of positive integer x. By
definition, a positive integer 0 <= x < 2^L is said to be the length-L
prefix of positive integer 0 <= y < 2^BITS if LSB(x, L) is equal to the most
significant L bits of LSB(y, BITS), For example, 6 (110 in binary) is the
length-3 prefix of 25 (11001), but 7 (111) is not.¶
Each of the programmed points beta is a vector of elements of some finite
field. We distinguish two types of fields: One for inner nodes (denoted
Idpf.FieldInner), and one for leaf nodes (Idpf.FieldLeaf). (Our
instantiation of Poplar1 (Section 8.4) will use a much larger field for
leaf nodes than for inner nodes. This is to ensure the IDPF is "extractable" as
defined in [BBCGGI21], Definition 1.)¶
A concrete IDPF defines the types and constants enumerated in Table 14. In
the remainder we write Idpf.Vec as shorthand for the type
Union[Vec[Vec[Idpf.FieldInner]], Vec[Vec[Idpf.FieldLeaf]]]. (This type denotes
either a vector of inner node field elements or leaf node field elements.) The
scheme is comprised of the following algorithms:¶
Idpf.gen(alpha: Unsigned, beta_inner: Vec[Vec[Idpf.FieldInner]], beta_leaf:
Vec[Idpf.FieldLeaf]) -> (Bytes, Vec[Bytes]) is the randomized IDPF-key
generation algorithm. Its inputs are the index alpha and the values beta.
The value of alpha MUST be in range [0, 2^BITS). The output is a public
part that is sent to all Aggregators and a vector of private IDPF keys, one
for each aggregator.¶
Idpf.eval(agg_id: Unsigned, public_share: Bytes, key: Bytes, level: Unsigned,
prefixes: Vec[Unsigned]) -> Idpf.Vec is the deterministic, stateless
IDPF-key evaluation algorithm run by each Aggregator. Its inputs are the
Aggregator's unique identifier, the public share distributed to all of the
Aggregators, the Aggregator's IDPF key, the "level" at which to evaluate the
IDPF, and the sequence of candidate prefixes. It returns the share of the
value corresponding to each candidate prefix.¶
The output type depends on the value of level: If level < Idpf.BITS-1, the
output is the value for an inner node, which has type
Vec[Vec[Idpf.FieldInner]]; otherwise, if level == Idpf.BITS-1, then the
output is the value for a leaf node, which has type
Vec[Vec[Idpf.FieldLeaf]].¶
The value of level MUST be in range [0, BITS). The indexes in prefixes
MUST all be distinct and in range [0, 2^level).¶
Applications MUST ensure that the Aggregator's identifier is equal to the
integer in range [0, SHARES) that matches the index of key in the sequence
of IDPF keys output by the Client.¶
In addition, the following method is derived for each concrete Idpf:¶
def current_field(Idpf, level):
return Idpf.FieldInner if level < Idpf.BITS-1 \
else Idpf.FieldLeaf
¶
Finally, an implementation note. The interface for IDPFs specified here is stateless, in the sense that there is no state carried between IDPF evaluations. This is to align the IDPF syntax with the VDAF abstraction boundary, which does not include shared state across across VDAF evaluations. In practice, of course, it will often be beneficial to expose a stateful API for IDPFs and carry the state across evaluations. See Section 8.3 for details.¶
| Parameter | Description |
|---|---|
| SHARES | Number of IDPF keys output by IDPF-key generator |
| BITS | Length in bits of each input string |
| VALUE_LEN | Number of field elements of each output value |
| KEY_SIZE | Size in bytes of each IDPF key |
| FieldInner | Implementation of Field (Section 6.1) used for values of inner nodes |
| FieldLeaf | Implementation of Field used for values of leaf nodes |
| Prg | Implementation of Prg (Section 6.2) |
This section specifies Poplar1, an implementation of the Vdaf interface
(Section 5). It is defined in terms of any Idpf (Section 8.1) for which
Idpf.SHARES == 2 and Idpf.VALUE_LEN == 2. The associated constants and types
required by the Vdaf interface are defined in Table 15. The methods
required for sharding, preparation, aggregation, and unsharding are described in
the remaining subsections. These methods make use of constants defined in
Table 16.¶
| Parameter | Value |
|---|---|
VERIFY_KEY_SIZE
|
Idpf.Prg.SEED_SIZE
|
ROUNDS
|
2
|
SHARES
|
2
|
Measurement
|
Unsigned
|
AggParam
|
Tuple[Unsigned, Vec[Unsigned]]
|
Prep
|
Tuple[Bytes, Unsigned, Idpf.Vec]
|
OutShare
|
Idpf.Vec
|
AggResult
|
Vec[Unsigned]
|
| Variable | Value |
|---|---|
| DST_SHARD_RAND: Unsigned | 1 |
| DST_CORR_INNER: Unsigned | 2 |
| DST_CORR_LEAF: Unsigned | 3 |
| DST_VERIFY_RAND: Unsigned | 4 |
The client's input is an IDPF index, denoted alpha. The programmed IDPF values
are pairs of field elements (1, k) where each k is chosen at random. This
random value is used as part of the secure sketching protocol of [BBCGGI21],
Appendix C.4. After evaluating their IDPF key shares on a given sequence of
candidate prefixes, the sketching protocol is used by the Aggregators to verify
that they hold shares of a one-hot vector. In addition, for each level of the
tree, the prover generates random elements a, b, and c and computes¶
A = -2*a + k
B = a^2 + b - k*a + c
¶
and sends additive shares of a, b, c, A and B to the Aggregators.
Putting everything together, the input-distribution algorithm is defined as
follows. Function encode_input_shares is defined in Section 8.2.6.¶
def measurement_to_input_shares(Poplar1, measurement, nonce):
prg = Poplar1.Idpf.Prg(gen_rand(Poplar1.Idpf.Prg.SEED_SIZE),
Poplar1.custom(DST_SHARD_RAND), b'')
# 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_offsets = vec_add(
Poplar1.Idpf.Prg.expand_into_vec(
Poplar1.Idpf.FieldInner,
corr_seed[0],
Poplar1.custom(DST_CORR_INNER),
byte(0) + nonce,
3 * (Poplar1.Idpf.BITS-1),
),
Poplar1.Idpf.Prg.expand_into_vec(
Poplar1.Idpf.FieldInner,
corr_seed[1],
Poplar1.custom(DST_CORR_INNER),
byte(1) + nonce,
3 * (Poplar1.Idpf.BITS-1),
),
)
corr_offsets += vec_add(
Poplar1.Idpf.Prg.expand_into_vec(
Poplar1.Idpf.FieldLeaf,
corr_seed[0],
Poplar1.custom(DST_CORR_LEAF),
byte(0) + nonce,
3,
),
Poplar1.Idpf.Prg.expand_into_vec(
Poplar1.Idpf.FieldLeaf,
corr_seed[1],
Poplar1.custom(DST_CORR_LEAF),
byte(1) + nonce,
3,
),
)
# 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), corr_offsets = corr_offsets[:3], corr_offsets[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))
The aggregation parameter encodes a sequence of candidate prefixes. When an
Aggregator receives an input share from the Client, it begins by evaluating its
IDPF share on each candidate prefix, recovering a data_share and auth_share
for each. The Aggregators use these and the correlation shares provided by the
Client to verify that the sequence of data_share values are additive shares of
a one-hot vector.¶
Aggregators MUST ensure the candidate prefixes are all unique and appear in
lexicographic order. (This is enforced in the definition of prep_init()
below.) Uniqueness is necessary to ensure the refined measurement (i.e., the sum
of the output shares) is in fact a one-hot vector. Otherwise, sketch
verification might fail, causing the Aggregators to erroneously reject a report
that is actually valid. Note that enforcing the order is not strictly necessary,
but this does allow uniqueness to be determined more efficiently.¶
The algorithms below make use of the auxiliary function decode_input_share()
defined in Section 8.2.6.¶
def prep_init(Poplar1, verify_key, agg_id, agg_param,
nonce, public_share, input_share):
(level, prefixes) = agg_param
(key, corr_seed, corr_inner, corr_leaf) = \
Poplar1.decode_input_share(input_share)
Field = Poplar1.Idpf.current_field(level)
# Ensure that candidate prefixes are all unique and appear in
# lexicographic order.
for i in range(1,len(prefixes)):
if prefixes[i-1] >= prefixes[i]:
raise ERR_INPUT # out-of-order prefix
# Evaluate the IDPF key at the given set of prefixes.
value = Poplar1.Idpf.eval(
agg_id, public_share, key, level, prefixes)
# Get shares of the correlated randomness for computing the
# Aggregator's share of the sketch for the given level of the IDPF
# tree.
if level < Poplar1.Idpf.BITS - 1:
corr_prg = Poplar1.Idpf.Prg(corr_seed,
Poplar1.custom(DST_CORR_INNER),
byte(agg_id) + nonce)
# Fast-forward the PRG state to the current level.
corr_prg.next_vec(Field, 3 * level)
else:
corr_prg = Poplar1.Idpf.Prg(corr_seed,
Poplar1.custom(DST_CORR_LEAF),
byte(agg_id) + nonce)
(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].
verify_rand_prg = Poplar1.Idpf.Prg(verify_key,
Poplar1.custom(DST_VERIFY_RAND),
nonce + I2OSP(level, 2))
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)
Aggregation parameters are valid for a given input share if no aggregation parameter with the same level has been used with the same input share before. The whole preparation phase MUST NOT be run more than once for a given combination of input share and level.¶
def is_valid(agg_param, previous_agg_params):
(level, _) = agg_param
return all(
level != other_level
for (other_level, _) in previous_agg_params
)
Aggregation involves simply adding up the output shares.¶
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)
Finally, the Collector unshards the aggregate result by adding up the aggregate shares.¶
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))
This section defines methods for serializing input shares, as required by the
Vdaf interface. Optional serialization of the aggregation parameter is also
specified below.¶
Implementation note: The aggregation parameter includes the level of the IDPF tree and the sequence of indices to evaluate. For implementations that perform per-report caching across executions of the VDAF, this may be more information than is strictly needed. In particular, it may be sufficient to convey which indices from the previous execution will have their children included in the next. This would help reduce communication overhead.¶
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) > 2^32 - 1:
raise ERR_INPUT # too many prefixes
encoded = Bytes()
encoded += I2OSP(level, 2)
encoded += I2OSP(len(prefixes), 4)
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 decode_agg_param(Poplar1, encoded):
encoded_level, encoded = encoded[:2], encoded[2:]
level = OS2IP(encoded_level)
encoded_prefix_count, encoded = encoded[:4], encoded[4:]
prefix_count = OS2IP(encoded_prefix_count)
l = floor(((level+1) * prefix_count + 7) / 8)
encoded_packed, encoded = encoded[:l], encoded[l:]
packed = OS2IP(encoded_packed)
prefixes = []
m = 2^(level+1) - 1
for i in range(prefix_count):
prefixes.append(packed >> ((level+1) * i) & m)
if len(encoded) != 0:
raise ERR_INPUT
return (level, prefixes)
¶
TODO(issue#32) Consider replacing the generic Prg object here with some
fixed-key mode for AES (something along the lines of ia.cr/2019/074). This
would allow us to take advantage of hardware acceleration, which would
significantly improve performance. We use SHA-3 primarily to instantiate
random oracles, but the random oracle model may not be required for IDPF. More
investigation is needed.¶
In this section we specify a concrete IDPF, called IdpfPoplar, suitable for instantiating Poplar1. The scheme gets its name from the name of the protocol of [BBCGGI21].¶
TODO We should consider giving IdpfPoplar a more distinctive name.¶
The constant and type definitions required by the Idpf interface are given in
Table 17.¶
| Parameter | Value |
|---|---|
| SHARES |
2
|
| BITS | any positive integer |
| VALUE_LEN | any positive integer |
| KEY_SIZE |
Prg.SEED_SIZE
|
| FieldInner |
Field64 (Table 3) |
| FieldLeaf |
Field255 (Table 5) |
| Prg | any implementation of Prg (Section 6.2) |
TODO Describe the construction in prose, beginning with a gentle introduction to the high level idea.¶
The description of the IDPF-key generation algorithm makes use of auxiliary
functions extend(), convert(), and encode_public_share() defined in
Section 8.3.3. In the following, we let Field2 denote the
field GF(2).¶
def gen(IdpfPoplar, 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):
Field = IdpfPoplar.current_field(level)
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], ctrl[0].conditional_select(seed_cw))
x1 = xor(s1[keep], ctrl[1].conditional_select(seed_cw))
(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)
# Implementation note: Here we negate the correction word if
# the control bit `ctrl[1]` is set. We avoid branching on the
# value in order to reduce leakage via timing side channels.
mask = Field(1) - Field(2) * Field(ctrl[1].as_unsigned())
for i in range(len(w_cw)):
w_cw[i] *= mask
correction_words.append((seed_cw, ctrl_cw, w_cw))
public_share = IdpfPoplar.encode_public_share(correction_words)
return (public_share, init_seed)
TODO Describe in prose how IDPF-key evaluation algorithm works.¶
The description of the IDPF-evaluation algorithm makes use of auxiliary
functions extend(), convert(), and decode_public_share() defined in
Section 8.3.3.¶
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):
Field = IdpfPoplar.current_field(level)
(seed_cw, ctrl_cw, w_cw) = correction_word
(s, t) = IdpfPoplar.extend(prev_seed)
s[0] = xor(s[0], prev_ctrl.conditional_select(seed_cw))
s[1] = xor(s[1], prev_ctrl.conditional_select(seed_cw))
t[0] += ctrl_cw[0] * prev_ctrl
t[1] += ctrl_cw[1] * prev_ctrl
next_ctrl = t[bit]
(next_seed, y) = IdpfPoplar.convert(level, s[bit])
# Implementation note: Here we add the correction word to the
# output if `next_ctrl` is set. We avoid branching on the value of
# the control bit in order to reduce side channel leakage.
mask = Field(next_ctrl.as_unsigned())
for i in range(len(y)):
y[i] += w_cw[i] * mask
return (next_seed, next_ctrl, y)
def extend(IdpfPoplar, seed):
prg = IdpfPoplar.Prg(seed, format_custom(1, 0, 0), b'')
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):
prg = IdpfPoplar.Prg(seed, format_custom(1, 0, 1), b'')
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()
control_bits = list(itertools.chain.from_iterable(
cw[1] for cw in correction_words
))
encoded += pack_bits(control_bits)
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:]
control_bits = unpack_bits(encoded_ctrl, 2 * IdpfPoplar.BITS)
correction_words = []
for level in range(IdpfPoplar.BITS):
Field = IdpfPoplar.current_field(level)
ctrl_cw = (
control_bits[level * 2],
control_bits[level * 2 + 1],
)
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))
leftover_bits = encoded_ctrl[-1] >> (
((IdpfPoplar.BITS + 3) % 4 + 1) * 2
)
if leftover_bits != 0 or len(encoded) != 0:
raise ERR_DECODE
return correction_words
Here, pack_bits() takes a list of bits, packs each group of eight bits into a
byte, in LSB to MSB order, padding the most significant bits of the last byte
with zeros as necessary, and returns the byte array. unpack_bits() performs
the reverse operation: it takes in a byte array and a number of bits, and
returns a list of bits, extracting eight bits from each byte in turn, in LSB to
MSB order, and stopping after the requested number of bits. If the byte array
has an incorrect length, or if unused bits in the last bytes are not zero, it
throws an error.¶
By default, Poplar1 is instantiated with IdpfPoplar (VALUE_LEN == 2) and
PrgSha3 (Section 6.2.1). This VDAF is suitable for any positive value of BITS.
Test vectors can be found in Appendix "Test Vectors".¶
VDAFs have two essential security goals:¶
Formal definitions of privacy and robustness can be found in [DPRS23]. 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:¶
Establishing secure channels:¶
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 measurements for the remaining Clients. Applications can guard against these risks by adding additional controls on report 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.¶
The Aggregators are responsible for exchanging the verification key in advance of executing the VDAF. Any procedure is acceptable as long as the following conditions are met:¶
Meeting these conditions is required in order to leverage security analysis in the framework of [DPRS23]. Their definition of robustness allows the attacker, playing the role of a cohort of malicious Clients, to submit arbitrary reports to the Aggregators and eavesdrop on their communications as they process them. Security in this model is achievable as long as the verification key is kept secret from the attacker.¶
The privacy definition of [DPRS23] considers an active attacker that controls the network and a subset of Aggregators; in addition, the attacker is allowed to choose the verification key used by each honest Aggregator over the course of the experiment. Security is achievable in this model as long as the key is picked at the start of the experiment, prior to any reports being generated. (The model also requires nonces to be generated at random; see Section 9.2 below.)¶
Meeting these requirements is relatively straightforward. For example, the Aggregators may designate one of their peers to generate the verification key and distribute it to the others. To assure Clients of key commitment, the Clients and (honest) Aggregators could bind reports to a shared context string derived from the key. For instance, the "task ID" of DAP [DAP] could be set to the hash of the verification key; then as long as honest Aggregators only consume reports for the task indicated by the Client, forging a new key after the fact would reduce to finding collisions in the underlying hash function. (Keeping the key secret from the Clients would require the hash function to be one-way.) However, since rotating the key implies rotating the task ID, this scheme would not allow key rotation over the lifetime of a task.¶
The sharding and preparation steps of VDAF execution depend on a nonce associated with the Client's report. To ensure privacy of the underlying measurement, the Client MUST generate this nonce using a CSPRNG. This is required in order to leverage security analysis for the privacy definition of [DPRS23], which assumes the nonce is chosen at random prior to generating the report.¶
Other security considerations may require the nonce to be non-repeating. For example, to achieve differential privacy it is necessary to avoid "over exposing" a measurement by including it too many times in a single batch or across multiple batches. It is RECOMMENDED that the nonce generated by the Client be used by the Aggregators for replay protection.¶
As described in Section 4.3 and Section 5.3 respectively, DAFs and VDAFs may impose restrictions on the re-use of input shares. This is to ensure that correlated randomness provided by the Client through the input share is not used more than once, which might compromise confidentiality of the Client's measurements.¶
Protocols that make use of VDAFs therefore MUST call Vdaf.is_valid
on the set of all aggregation parameters used for a Client's input share, and
only proceed with the preparation and aggregation phases if that function call
returns True.¶
A codepoint for each (V)DAF in this document is defined in the table below. Note
that 0xFFFF0000 through 0xFFFFFFFF are reserved for private use.¶
| Value | Scheme | Type | Reference |
|---|---|---|---|
0x00000000
|
Prio3Count | VDAF | Section 7.4.1 |
0x00000001
|
Prio3Sum | VDAF | Section 7.4.2 |
0x00000002
|
Prio3Histogram | VDAF | Section 7.4.3 |
0x00000003 to 0x00000FFF
|
reserved for Prio3 | VDAF | n/a |
0x00001000
|
Poplar1 | VDAF | Section 8.4 |
0xFFFF0000 to 0xFFFFFFFF
|
reserved | n/a | n/a |
OPEN ISSUE Currently the scheme includes the PRG. This means that we need bits of the codepoint to differentiate between PRGs. We could instead make the PRG generic (e.g., Prio3Count(Aes128) instead of Prio3Aes128Count) and define a separate codepoint.¶
The security considerations in Section 9 are based largely on the security analysis of [DPRS23]. Thanks to Hannah Davis and Mike Rosulek, who lent their time to developing definitions and security proofs.¶
Thanks to Henry Corrigan-Gibbs, Armando Faz-Hernández, Simon Friedberger, Tim Geoghegan, Mariana Raykova, Jacob Rothstein, and Christopher Wood for useful feedback on and contributions to the spec.¶
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 Prio3Sum, 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: >-
d8700d17841945acb2891eaadb82b14c71098bcf4a708cf5bfff34c19f1fcb389524
33062a90142b69f3de7e58566e86
input_share_1: >-
0101010101010101010101010101010101010101010101010101010101010101
round_0:
prep_share_0: >-
bc5ed32148ccdfe8149c04ac5b2e26bbb74c2c01d26b8e9a4f26452740c8b6e9
prep_share_1: >-
43a12cddb7332019fa93469841dbb1aba89845e53745d13445a31850c99c4ea8
prep_message: >-
out_share_0:
- d8700d17841945ac
out_share_1:
- 278ff2e77be6ba56
agg_share_0: >-
d8700d17841945ac
agg_share_1: >-
278ff2e77be6ba56
agg_result: 1
¶
bits: 8
verify_key: "01010101010101010101010101010101"
upload_0:
measurement: 100
nonce: "01010101010101010101010101010101"
public_share: >-
905f744fdd73d539b852697c311bcc977e2eef2723dabfbae02c93ce1f196ae8
input_share_0: >-
9129dfc509b86e7947f238dd725ae3cc8393f640c44fd72eb0dd39bf28fbbf8e9455
f069ca3089119d684ae29c280e954061350c04c90916f27895fe374ebd6badc69ad0
e1fda3e02a85750c9a27b4f34a4dd09ce0c2cef5599c0a0a40c2c3fdea329eaa925f
a1a65b7f947e9ec0b395412e56245f4d9fa5993f3e6a7263d71d73ac5dbb78a3104f
23f0c67893b390b1b14a6211c54651645548d145213de5c87acd21d6218248991394
708b9d823c9d9d7e16996f32d4578c2ecec9b0c2673c4d1ee156ffac2415369c1583
21482ba0d8fa021f3c00185a99c74d619637f463e6d5bb22bb45cf3d43a9ba332148
a61b42c26567563b8d9d612a8e85be68dd542838a2e1bd03e471ccf5514e333b3837
0a41c8df749dd43a86ee2708c1797dd10fb2d655700c3d74e4c9a9726eafa553035d
846162f981925e83d8dac7f08784339c3b89f54461ae764239bb521b6e3419ebe8d5
6638968479a53a9ec5453b3f94dad0e3b41e71bc273222733428a18f3e66bdb42081
55aa11215ce9c91bbf7664e10992cc2d8c30aa0abf524b6fc545a599a43b66d676a0
2be73e6fc8c24e0ad17eceba6e081d8d8e7882eae163cc2afcc1e76cb7b3936168e0
a71ce3f73999e7769ec43108075657cc36f5108a0b1713777bc6b561fed05dd0869d
596c8cc079628e94440df17b63486d5b1e6af79dd1378f45bbdf68497d064e0c9f63
b768bdd7e0761ec441b1a529e742d4e13f4755cdc99569236592dae907aab4ba5304
cb7224469fb8bd76f2b348026d5d6312fc8c52f891461b08c5b05fdae2e1c745ce94
63b92efb59a12cca026c6afb195322ef69c5eb452788742075e41231991d324f24f6
21f6b79d6da2472438b455379856f7623ca9a4419d61eababea02634010101010101
01010101010101010101
input_share_1: >-
01010101010101010101010101010101010101010101010101010101010101010101
0101010101010101010101010101
round_0:
prep_share_0: >-
b11ad355381629c48921bae8ff502c7e9b214980f51d58fd3b7293542ddb43f83f
f15e16a0d7b38ed62357f1dbae34d4905f744fdd73d539b852697c311bcc97
prep_share_1: >-
4ee52caac7e9d61f76de451700afd383f622613281a91f1113882cd7646c207484
da515e63d02452fe32316648fd84277e2eef2723dabfbae02c93ce1f196ae8
prep_message: >-
fab4374d7e6549b8ae110e76b4eb315d
out_share_0:
- 36a9b3cc584eefb20e6f6eca8c54caaf
out_share_1:
- c9564c33a7b11031f190913573ab35b6
agg_share_0: >-
36a9b3cc584eefb20e6f6eca8c54caaf
agg_share_1: >-
c9564c33a7b11031f190913573ab35b6
agg_result: 100
¶
buckets: [1, 10, 100]
verify_key: "01010101010101010101010101010101"
upload_0:
measurement: 50
nonce: "01010101010101010101010101010101"
public_share: >-
d995029335b0240dbdca1cbfb1211ecc67fa8edb6f0f74882a626d290baa199c
input_share_0: >-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_share_1: >-
01010101010101010101010101010101010101010101010101010101010101010101
0101010101010101010101010101
round_0:
prep_share_0: >-
2d3f3200d6afb340b09dfc30b92ea4146ad4f5e2677515ed1c4e6ebcc8b6809ef5
762a4ee715e20e05b219f8529dbfc1d995029335b0240dbdca1cbfb1211ecc
prep_share_1: >-
d2c0cdff29504ca34f6203cf46d15bed9083e4170397a5da2d55754b9a7bbf3803
0a54eebf540cd5ff5e6b07d451ce7067fa8edb6f0f74882a626d290baa199c
prep_message: >-
c3885ffa3a60ba7f6413596d328cfd27
out_share_0:
- 22d711113962f5ffe2f3119db262a769
- 40036dd11c6ac7a0ea844da70ed1cb57
- 5adaee861cb1075340a455a13bbadd62
- 70e2ab78349dbf2f7aca71d9a77efb01
out_share_1:
- dd28eeeec69d09e41d0cee624d9d5898
- bffc922ee3953843157bb258f12e34aa
- a5251179e34ef890bf5baa5ec44522a0
- 8f1d5487cb6240b485358e2658810500
agg_share_0: >-
22d711113962f5ffe2f3119db262a76940036dd11c6ac7a0ea844da70ed1cb575adaee
861cb1075340a455a13bbadd6270e2ab78349dbf2f7aca71d9a77efb01
agg_share_1: >-
dd28eeeec69d09e41d0cee624d9d5898bffc922ee3953843157bb258f12e34aaa52511
79e34ef890bf5baa5ec44522a08f1d5487cb6240b485358e2658810500
agg_result: [0, 0, 1, 0]
¶
verify_key: "01010101010101010101010101010101"
agg_param: (0, [0, 1])
upload_0:
round_0:
prep_share_0: >-
45d4d76583ff3d4bfadb0a1a136a6df5139a62a553367029
prep_share_1: >-
a3f8aa69980f13c2749bf9b3c23e21e19e39198fb3f8e2ed
prep_message: >-
e9cd81cf1c0e510d6f7703ced5a88fd5b1d37c35072f5316
round_1:
prep_share_0: >-
23568c2a012e01c9
prep_share_1: >-
dca973d4fed1fe38
prep_message: >-
out_share_0:
- fcf3039f72d70136
- 7a930346dcd33ac4
out_share_1:
- 030cfc5f8d28fecb
- 856cfcb8232cc53e
agg_share_0: >-
fcf3039f72d701367a930346dcd33ac4
agg_share_1: >-
030cfc5f8d28fecb856cfcb8232cc53e
agg_result: [0, 1]
¶
verify_key: "01010101010101010101010101010101"
agg_param: (1, [0, 1, 2, 3])
upload_0:
round_0:
prep_share_0: >-
7ff2aa9b62d4699bb13b5d89054dc718ef0ef340d31e71e7
prep_share_1: >-
152587e270e3ad1f7166f605136db06284feef9e0c60ec4a
prep_message: >-
9518327dd3b816ba22a2538f18bb7779740de2dfdf7f5e30
round_1:
prep_share_0: >-
edbf9effc5523ad6
prep_share_1: >-
124060ff3aadc52b
prep_message: >-
out_share_0:
- e6b4675d5645143d
- 6751646bdd854e4f
- 782cd9dd3b220cbe
- 07af78ced3d6e89f
out_share_1:
- 194b98a1a9baebc4
- 98ae9b93227ab1b2
- 87d32621c4ddf343
- f85087302c291763
agg_share_0: >-
e6b4675d5645143d6751646bdd854e4f782cd9dd3b220cbe07af78ced3d6e89f
agg_share_1: >-
194b98a1a9baebc498ae9b93227ab1b287d32621c4ddf343f85087302c291763
agg_result: [0, 0, 0, 1]
¶
verify_key: "01010101010101010101010101010101"
agg_param: (2, [0, 2, 4, 6])
upload_0:
round_0:
prep_share_0: >-
6d956ea3b01727f03e3c4b1cad1461c3fae8fc676763d8ef
prep_share_1: >-
9b8bfdb242f67c7aed868f197e62d46f4450ac65b089a5b6
prep_message: >-
09216c56f30da4692bc2da372b7736313f39a8ce17ed7ea4
round_1:
prep_share_0: >-
a22d3f45a76cf4f3
prep_share_1: >-
5dd2c0b958930b0e
prep_message: >-
out_share_0:
- bb4d5c8b44fea90f
- 01c9989f7fa25aef
- 46f355d9bc179269
- 50600db413c684ce
out_share_1:
- 44b2a373bb0156f2
- fe36675f805da512
- b90caa2543e86d98
- af9ff24aec397b34
agg_share_0: >-
bb4d5c8b44fea90f01c9989f7fa25aef46f355d9bc17926950600db413c684ce
agg_share_1: >-
44b2a373bb0156f2fe36675f805da512b90caa2543e86d98af9ff24aec397b34
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: >-
21acceb054f9fe345aa353803f2debb5a307dad06b26dc6a7c7b79a1a665448b20
88a892dad00cba3ab6ab9b9283b734937b281810100158daca6188dbbc4e155e3a
0697f2291a6f723b6085dedf9a23817a554358e9f191aaa2a2425c938238
prep_share_1: >-
11fbc7788b2224efb1c760840c8ef96f400e6762e8bcfa000823120f2fb12e3838
7ab34acd8cbbb78016048aaf737e167c8b7660581df35eb1af7b95de07d29526de
fc2f23c59aff034e7feec503ae2e064db27e2c0f57ce905cfba9fcc3dab0
prep_message: >-
33a89628e01c23240c6ab4044bbce524e316423353e3d66a849e8bb0d61672c359
035bdda85cc871baccb02641f7354b10069e78682df4b78c79dd1eb9c420aa0519
02c715eeb56e7589e074a3e3485187c807c184f949603aff9dec59575cfb
round_1:
prep_share_0: >-
3dc137d56f1e1eb3f09ea7300d66f972d74efd3f1efd667bd6079dc9b16179d6
prep_share_1: >-
423ec82a90e1e14c0f6158cff299068d28b102c0e102998429f862364e9e8617
prep_message: >-
out_share_0:
- 5ea259ad47a621cc04bc4804ecc9d302ccd2225b37e82770912154e4c0495f6b
- 6f8fb25f78167a93352cd07ca7bd6ea4f12f457b2968a62f4fc721fd12946030
- 066e1fbff795f0eab7023023e8662c0d01804d169505f1f58bb005b1ae8d7906
- 241a08089dd8e7c318987d598d3fe8cb2c7870b6d3a46228307f5cc50ee1b81b
- 16d9502d5eb5ba9bd5e5c55e0385b35ee3a944585225848fd5cb82ce78ea0e77
- 2ccf7b85827ba848284a28c82e5b8bb9c591b4b21628f374c3fc2cf79e3e8560
- 5e16bb7dbf1ca16bf038dbc8702e4a1ee79074856c05d742e3d43e6f94d8022c
out_share_1:
- 215da652b859de33fb43b7fb13362cfd332ddda4c817d88f6edeab1b3fb6a082
- 10704da087e9856ccad32f835842915b0ed0ba84d69759d0b038de02ed6b9fbd
- 7991e040086a0f1548fdcfdc1799d3f2fe7fb2e96afa0e0a744ffa4e517286e7
- 5be5f7f76227183ce76782a672c01734d3878f492c5b9dd7cf80a33af11e47d2
- 6926afd2a14a45642a1a3aa1fc7a4ca11c56bba7adda7b702a347d318715f176
- 5330847a7d8457b7d7b5d737d1a474463a6e4b4de9d70c8b3c03d30861c17a8e
- 21e9448240e35e940fc724378fd1b5e1186f8b7a93fa28bd1c2bc1906b27fdc1
agg_share_0: >-
5ea259ad47a621cc04bc4804ecc9d302ccd2225b37e82770912154e4c0495f6b6f8fb2
5f78167a93352cd07ca7bd6ea4f12f457b2968a62f4fc721fd12946030066e1fbff795
f0eab7023023e8662c0d01804d169505f1f58bb005b1ae8d7906241a08089dd8e7c318
987d598d3fe8cb2c7870b6d3a46228307f5cc50ee1b81b16d9502d5eb5ba9bd5e5c55e
0385b35ee3a944585225848fd5cb82ce78ea0e772ccf7b85827ba848284a28c82e5b8b
b9c591b4b21628f374c3fc2cf79e3e85605e16bb7dbf1ca16bf038dbc8702e4a1ee790
74856c05d742e3d43e6f94d8022c
agg_share_1: >-
215da652b859de33fb43b7fb13362cfd332ddda4c817d88f6edeab1b3fb6a08210704d
a087e9856ccad32f835842915b0ed0ba84d69759d0b038de02ed6b9fbd7991e040086a
0f1548fdcfdc1799d3f2fe7fb2e96afa0e0a744ffa4e517286e75be5f7f76227183ce7
6782a672c01734d3878f492c5b9dd7cf80a33af11e47d26926afd2a14a45642a1a3aa1
fc7a4ca11c56bba7adda7b702a347d318715f1765330847a7d8457b7d7b5d737d1a474
463a6e4b4de9d70c8b3c03d30861c17a8e21e9448240e35e940fc724378fd1b5e1186f
8b7a93fa28bd1c2bc1906b27fdc1
agg_result: [0, 0, 0, 0, 0, 1, 0]
¶