Crypto Forum J. Chen
InternetDraft Apple Inc.
Intended status: Informational C. Patton
Expires: 5 September 2024 Cloudflare
4 March 2024
Private Inexpensive Norm Enforcement (PINE) VDAF
draftchencfrgvdafpine00
Abstract
This document describes PINE, a Verifiable Distributed Aggregation
Function (VDAF) for secure aggregation of highdimensional, real
valued vectors with bounded L2norm. PINE is intended to facilitate
private and robust federated machine learning.
About This Document
This note is to be removed before publishing as an RFC.
The latest revision of this draft can be found at
https://junyechen1996.github.io/draftchencfrgvdafpine/draftchen
cfrgvdafpine.html. Status information for this document may be
found at https://datatracker.ietf.org/doc/draftchencfrgvdafpine/.
Discussion of this document takes place on the Crypto Forum Research
Group mailing list (mailto:cfrg@ietf.org), which is archived at
https://mailarchive.ietf.org/arch/search/?email_list=cfrg. Subscribe
at https://www.ietf.org/mailman/listinfo/cfrg/.
Source for this draft and an issue tracker can be found at
https://github.com/junyechen1996/draftchencfrgvdafpine.
Status of This Memo
This InternetDraft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
InternetDrafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
working documents as InternetDrafts. The list of current Internet
Drafts is at https://datatracker.ietf.org/drafts/current/.
InternetDrafts are draft documents valid for a maximum of six months
and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use InternetDrafts as reference
material or to cite them other than as "work in progress."
Chen & Patton Expires 5 September 2024 [Page 1]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
This InternetDraft will expire on 5 September 2024.
Copyright Notice
Copyright (c) 2024 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents (https://trustee.ietf.org/
licenseinfo) in effect on the date of publication of this document.
Please review these documents carefully, as they describe your rights
and restrictions with respect to this document.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Conventions and Definitions . . . . . . . . . . . . . . . . . 5
3. PINE Overview . . . . . . . . . . . . . . . . . . . . . . . . 6
4. The PINE Proof System . . . . . . . . . . . . . . . . . . . . 7
4.1. Measurement Encoding . . . . . . . . . . . . . . . . . . 9
4.1.1. Encoding RangeChecked Results . . . . . . . . . . . 9
4.1.2. Encoding Gradient and L2Norm Check . . . . . . . . . 9
4.1.3. Running the Wraparound Checks . . . . . . . . . . . . 10
4.1.4. Encoding the RangeChecked, Wraparound Check
Results . . . . . . . . . . . . . . . . . . . . . . . 10
4.2. The FLP Circuit . . . . . . . . . . . . . . . . . . . . . 11
4.2.1. Range Check . . . . . . . . . . . . . . . . . . . . . 12
4.2.2. Bit Check . . . . . . . . . . . . . . . . . . . . . . 12
4.2.3. L2 Norm Check . . . . . . . . . . . . . . . . . . . . 12
4.2.4. Wraparound Check . . . . . . . . . . . . . . . . . . 13
4.2.5. Putting All Checks Together . . . . . . . . . . . . . 13
5. The PINE VDAF . . . . . . . . . . . . . . . . . . . . . . . . 14
5.1. Sharding . . . . . . . . . . . . . . . . . . . . . . . . 15
5.2. Preparation . . . . . . . . . . . . . . . . . . . . . . . 15
5.3. Aggregation . . . . . . . . . . . . . . . . . . . . . . . 15
5.4. Unsharding . . . . . . . . . . . . . . . . . . . . . . . 16
6. Variants . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7. PINE Auxiliary Functions . . . . . . . . . . . . . . . . . . 16
8. Security Considerations . . . . . . . . . . . . . . . . . . . 16
9. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 16
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 16
10.1. Normative References . . . . . . . . . . . . . . . . . . 16
10.2. Informative References . . . . . . . . . . . . . . . . . 16
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 17
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 17
Chen & Patton Expires 5 September 2024 [Page 2]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
1. Introduction
The goal of federated machine learning [MR17] is to enable training
of machine learning models from data stored on users' devices. The
bulk of the computation is carried out ondevice: each user trains
the model on its data locally, then sends a model update to a central
server. These model updates are commonly referred to as "gradients"
[Lem12]. The server aggregates the gradients, applies them to the
central model, and sends the updated model to the users to repeat the
process.
data

v
++ gradients ++
 Clients + >  Server 
++ + ++
++  
++ 
^ 
 updated model 
++
Figure 1: Federated learning
Federated learning improves user privacy by ensuring the training
data never leaves users' devices. However, it requires computing an
aggregate of the gradients sent from devices, which may still reveal
a significant amount of information about the underlying data. One
way to mitigate this risk is to distribute the aggregation step
across multiple servers such that no server sees any gradient in the
clear.
With a Verifiable Distributed Aggregation Function [VDAF], this is
achieved by having each user shard their gradient into a number of
secret shares, one for each aggregation server. Each server
aggregates their shares locally, then combines their share of the
aggregate with the other servers to get the aggregate result.
Chen & Patton Expires 5 September 2024 [Page 3]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
data

v gradient aggregate
++ shares ++ shares ++
 Clients + >  Aggregators + >  Collector 
++ + ++  ++
++  ++ 
++ 
^ 
 updated model 
++
Figure 2: Federated learning with a VDAF
Along with keeping the gradients private, it is desirable to ensure
robustness of the overall computation by preventing clients from
"poisoning" the aggregate and corrupting the trained machine learning
model. A client's gradient is typically expressed as a vector of
real numbers. A common goal is to ensure each gradient has a bounded
"L2norm" (sometimes called Euclidean norm): the square root of the
sum of the squares of each entry of the input vector. Bounding the
L2 norm is used in federated learning to limit the contribution of
each client to the aggregate, without over constraining the
distribution of inputs. [CP: Add a relevant reference.]
In theory, Prio3 (Section 7 of [VDAF]) could be adapted to support
this functionality, but for highdimensional data, the concrete cost
in terms of runtime and communication would be prohibitively high.
The basic idea is simple. An FLP ("Fully Linear Proof", see
Section 7.3 of [VDAF]) could be used to compute the L2 norm of the
secret shared gradient and check that the result is in the desired
range, all without learning the gradient or its norm. This
computation, on its own, can be done efficiently: the challenge lies
in ensuring that the computation itself was carried out correctly,
while properly accounting for the relevant mathematical details of
the proof system and the range of possible inputs.
This document describes PINE ("Private Inexpensive Norm
Enforcement"), a VDAF for secure aggregation of gradients with
bounded L2norm [ROCT23]. Its design is based largely on Prio3 in
that the norm is computed and verified using an FLP. However, PINE
uses a new technique for verifying the correctness of the norm
computation that is incompatible with Prio3.
We give an overview of this technique in Section 3. In Section 4 we
specify an FLP circuit and accompanying encoding scheme for computing
and verifying the L2 norm of each gradient. Finally, in Section 5 we
specify the complete multiparty, 1round VDAF.
Chen & Patton Expires 5 September 2024 [Page 4]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
NOTE As of this draft, the algorithms are not yet fully specified.
We are still working out some of the minor details. In the
meantime, please refer to the reference code on which the spec
will be based: https://github.com/junyechen1996/draftchencfrg
vdafpine/tree/main/poc
2. Conventions and Definitions
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in
BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
This document uses the same parameters and conventions specified for:
* Clients, Aggregators, and Collectors from Section 5 of [VDAF].
* Finite fields from Section 6.1 of [VDAF]. All fields in this
document have prime order.
* XOFs ("eXtendable Output Functions") from Section 6.2 of [VDAF].
A floating point number, denoted float, is a IEEE754 compatible
float64 value [IEEE7542019].
A "gradient" is a vector of floating point numbers. Each coordinate
of this vector is called an "entry". The "L2 norm", or simply
"norm", of a gradient is the square root of the sum of the squares of
its entries.
The "dot product" of two vectors is to compute the sum of element
wise multiplications of the two vectors.
The userspecified parameters to initialize PINE are defined in
Table 1.
Chen & Patton Expires 5 September 2024 [Page 5]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
+===============+=======+=================================+
 Parameter  Type  Description 
+===============+=======+=================================+
 l2_norm_bound  float  The L2 norm upper bound 
   (inclusive). 
++++
 dimension  int  Dimension of each gradient. 
++++
 num_frac_bits  int  The number of bits of precision 
   to use when encoding each 
   gradient entry into the field. 
++++
Table 1: User parameters for PINE.
3. PINE Overview
This section provides an overview of the main technical contribution
of [ROCT23] that forms the basis of PINE. To motivate their idea,
let us first say how Prio3 from Section 7 of [VDAF] would be used to
aggregate vectors with bounded L2 norm.
Prio3 uses an FLP ("Fully Linear Proof"; see Section 7.3 of [VDAF])
to verify properties of a secret shared measurement without revealing
the measurement to the Aggregators. The property to be verified is
expressed as an arithmetic circuit over a finite field (Section 7.3.2
of [VDAF]). Let q denote the field modulus.
In our case, the circuit would take (a share of) the gradient as
input, compute the squared L2norm (the sum of the squares of the
entries of the gradient), and check that the result is in the desired
range. Note that we do not compute the exact norm: it is
mathematically equivalent to compute the squared norm and check that
it is smaller than the square of the bound.
Crucially, arithmetic in this computation is modulo q. This means
that, for a given gradient, the norm may have a different result when
computed in our finite field than in the ring of integers. For
example, suppose our bound is 10: the gradient [99, 0, 7] has squared
L2norm of 9850 over the integers (out of range), but only 6 modulo q
= 23 (in range). This circuit would therefore deem the gradient
valid, when in fact it is invalid.
Thus the central challenge of adapting FLPs to this problem is to
prevent the norm computation from "wrapping around" the field
modulus.
Chen & Patton Expires 5 September 2024 [Page 6]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
One way to achieve this is to ensure that each gradient entry is in a
range that ensures the norm is sufficiently small. However, this
approach has high communication cost (roughly num_frac_bits *
dimension field elements per entry), which becomes prohibitive for
highdimensional data.
PINE uses a different strategy: rather than prevent wraparounds, we
can try to detect whether a wraparound has occurred.
[ROCT23] devises a probabilistic test for this purpose. A random
vector over the field is generated (via a procedure described in
Section 4.1.3) where each entry is sampled independently from a
particular probability distribution. To test for wraparound, compute
the dot product of this vector and the gradient, and check if the
result is in a specific range determined by parameters in Table 1.
If the norm wraps around the field modulus, then the dot product is
likely to be large. In fact, [ROCT23] show that this test correctly
detects wraparounds with probability 1/2. To decrease the false
negative probability (that is, the probability of misclassifying an
invalid gradient as valid), we simply repeat this test a number of
times, each time with a vector sampled from the same distribution.
However, [ROCT23] also show that each wraparound test has a nonzero
false positive probability (the probability of misclassifying a valid
gradient as invalid). We refer to this probability as the "zero
knowledge error", or in short, "ZK error". This creates a problem
for privacy, as the Aggregators learn information about a valid
gradient they were not meant to learn: whether its dot product with a
known vector is in a particular range. [CP: We need a more intuitive
explanation of the information that's leaked.] The parameters of
PINE are chosen carefully in order to ensure this leakage is
negligible.
4. The PINE Proof System
This section specifies a randomized encoding of gradients and FLP
circuit (Section 7.3 of [VDAF]) for checking that (1) the gradient's
squared L2norm falls in the desired range and (2) the squared
L2norm does not wrap around the field modulus. We specify the
encoding and validity circuit in a class PineValid.
The encoding algorithm takes as input the gradient and an XOF seed
used to derive the random vectors for the wraparound tests. The seed
must be known both to the Client and the Aggregators: Section 5
describes how the seed is derived from shares of the gradient.
Chen & Patton Expires 5 September 2024 [Page 7]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
Operational parameters for the proof system are summarized below in
Table 2.
+=======================+=======+=============================+
 Parameter  Type  Description 
+=======================+=======+=============================+
 alpha  float  Parameter in wraparound 
   check that determines the 
   ZK error. The higher alpha 
   is, the lower ZK error is. 
++++
 num_wr_checks  int  Number of wraparound checks 
   to run. 
++++
 num_wr_successes  int  Minimum number of 
   wraparound checks that a 
   Client must pass. 
++++
 encoded_sq_norm_bound  Field  The square of l2_norm_bound 
   encoded into a field 
   element. 
++++
 wr_check_bound  Field  The bound of the range 
   check for each wraparound 
   check. 
++++
 num_bits_for_sq_norm  int  Number of bits to encode 
   the squared L2norm. 
++++
 num_bits_for_wr_check  int  Number of bits to encode 
   the range check in each 
   wraparound check. 
++++
 bit_checked_len  int  Number of field elements in 
   the encoded measurement 
   that are expected to be 
   bits. 
++++
 chunk_length  int  Parameter of the FLP. 
++++
Table 2: Operational parameters of the PINE FLP.
Chen & Patton Expires 5 September 2024 [Page 8]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
4.1. Measurement Encoding
The measurement encoding is done in two stages: * Section 4.1.2
involves encoding floating point numbers in the Client gradient into
field elements Section 4.1.2.1, and encoding the results for L2norm
check Section 4.1.2.2, by computing the bit representation of the
squared L2norm, modulo q, of the encoded gradient. The result of
this step allows Aggregators to check the squared L2norm of the
Client's gradient, modulo q, falls in the desired range of [0,
encoded_sq_norm_bound]. * Section 4.1.4 involves encoding the results
of running wraparound checks Section 4.1.3, based on the encoded
gradient from the previous step, and the random vectors derived from
a short, random seed using an XOF. The result of this step, along
with the encoded gradient and the random vector that the Aggregators
derive on their own, allow the Aggregators to run wraparound checks
on their own.
4.1.1. Encoding RangeChecked Results
Encoding rangechecked results is a common subroutine during
measurement encoding. The goal is to allow the Client to prove a
value is in the desired range of [B1, B2], over the field modulus q
(see Figure 1 in [ROCT23]). The Client computes the "v bits", the
bit representation of value  B1 (modulo q), and the "u bits", the
bit representation of B2  value (modulo q). The number of bits for
the v and u bits is ceil(log2(B2  B1 + 1)).
As an optimization for communication cost per Remark 3.2 in [ROCT23],
the Client can skip sending the u bits if B2  B1 + 1 (modulo q) is a
power of 2. This is because the available v bits can naturally bound
value  B1 to be B2  B1.
4.1.2. Encoding Gradient and L2Norm Check
We define a function PineValid.encode_gradient_and_norm(self,
measurement: list[float]) > list[Field] that implements this
encoding step.
4.1.2.1. Encoding of Floating Point Numbers into Field Elements
TODO Specify how floating point numbers are represented as field
elements.
4.1.2.2. Encoding the RangeChecked, Squared Norm
TODO Specify how the Client encodes the norm such that the
Aggregators can check that it is in the desired range.
Chen & Patton Expires 5 September 2024 [Page 9]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
TODO Put full implementation of encode_gradient_and_norm() here.
4.1.3. Running the Wraparound Checks
Given the encoded gradient from Section 4.1.2 and the XOF to generate
the random vectors, the Client needs to run the wraparound check
num_wr_checks times. Each wraparound check works as follows.
The Client generates a random vector with the same dimension as the
gradient's dimension. Each entry of the random vector is a field
element of 1 with probability 1/4, or 0 with probability 1/2, or q1
with probability 1/4, over the field modulus q. The Client samples
each entry by sampling from the XOF output stream two bits at a time:
* If the bits are 00, set the entry to be q1. * If the bits are 01
or 10, set the entry to be 0. * If the bits are 11, set the entry to
be 1.
Finally, the Client computes the dot product of the encoded gradient
and the random vector, modulo q.
Note the Client does not send this dot product to the Aggregators.
The Aggregators will compute the dot product themselves, based on the
encoded gradient and the random vector derived on their own.
4.1.4. Encoding the RangeChecked, Wraparound Check Results
We define a function PineValid.encode_wr_checks(self,
encoded_gradient: list[Field], wr_joint_rand_xof: Xof) >
tuple[list[Field], list[Field]] that implements this encoding step.
It returns the tuple of rangechecked, wraparound check results that
will be sent to the Aggregators, and the wraparound check results
(i.e., the dot products from Section 4.1.3) that will be passed as
inputs to the FLP circuit.
The Client obtains the wraparound check results, as described in
Section 4.1.3. For each check, the Client runs the range check on
the result to see if it is in the range of [wr_check_bound + 1,
wr_check_bound]. Note we choose wr_check_bound, such that
wr_check_bound is a power of 2, so the Client does not have to send
the u bits in range check. The Client also keeps track of a success
bit wr_check_g, which is a 1 if the wraparound check result is in
range, and 0 otherwise.
The Client counts how many wraparound checks it has passed. If it
has passed fewer than num_wr_successes of them, it should retry, by
using a new XOF seed to regenerate the random vectors and rerun
wraparound checks Section 4.1.3.
Chen & Patton Expires 5 September 2024 [Page 10]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
The rangechecked results and the success bits are sent to the
Aggregators, and the wraparound check results are passed to the FLP
circuit.
4.2. The FLP Circuit
Evaluation of the validity circuit begins by unpacking the encoded
measurement into the following components:
* The first dimension entries are the encoded_gradient, the field
elements encoded from the floating point numbers.
* The next bit_checked_len entries are expected to be bits, and
should contain the bits for the rangechecked L2norm, the bits
for the rangechecked wraparound check results, and the success
bits in wraparound checks.
* The last num_wr_checks are the wraparound check results, i.e., the
dot products of the encoded gradient and the random vectors.
It also unpacks the "joint randomness" that is shared between the
Client and Aggregators, to compute random linear combinations of all
the checks:
* The first joint randomness field element is to reduce over the bit
checks at all bit entries.
* The second joint randomness field element is to reduce over all
the quadratic checks in wraparound check.
* The last joint randomness field element is to reduce over all the
checks, which include the reduced bit check result, the L2 norm
equality check, the L2 norm range check, the reduced quadratic
checks in wraparound check, and the success count check for
wraparound check.
In the subsections below, we outline the various checks computed by
the validity circuit, which includes the bit check on all the bit
entries Section 4.2.2, the L2 norm check Section 4.2.3, and the
wraparound check Section 4.2.4. Some of the auxiliary functions in
these checks are defined in Section 7.
Chen & Patton Expires 5 September 2024 [Page 11]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
4.2.1. Range Check
In order to verify the rangechecked results reported by the Client
as described in Section 4.1.1, the Aggregators will verify (1) v bits
and u bits are indeed composed of bits, as described in
Section 4.2.2, and (2) the verify the decoded value from the v bits,
and the decoded value from the u bits sum up to B2  B1 (modulo q).
If the Client skips sending the u bits as an optimization mentioned
in Section 4.1.4, then the Aggregators only need to verify the
decoded value from the v bits is equal to B2  B1 (modulo q).
4.2.2. Bit Check
The purpose of bit check on a field element is to prevent any
computation involving the field element from going out of range. For
example, if we were to compute the squared L2norm from the bit
representation claimed by the Client, bit check ensures the decoded
value from the bit representation is at most 2^(num_bits_for_norm) 
1.
To run bit check on an array of field elements bit_checked, we use a
similar approach as Section 7.3.1.1 of [VDAF], by constructing a
polynomial from a random linear combination of the polynomial x(x1)
evaluated at each element of bit_checked. We then evaluate the
polynomial at a random point r_bit_check, i.e., the joint randomness
for bit check:
f(r_bit_check) = bit_checked[0] * (bit_checked[0]  1) + \
r_bit_check * bit_checked[1] * (bit_checked[1]  1) + \
r_bit_check^2 * bit_checked[2] * (bit_checked[2]  1) + ...
If one of the entries in bit_checked is not a bit, then
f(r_bit_check) is nonzero with high probability.
TODO Put PineValid.eval_bit_check() implementation here.
4.2.3. L2 Norm Check
The purpose of L2 norm check is to check the squared L2norm of the
encoded gradient is in the range of [0, encoded_sq_norm_bound].
The validity circuit verifies two properties of the L2 norm reported
by the Client:
* Equality check: The squared norm computed from the encoded
gradient is equal to the bit representation reported by the
Client. For this, the Aggregators compute their shares of the
Chen & Patton Expires 5 September 2024 [Page 12]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
squared norm from their shares of the encoded gradient, and also
decode their shares of the bit representation of the squared norm
(as defined above in Section 4.2.2), and check that the values are
equal.
* Range check: The squared norm reported by the Client is in the
desired range [0, encoded_sq_norm_bound]. For this, the
Aggregators run the range check described in Section 4.2.1.
TODO Put PineValid.eval_norm_check() implementation here.
4.2.4. Wraparound Check
The purpose of wraparound check is to check the squared L2norm of
the encoded Client gradient hasn't overflown the field modulus q.
The validity circuit verifies two properties for wraparound checks:
* Quadratic check (See bullet point 3 in Figure 2 of [ROCT23]):
Recall in Section 4.1.4, the Client keeps track of a success bit
for each wraparound check, i.e., whether it has passed that check.
For each check, the Aggregators then verify a quadratic constraint
that, either the success bit is a 0 (i.e., the Client has failed
that check), or the success bit is a 1, and the rangechecked
result reported by the Client is correct, based on the wraparound
check result (i.e., the dot product) computed by the Aggregators
from the encoded gradient and the random vector. For this, the
Aggregators multiply their shares of the success bit, and the
difference of the rangechecked result reported by the Client, and
that computed by the Aggregators. We then construct a polynomial
from a random linear combination of the quadratic check at each
wraparound check, and evaluate it at a random point r_wr_check,
the joint randomness.
* Success count check: The number of successful wraparound checks,
by summing the success bits, is equal to the constant
num_wr_successes. For this, the Aggregators sum their shares of
the success bits for all wraparound checks.
TODO Put PineValid.eval_wr_check() implementation here.
4.2.5. Putting All Checks Together
Finally, we will construct a polynomial from a random linear
combination of all the checks from PineValid.eval_bit_checks(),
PineValid.eval_norm_check(), and PineValid.eval_wr_check(), and
evaluate it at the final joint randomness r_final. The full
implementation of PineValid.eval() is as follows:
Chen & Patton Expires 5 September 2024 [Page 13]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
TODO Specify the implementation of Valid from Section 7.3.2 of
[VDAF].
5. The PINE VDAF
This section describes PINE VDAF for [ROCT23], a oneround VDAF with
no aggregation parameter. It takes a set of Client gradients
expressed as vectors of floating point values, and computes an
elementwise summation of valid gradients with bounded L2norm
configured by the user parameters in Table 1. The VDAF largely uses
the encoding and validation schemes in Section 4, and also specifies
how the joint randomness shared between the Client and Aggregators is
derived. There are two kinds of joint randomness used:
* "Verification joint randomness": These are the field elements used
by the Client and Aggregators to evaluate the FLP circuit. The
verification joint randomness is derived similar to the joint
randomness in Prio3 Section 7.2.1.2 of [VDAF]: the XOF is applied
to each secret share of the encoded measurement to derive the
"part"; and the parts are hashed together, using the XOF once
more, to get the seed for deriving the joint randomness itself.
* "Wraparound joint randomness": This is used to generate the random
vectors in the wraparound checks that both the Clients and
Aggregators need to derive on their own. It is generated in much
the same way as the verification joint randomness, except that
only the gradient and the rangechecked norm are used to derive
the parts.
In order for the Client to shard its gradient into input shares for
the Aggregators, the Client first encodes its gradient into field
elements, and encodes the rangechecked L2norm, according to
Section 4.1.2. Next, it derives the wraparound joint randomness for
the wraparound checks as described above, and uses that to encode the
rangechecked, wraparound check results as described in
Section 4.1.4}. The encoded gradient, rangechecked norm, and range
checked wraparound check results will be secretshared to (1) be sent
as input shares for the Aggregators, and (2) derive the verification
joint randomness as described above. The Client then generates the
proof with the FLP and secret shares it. The secretshared proof,
along with the input shares, and the joint randomness parts for both
wraparound and verification joint randomness, are sent to the
Aggregators.
Then the Aggregators carry out a multiparty computation to obtain
the output shares (the secret shares of the encoded Client gradient),
and also reject Client gradients that have invalid L2norm. Each
Aggregator first needs to derive wraparound and verification joint
Chen & Patton Expires 5 September 2024 [Page 14]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
randomness. Similar to Prio3 preparation Section 7.2.2 of [VDAF],
the Aggregator does not derive every joint randomness part like the
Client does. It only derives the joint randomness part from its
secret share via the XOF, and applies its part and and other
Aggregators' parts sent by the Client to the XOF to obtain the joint
randomness seed. Then each Aggregator runs the wraparound checks
Section 4.1.3 with its share of encoded gradient and the wraparound
joint randomness, and queries the FLP with its input share, proof
share, the wraparound check results, and the verification joint
randomness. All Aggregators then exchange the results from the FLP
and decide whether to accept that Client gradient.
Next, each Aggregator sums up their shares of the encoded gradients
and sends the aggregate share to the Collector. Finally, the
Collector sums up the aggregate shares to obtain the aggregate
result, and decodes it into an array of floating point values.
Like Prio3 Section 7.1.2 of [VDAF], PINE supports generation and
verification of multiple FLPs. The goal is to improve robustness of
PINE (Corollary 3.13 in [ROCT23]) by generating multiple unique
proofs from the Client, and only accepting the Client gradient if all
proofs have been verified by the Aggregators. The benefit is that
one can improve the communication cost between Clients and
Aggregators, by instantiating PINE FLP with a smaller field, but
repeating the proof generation (Flp.prove) and validation (Flp.query)
multiple times.
The remainder of this section is structured as follows. We will
specify the exact algorithms for Client sharding Section 5.1,
Aggregator preparation Section 5.2 and aggregation Section 5.3, and
Collector unsharding Section 5.4.
5.1. Sharding
TODO Specify the implementation of Vdaf.shard().
5.2. Preparation
TODO Specify the implementations of Vdaf.prep_init(),
.prep_shares_to_prep(), and .prep_next().
5.3. Aggregation
TODO Specify the implementation of Vdaf.aggregate().
Chen & Patton Expires 5 September 2024 [Page 15]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
5.4. Unsharding
TODO Specify the implementation of Vdaf.unshard().
6. Variants
TODO Specify concrete parameterizations of VDAFs, including the
choice of field, number of proofs, and valid ranges for the
parameters in Table 1.
7. PINE Auxiliary Functions
TODO Put all auxiliary functions here, including range_check(),
parallel_sum().
8. Security Considerations
Our security considerations for PINE are the same as those for Prio3
described in Section 9 of [VDAF].
9. IANA Considerations
TODO Ask IANA to allocate an algorithm ID from the VDAF algorithm
ID registry.
10. References
10.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, .
[VDAF] Barnes, R., Cook, D., Patton, C., and P. Schoppmann,
"Verifiable Distributed Aggregation Functions", Work in
Progress, InternetDraft, draftirtfcfrgvdaf08, 20
November 2023, .
10.2. Informative References
Chen & Patton Expires 5 September 2024 [Page 16]
InternetDraft Private Inexpensive Norm Enforcement (PI March 2024
[BBCGGI19] Boneh, D., Boyle, E., CorriganGibbs, H., Gilboa, N., and
Y. Ishai, "ZeroKnowledge Proofs on SecretShared Data via
Fully Linear PCPs", CRYPTO 2019 , 2019,
.
[IEEE7542019]
"IEEE Standard for FloatingPoint Arithmetic", 2019,
.
[Lem12] LemarĂ©chal, C., "Cauchy and the gradient method", 2012,
.
[MR17] McMahan, B. and D. Ramage, "Federated Learning:
Collaborative Machine Learning without Centralized
Training Data", 2017, .
[ROCT23] Rothblum, G. N., Omri, E., Chen, J., and K. Talwar, "PINE:
Efficient NormBound Verification for SecretShared
Vectors", 2023, .
[Tal22] Talwar, K., "Differential Secrecy for Distributed Data and
Applications to Robust Differentially Secure Vector
Summation", 2022, .
Acknowledgments
Guy Rothblum Apple Inc. gn_rothblum@apple.com
Kunal Talwar Apple Inc. ktalwar@apple.com
Authors' Addresses
Junye Chen
Apple Inc.
Email: junyec@apple.com
Christopher Patton
Cloudflare
Email: chrispatton+ietf@gmail.com
Chen & Patton Expires 5 September 2024 [Page 17]