Open Access Published by De Gruyter February 19, 2019

Generic constructions of PoRs from codes and instantiations

Julien Lavauzelle ORCID logo and Franรงoise Levy-dit-Vehel

Abstract

In this paper, we show how to construct โ€“ from any linear code โ€“ a Proof of Retrievability ( ๐–ฏ๐—ˆ๐–ฑ ) which features very low computation complexity on both the client ( ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ ) and the server ( ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ ) sides, as well as small client storage (typically 512โ€‰bits). We adapt the security model initiated by Juels and Kaliski [PoRs: Proofs of retrievability for large files, Proceedings of the 2007 ACM Conference on Computer and Communications Securityโ€”CCS 2007, ACM, New York 2007, 584โ€“597] to fit into the framework of Paterson, Stinson and Upadhyay [A coding theory foundation for the analysis of general unconditionally secure proof-of-retrievability schemes for cloud storage, J. Math. Cryptol. 7 2013, 3, 183โ€“216], from which our construction evolves. We thus provide a rigorous treatment of the security of our generic design; more precisely, we sharply bound the extraction failure of our protocol according to this security model. Next we instantiate our formal construction with codes built from tensor-products as well as with Reedโ€“Muller codes and lifted codes, yielding ๐–ฏ๐—ˆ๐–ฑ s with moderate communication complexity and (server) storage overhead, in addition to the aforementioned features.

MSC 2010: 11T71

1 Introduction

1.1 Motivation

Cloud computing and storage has evolved quite spectacularly over the past decade. Especially, data outsourcing allows users and companies to lighten their storage burden and maintenance cost. Though, it raises several issues: for example, how can someone check efficiently that he can retrieve without any loss a massive file that he had uploaded on a distant server and erased from his personal system?

Proofs of retrievability ( ๐–ฏ๐—ˆ๐–ฑ s) address this issue. They are cryptographic protocols involving two parts: a client (or a verifier) and a server (or a prover). ๐–ฏ๐—ˆ๐–ฑ s usually consist in the following phases. First, a key generation process creates secret material related to the file, meant to be kept by the client only. Then the file is initialised, that is, it is encoded and/or encrypted according to the secret data held by the client. This processed file is uploaded to the server. In order to check retrievability, the client can run a verification procedure, which is the core of the ๐–ฏ๐—ˆ๐–ฑ . Finally, if the client is convinced that the server still holds his file, the client can proceed at any time to the extraction of the file.

Several parameters must be taken into account. Plainly, the verification process has to feature a low communication complexity, as the main goal is to avoid downloading a large part of the file to only check its extractability. Second, the storage overhead induced by the protocol must be low, as large server overhead would imply high fees for the customer. Third, the computation cost of the verification procedure must be low, both for the client (which is likely to own a lightweight device) and the server (whose computation work could also be expensive for the client).

Notice that proofs of data possession ( ๐–ฏ๐–ฃ๐–ฏ ) represent protocols close to what is needed in ๐–ฏ๐—ˆ๐–ฑ s. However, in ๐–ฏ๐–ฃ๐–ฏ s, one does not require the client to be able to extract the file from the server. Instances of ๐–ฏ๐–ฃ๐–ฏ s are given by Ateniese et al. [2]. Besides, protocols of Lillibridge et al. [8] and Naor and Rothblum [10] are very often seen as precursors for ๐–ฏ๐—ˆ๐–ฑ s. For instance, the work of Naor and Rothblum [10] considers a setting in which the client directly accesses the file stored by the prover/server (while the actual ๐–ฏ๐—ˆ๐–ฑ definition uses โ€œan arbitrary program as opposed to a simple memory layout and this program may answer these questions in an arbitrary mannerโ€ [14]).

1.2 Previous work

Juels and Kaliski [6] gave the first formal definition of ๐–ฏ๐—ˆ๐–ฑ s. They also proposed a first construction based on so-called sentinels (namely, random parts of the file to be checked during the verification step) the client keeps secretly on his device. Additionally, an erasure code ensures the integrity of the file to be extracted. This seminal work also raised several interesting points. On the one hand, it revealed that (i) the client must store secret data to be used in the verification step and (ii) coding is needed in order to retrieve the file without erasures or errors. On the other hand, in Juels and Kaliskiโ€™s construction, the verification step can only be performed a finite number of times since sentinels cannot be reused endlessly.

As a consequence, Shacham and Waters proposed to consider unbounded-use ๐–ฏ๐—ˆ๐–ฑ s in [14], where they built two kinds of ๐–ฏ๐—ˆ๐–ฑ s. The first one is based on linear combinations of authenticators produced via pseudo-random functions; its security was proved using cryptographic tools such as unforgeable MAC scheme, semantically secure symmetric encryption and secure PRFs. The second one is a publicly verifiable scheme based on the Diffieโ€“Hellman problem in bilinear groups.

Bowers, Juels and Oprea [3] adopted a coding-theoretic approach (inner code, outer code) to compare variants of Shachamโ€“Waters and Juelsโ€“Kaliski schemes. They focused on the efficiency of the schemes, and proved that, despite bounded use, new variants of Juelsโ€“Kaliski construction are highly competitive compared to other existing schemes.

In [11], Paterson, Stinson and Upadhyay provide a general framework for ๐–ฏ๐—ˆ๐–ฑ s in the unconditional security model. They show that retrievability of the file can be expressed as error correction of a so-called response code. That allows them to precisely quantify the extraction success as a function of the success probability of a proving algorithm: indeed, in this setting, extraction can be naturally seen as nearest-neighbour decoding in the response code. They notably apply their framework to prove the security of a modified version of the Shachamโ€“Waters scheme. Also, notice that, prior to [11], Dodis, Vahan and Wichs [4] proposed another coding-theoretic model for ๐–ฏ๐—ˆ๐–ฑ s that allowed them to build efficient bounded-use and unbounded-use ๐–ฏ๐—ˆ๐–ฑ schemes.

With practicality in mind, other features have been deployed on ๐–ฏ๐—ˆ๐–ฑ s. For instance, Wang et al. [15] presented a ๐–ฏ๐—ˆ๐–ฑ construction based on Merkle hash trees, which allows efficient file updates on the server. Their scheme is provably secure under cryptographic assumptions (hardness of Diffieโ€“Hellman in bilinear groups, unforgeable signatures, etc.) and has been improved by Mo, Zhou and Chen [9] in order to prevent unbalanced trees. More recently, other features have been proposed for ๐–ฏ๐—ˆ๐–ฑ s, such as multi-prover ๐–ฏ๐—ˆ๐–ฑ s (see [12]) or public verifiability (for instance in [13]).

1.3 Our approach

As we remarked before, most ๐–ฏ๐—ˆ๐–ฑ schemes rely on two techniques: (i) the client locally stores secret data in order to check the integrity of the file, and (ii) the client encodes the file in order to repair a small number of erasures and errors that could have been missed during the verification step.

In this work, we propose to build ๐–ฏ๐—ˆ๐–ฑ schemes using codes that fulfil the two previous goals, when equipped with a suitable family of efficiently computable random permutations. More precisely, our idea is the following. Given a file F, a code ๐’ž and a family of random permutations ฯƒ K , the client sends to the server an encoded and scrambled version ฯƒ K โข ( ๐’ž โข ( F ) ) of his file. Then the verification step consists in checking โ€œshortโ€ relations among descrambled symbols of w = ๐’ž โข ( F ) , which come, for instance, from low-weight parity-check equations for ๐’ž . Moreover, during the extraction step, the code ๐’ž provides the redundancy necessary to repair erasures and potential unnoticed errors.

In the present work, we develop a seminal idea that appeared in [7], where the authors proposed a construction of ๐–ฏ๐—ˆ๐–ฑ s based on lifted codes. We here provide a more generic construction and give a deeper analysis of its security.

While our scheme does not feature updatability nor public verifiability, we emphasise the genericity of our construction, which is based on well-studied algebraic and combinatorial structures, namely, codes and their parity-check equations. Moreover, since the code ๐’ž is public, the client must only store the secret material associated to the random permutations ฯƒ K , which consist in a few bytes. Besides, an honest server simply needs to read pieces of w during the verification step, and therefore has very low computational burden compared to many other ๐–ฏ๐—ˆ๐–ฑ schemes.

1.4 Organisation

Section 2 is devoted to the definition and security model of proofs of retrievability. Despite the great disparity of models in ๐–ฏ๐—ˆ๐–ฑ literature, we try to keep close to the definitions given in [6, 11] for the sake of uniformity.

Section 3 presents our construction of ๐–ฏ๐—ˆ๐–ฑ . Precisely, in Section 3.1, we introduce objects called verification structures for a code ๐’ž that will be used in the definition of our ๐–ฏ๐—ˆ๐–ฑ scheme (Section 3.2). A rigorous analysis of our scheme is the purpose of the remainder of that section.

The performance of our generic construction is given in Section 4. We then provide several instances in Section 5, proving the practicality of our ๐–ฏ๐—ˆ๐–ฑ schemes for some classes of codes.

2 Proofs of retrievability

2.1 Definition of underlying protocols

We recall that, in proofs of retrievability, a user wants to estimate if a message m can be retrieved from a encoded version w of the message stored on a server. In all what follows, the user will be known as the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ (wants to verify the retrievability of the message) while the server is the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ (aims at proving the retrievability). The message space is denoted by โ„ณ while ๐’ฒ , the (server) file space, is the set of encoded versions of the messages. We also denote by ๐’ฆ the set of secret values (or keys) kept by the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ , and by โ„› the space of responses to challenges.

Throughout the paper, the symbols โ† R and โ† respectively denote the output of randomised and deterministic algorithms.

Definition 2.1.

A keyed proof of retrievability ( ๐–ฏ๐—ˆ๐–ฑ ) is a tuple of algorithms ( ๐–ช๐–พ๐—’๐–ฆ๐–พ๐—‡ , ๐–จ๐—‡๐—‚๐— , ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—’ , ๐–ค๐—‘๐—๐—‹๐–บ๐–ผ๐— ) running as follows:

  1. (1)

    The key generation algorithm ๐–ช๐–พ๐—’๐–ฆ๐–พ๐—‡ generates uniformly at random a key ฮบ โ† R ๐’ฆ . The key ฮบ is secretly kept by the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ .

  2. (2)

    The initialisation algorithm ๐–จ๐—‡๐—‚๐— is a deterministic algorithm which takes, as input, a message m โˆˆ โ„ณ and a key ฮบ โˆˆ ๐’ฆ , and outputs a file w โˆˆ ๐’ฒ . ๐–จ๐—‡๐—‚๐— is run by the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ which initially holds the message m. After the process, the file w is sent to the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ , and the message m is erased on ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ โ€™s side. Upon receipt of w, the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ sets a deterministic algorithm ๐–ฏ ( w ) that will be run during the verification procedure.

  3. (3)

    The verification algorithm ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—’ is a randomised algorithm initiated by the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ which needs a secret key ฮบ โˆˆ ๐’ฆ and interacts with the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ . ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—’ is depicted in Figure 1 and works as follows:

    1. (i)

      the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ runs a random query generator that outputs a challenge u โ† R ๐’ฌ (the set ๐’ฌ being the so-called query set);

    2. (ii)

      the challenge u is sent to the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ ;

    3. (iii)

      the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ outputs a response r u โ† ๐–ฏ ( w ) โข ( u ) โˆˆ โ„› ;

    4. (iv)

      the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ checks the validity of r u according to u and ฮบ; the algorithm ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—’ finally outputs the Boolean value ๐–ข๐—๐–พ๐–ผ๐—„ โข ( u , r u , ฮบ ) .

  4. (4)

    The extraction algorithm ๐–ค๐—‘๐—๐—‹๐–บ๐–ผ๐— is run by the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ . It takes, as input, ฮบ and r = ( r u : u โˆˆ ๐’ฌ ) โˆˆ โ„› ๐’ฌ and outputs either a message m โ€ฒ โˆˆ โ„ณ or a failure symbol โŠฅ . We say that extraction succeeds if ๐–ค๐—‘๐—๐—‹๐–บ๐–ผ๐— โข ( r , ฮบ ) = m .

The vector r = ( r u โ† ๐–ฏ ( w ) ( u ) ) u โˆˆ ๐’ฌ โˆˆ โ„› ๐’ฌ is called the response word associated to ๐–ฏ ( w ) .

Figure 1 Definition of the algorithm ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—’{\mathsf{Verify}}.

Figure 1

Definition of the algorithm ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—’ .

Note that, in assuming that the response algorithm ๐–ฏ ( w ) is deterministic and non-adaptive[1], we follow the work of Paterson, Stinson and Upadhyay [11]. The authors justify determinism of response algorithms by the fact that any probabilistic prover can be replaced by a deterministic prover whose success probability is at least as good as the probabilistic one.

In Definition 2.1, we can see that a deterministic algorithm ๐–ฏ ( w ) can be represented by the vector of its outputs r = ( ๐–ฏ ( w ) โข ( u ) ) u โˆˆ ๐’ฌ , called the response word of ๐–ฏ ( w ) . Therefore, we can assume that, before the verification step, the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ produces a word r ( w ) โˆˆ โ„› ๐’ฌ related to the file w he holds. In other words, we model provers as algorithms ๐–ฏ which, given as input w, return a word r โˆˆ โ„› ๐’ฌ .

Following [11], we also assume in this chapter that the extraction algorithm ๐–ค๐—‘๐—๐—‹๐–บ๐–ผ๐— is deterministic, though, in general, it can be randomised. Finally, notice that proofs of retrievability aim at proving the extractability of a file. The extraction algorithm is therefore a tool to retrieve the whole file. Hence its computational efficiency is not a crucial feature.

Table 1 summarises the information held by each entity after the initialisation step. Table 2 reports the inputs and outputs of the algorithms involved in a ๐–ฏ๐—ˆ๐–ฑ .

Table 1

Information held by each entity after the initialisation step.

๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹
ฮบ w
Table 2

Inputs and outputs of the algorithms involved in a ๐–ฏ๐—ˆ๐–ฑ .

Algorithm ๐–ช๐–พ๐—’๐–ฆ๐–พ๐—‡ ๐–จ๐—‡๐—‚๐— ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—’ ๐–ข๐—๐–พ๐–ผ๐—„ ๐–ค๐—‘๐—๐—‹๐–บ๐–ผ๐—
Input 1 ฮป m, ฮบ r, ฮบ u, r u , ฮบ r, ฮบ
Output ฮบ w True or False True or False m โ€ฒ or โŠฅ

2.2 Security models

One should first notice that, despite many efforts, proofs of retrievability lack a general agreement on the definition of their security model. Nevertheless, our definitions remain very close to the ones given in the original work of Juels and Kaliski [6].

For a response word r โˆˆ โ„› ๐’ฌ given by the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ and a key ฮบ โˆˆ ๐’ฆ kept by the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ , we first define the success of r according to ฮบ as

succ ( r , ฮบ ) : = Pr u ( ๐–ข๐—๐–พ๐–ผ๐—„ ( u , r u , ฮบ ) = ๐šƒ๐š›๐šž๐šŽ ) ,

where the probability is taken over the internal randomness of ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—’ . A first security model can be defined as follows.

Definition 2.2 (Security model, strong version).

Let ฮต , ฯ„ โˆˆ [ 0 , 1 ] . A proof of retrievability ( ๐–ช๐–พ๐—’๐–ฆ๐–พ๐—‡ , ๐–จ๐—‡๐—‚๐— , ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—’ , ๐–ค๐—‘๐—๐—‹๐–บ๐–ผ๐— ) is strongly ( ฮต , ฯ„ ) -sound if, for every initial file m โˆˆ โ„ณ , every uploaded file w โˆˆ ๐’ฒ and every prover ๐–ฏ : ๐’ฒ โ†’ โ„› ๐’ฌ , we have

(2.1) Pr โก ( ๐–ค๐—‘๐—๐—‹๐–บ๐–ผ๐— โข ( r , ฮบ ) โ‰  m succ โก ( r , ฮบ ) โ‰ฅ 1 - ฮต | ฮบ โ† R ๐–ช๐–พ๐—’๐–ฆ๐–พ๐—‡ โข ( 1 ฮป ) w โ† ๐–จ๐—‡๐—‚๐— โข ( m , ฮบ ) r โ† ๐–ฏ โข ( w ) ) โ‰ค ฯ„ ,

the probability being taken over the internal randomness of ๐–ช๐–พ๐—’๐–ฆ๐–พ๐—‡ under the constraint that w = ๐–จ๐—‡๐—‚๐— โข ( m , ฮบ ) .

A remark concerning parameters ฮต and ฯ„

In proofs of retrievability, we aim at making the extraction of the desired file m as sure as possible when the audit succeeds. Hence it is desirable to have ฯ„ small. On the other hand, the parameter ฮต measures the rate of unsuccessful audits which leads the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ to believe the extraction will fail. Therefore, one does not necessarily need to look for large values of ฮต, though, in practice, large ฮต afford more flexibility, for instance, if communication errors occur between the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ and the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ during the verification procedure.

Definition 2.2 provides a strong security model, in the sense that (i) it does not require any bound on the response algorithms given by the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ and (ii) the probability in (2.1) is taken over fixed messages m (informally, it means the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ knows m).

However, keyed proofs of retrievability are usually insecure according to the security model given in Definition 2.2. For instance, in [11], Paterson, Stinson and Upadhyay noticed that in the Shachamโ€“Waters scheme [14], given the knowledge of m and w, an unbounded ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ may be able to

  1. (i)

    compute (or at least randomly guess) a key ฮบ such that ๐–จ๐—‡๐—‚๐— โข ( m , ฮบ ) = w ,

  2. (ii)

    build m โ€ฒ โ‰  m such that ๐–จ๐—‡๐—‚๐— โข ( m โ€ฒ , ฮบ ) = w โ€ฒ ,

  3. (iii)

    set ๐–ฏ ( w โ€ฒ ) = r โ€ฒ which (a) successfully passes every audit and (b) leads to the extraction of m โ€ฒ โ‰  m .

Hence we choose to use a weaker but still realistic security model, where, informally, the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ only knows what he stores (that is, w) and has no information on the initial message m. The following security model thus remains conform with the one given by Paterson, Stinson and Upadhyay [11].

Definition 2.3 (Security model, weak version).

Let ฮต , ฯ„ โˆˆ [ 0 , 1 ] . A proof of retrievability ( ๐–ช๐–พ๐—’๐–ฆ๐–พ๐—‡ , ๐–จ๐—‡๐—‚๐— , ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—’ , ๐–ค๐—‘๐—๐—‹๐–บ๐–ผ๐— ) is weakly ( ฮต , ฯ„ ) -sound (or simply ( ฮต , ฯ„ ) -sound) if, for every polynomial-time prover ๐–ฏ : ๐’ฒ โ†’ โ„› ๐’ฌ and every uploaded file w โˆˆ ๐’ฒ , we have

(2.2) Pr โก ( ๐–ค๐—‘๐—๐—‹๐–บ๐–ผ๐— โข ( r , ฮบ ) โ‰  m succ โก ( r , ฮบ ) โ‰ฅ 1 - ฮต | m โ† R โ„ณ ฮบ โ† R ๐–ช๐–พ๐—’๐–ฆ๐–พ๐—‡ โข ( 1 ฮป ) w โ† ๐–จ๐—‡๐—‚๐— โข ( m , ฮบ ) r โ† ๐–ฏ โข ( w ) ) โ‰ค ฯ„ .

In equation (2.2), the randomness comes from pairs ( m , ฮบ ) โˆˆ โ„ณ ร— ๐’ฆ picked uniformly at random among those satisfying w = ๐–จ๐—‡๐—‚๐— โข ( m , ฮบ ) .

Since we deal with values of ฯ„ very close to 0, we also say that a strongly ( ฮต , ฯ„ ) -sound ๐–ฏ๐—ˆ๐–ฑ admits ฮป = - log 2 โก ( ฯ„ ) bits of security against ฮต-adversaries.

Informally, saying that a ๐–ฏ๐—ˆ๐–ฑ is not weakly sound amounts to finding a polynomial-time deterministic algorithm ๐–ฏ which

  • โ€ข

    takes, as input, a file w โˆˆ ๐’ฒ and outputs a response word r โˆˆ โ„› ๐’ฌ ,

  • โ€ข

    makes the extraction fail with non-negligible probability (over messages m and keys ฮบ such that the corresponding response words are successfully audited).

3 Our generic construction

Schematically, in the initialisation phase of our construction, the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹

  1. (i)

    encodes his file according to a code ๐’ž ,

  2. (ii)

    scrambles the resulting codeword using a tuple of permutations over the base field,

  3. (iii)

    uploads the result to the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ .

As we explained in the introduction, the verification step then consists in checking that the server is still able to give answers that, once descrambled, satisfy low-weight parity-check equations for ๐’ž .

For this purpose, we next introduce objects called verification structures for codes, which will be used in the definition of our generic ๐–ฏ๐—ˆ๐–ฑ scheme.

3.1 Verification structures: A tool for our PoR scheme

We here consider ๐”ฝ q , the finite field with q elements. From well-known coding theory terminology, the support of a word w โˆˆ ๐”ฝ q n is supp ( w ) : = { i โˆˆ [ 1 , n ] , w i โ‰  0 } , and its weight is wt ( w ) : = | supp ( w ) | .

In this work, we need to consider codes whose alphabets are finite-dimensional spaces โ„› over ๐”ฝ q , typically โ„› = ๐”ฝ q s . Precisely, a code ๐’ž of length n over โ„› is a subset of โ„› n . A code ๐’ž โŠ† โ„› n is ๐”ฝ q -linear if ๐’ž is a vector space over ๐”ฝ q . When โ„› = ๐”ฝ q , we get the usual definition of linear codes over finite fields. Unless stated otherwise, we only consider ๐”ฝ q -linear codes, that we will refer to as codes.

We usually denote by k the dimension over ๐”ฝ q of a code ๐’ž . Its minimum distance d min โข ( ๐’ž ) is the smallest Hamming distance between two distinct codewords. If n is the length of ๐’ž , then d min โข ( ๐’ž ) / n โˆˆ [ 0 , 1 ] is the relative minimum distance of the code ๐’ž , while k / n represents its rate. If ๐’ž โŠ† ๐”ฝ q n , its dual code ๐’ž โŠฅ is defined as { h โˆˆ ๐”ฝ q n , โˆ‘ i = 1 n h i โข c i = 0 โข for all โข c โˆˆ ๐’ž } . Codewords in ๐’ž โŠฅ are also called parity-check equations for ๐’ž .

Definition 3.1 (Verification structure).

Let 1 โ‰ค โ„“ โ‰ค n and ๐’ž โŠ† ๐”ฝ q n be a code. Let also ๐’ฌ be a non-empty set of โ„“ -subsets of [ 1 , n ] . Set โ„› = ๐”ฝ q โ„“ . We define the restriction mapR associated to ๐’ฌ as

R : ๐’ฌ ร— ๐”ฝ q n โ†’ โ„› ,
( u , w ) โ†ฆ w | u .

Given an integer s โ‰ฅ 1 and a map V : ๐’ฌ ร— โ„› โ†’ ๐”ฝ q s , we say that ( ๐’ฌ , V ) is a verification structure for ๐’ž if the following holds:

  1. (1)

    For all i โˆˆ [ 1 , n ] , there exists u โˆˆ ๐’ฌ such that i โˆˆ u .

  2. (2)

    For all u โˆˆ ๐’ฌ , the map ๐”ฝ q n โ†’ ๐”ฝ q s given by a โ†ฆ V โข ( u , R โข ( u , a ) ) is surjective and vanishes on the code ๐’ž . Explicitly,

    V โข ( u , R โข ( u , c ) ) = 0 โ€ƒ for all โข c โˆˆ ๐’ž .

The map V is then called a verification map for ๐’ž , and the set ๐’ฌ a query set for ๐’ž . By convention, for w โˆˆ ๐”ฝ q n and r โˆˆ โ„› ๐’ฌ , we define

R โข ( w ) : = ( R ( u , w ) : u โˆˆ ๐’ฌ ) โˆˆ โ„› ๐’ฌ , V โข ( r ) : = ( V ( u , r u ) : u โˆˆ ๐’ฌ ) โˆˆ ( ๐”ฝ q s ) ๐’ฌ .

Finally, the code R ( ๐’ž ) : = { R ( c ) , c โˆˆ ๐’ž } is called the response code of ๐’ž .

Example 3.2 (Fundamental example).

Let ๐’ž be a code, and let โ„‹ be a set of parity-check equations for ๐’ž of Hamming weight โ„“ , whose supports are pairwise distinct. Define the query set ๐’ฌ = { supp ( h ) , h โˆˆ โ„‹ } and, for any u โˆˆ ๐’ฌ , h โข ( u ) to be the unique parity-check equation in โ„‹ whose support is u. Finally, we define a map V by

V : ๐’ฌ ร— โ„› โ†’ ๐”ฝ q , ( u , r ) โ†ฆ โˆ‘ i = 1 โ„“ h โข ( u ) u i โข r i .

Notice that we set s = 1 here. By construction, it is clear that ( ๐’ฌ , V ) is a verification structure for ๐’ž .

Example 3.3 (Toy example).

Let ๐’ž โŠ† ๐”ฝ 2 7 be a binary Hadamard code of length n = 7 and dimension k = 3 . In other words, ๐’ž is defined by a parity-check matrix

H = ( 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 1 ) .

According to Example 3.2, we define ๐’ฌ to be the set of supports of rows of H. In other words,

๐’ฌ = { { 1 , 2 , 3 } , { 1 , 4 , 5 } , { 1 , 6 , 7 } , { 2 , 5 , 6 } , { 2 , 4 , 7 } , { 3 , 4 , 6 } , { 3 , 5 , 7 } } .

Then the verification map V : ๐’ฌ ร— ๐”ฝ 2 3 โ†’ ๐”ฝ 2 can be defined as follows. If u = { u 1 , u 2 , u 3 } โˆˆ ๐’ฌ and b โˆˆ ๐”ฝ 2 u is indexed according to u, then we define

V โข ( u , b ) = โˆ‘ i = 1 3 b u i .

Now let m = ( m 1 , m 2 , m 3 ) โˆˆ ๐”ฝ 2 3 . The message m can be encoded into

c = ( m 1 , m 2 , m 1 + m 2 , m 3 , m 1 + m 3 , m 1 + m 2 + m 3 , m 2 + m 3 ) โˆˆ ๐’ž .

Hence the word r = R โข ( c ) โˆˆ ( ๐”ฝ 2 3 ) 7 is

r = ( ( c 1 c 2 c 3 ) , ( c 1 c 4 c 5 ) , ( c 1 c 6 c 7 ) , ( c 2 c 5 c 6 ) , ( c 2 c 4 c 7 ) , ( c 3 c 4 c 6 ) , ( c 3 c 5 c 7 ) ) = ( ( m 1 m 2 m 1 + m 2 ) , ( m 1 m 3 m 1 + m 3 ) , ( m 1 m 1 + m 2 + m 3 m 2 + m 3 ) , ( m 2 m 1 + m 3 m 1 + m 2 + m 3 ) , ( m 2 m 3 m 2 + m 3 ) , ( m 1 + m 2 m 3 m 1 + m 2 + m 3 ) , ( m 1 + m 2 m 1 + m 3 m 2 + m 3 ) ) .

For each vector-coordinate b โˆˆ ๐”ฝ 2 3 of r = R โข ( c ) , one can now check that โˆ‘ j b j = 0 . Hence we get V โข ( R โข ( c ) ) = 0 , as expected.

From now on, we denote by N = | ๐’ฌ | the length of the response code R โข ( ๐’ž ) of a code ๐’ž equipped with a verification structure ( ๐’ฌ , V ) .

3.2 Definition of our PoR scheme

Let ( ๐’ฌ , V ) be a verification structure for ๐’ž โŠ† ๐”ฝ q n , and let ฯƒ โˆˆ ๐”– โข ( ๐”ฝ q ) n , where ๐”– โข ( ๐”ฝ q ) denotes the set of permutations over ๐”ฝ q . Any n-tuple of permutations ฯƒ = ( ฯƒ 1 , โ€ฆ , ฯƒ n ) โˆˆ ๐”– โข ( ๐”ฝ q ) n naturally acts on c โˆˆ ๐”ฝ q n by

ฯƒ โข ( c ) โ†ฆ ( ฯƒ 1 โข ( c 1 ) , โ€ฆ , ฯƒ n โข ( c n ) ) ,

and we define ฯƒ ( ๐’ž ) = { ฯƒ ( c ) , c โˆˆ ๐’ž } . Let finally

V ฯƒ : ๐’ฌ ร— ๐”ฝ q โ„“ โ†’ ๐”ฝ q s ,
( u , y ) โ†ฆ V โข ( u , ฯƒ | u - 1 โข ( y ) ) ,

where ฯƒ | u - 1 โข ( y ) = ( ฯƒ u 1 - 1 โข ( y 1 ) , โ€ฆ , ฯƒ u โ„“ - 1 โข ( y โ„“ ) ) . The map V ฯƒ has been defined in order to satisfy

V ฯƒ โข ( u , R โข ( u , ฯƒ โข ( c ) ) ) = V โข ( u , R โข ( u , c ) )

for every ( c , u ) โˆˆ ๐’ž ร— ๐’ฌ .

Based on this, our ๐–ฏ๐—ˆ๐–ฑ construction is given in Figure 2.

Figure 2 Definition of our ๐–ฏ๐—ˆ๐–ฑ{\mathsf{PoR}} scheme.

Figure 2

Definition of our ๐–ฏ๐—ˆ๐–ฑ scheme.

Figure 3 Our extraction procedure ๐–ค๐—‘๐—๐—‹๐–บ๐–ผ๐—โข(r,ฯƒ){\mathsf{Extract}(r,\sigma)}.

Figure 3

Our extraction procedure ๐–ค๐—‘๐—๐—‹๐–บ๐–ผ๐— โข ( r , ฯƒ ) .

3.3 Analysis

3.3.1 Preliminary results

We first give results concerning verification structures and response codes. The following two lemmata are straightforward to prove.

Lemma 3.4.

Let ( Q , V ) be a verification structure for a code C โŠ† F q n . Then ( Q , V ฯƒ ) is a verification structure for ฯƒ โข ( C ) .

Lemma 3.5.

Let Q be any query-set for a code C โŠ† F q n whose elements have cardinality โ„“ โ‰ฅ 1 . Then its response code R โข ( C ) is an F q -linear code over the alphabet R โ‰ƒ F q โ„“ .

Remark 3.6.

By considering ฯƒ โข ( ๐’ž ) instead of ๐’ž , we loose the ๐”ฝ q -linearity, but one can check that verification structures still make sense and provide the result claimed in Lemma 3.4.

The next result states that the map ๐’ž โ†ฆ ฯƒ โข ( ๐’ž ) does not modify the distance between codewords.

Lemma 3.7.

Let C โŠ† F q n be a linear code, ( Q , V ) a verification structure for C , and ฯƒ โˆˆ S โข ( F q ) n . Then it holds that

  • โ€ข

    the distribution of distances in ๐’ž and ฯƒ โข ( ๐’ž ) are the same,

  • โ€ข

    the distribution of distances in R โข ( ๐’ž ) and R โข ( ฯƒ โข ( ๐’ž ) ) are the same.

Proof.

Since every ฯƒ i is one-to-one, for any c , c โ€ฒ โˆˆ ๐’ž , we get

d โข ( c , c โ€ฒ ) = | { i โˆˆ [ 1 , n ] , c i โ‰  c i โ€ฒ } | = | { i โˆˆ [ 1 , n ] , ฯƒ i ( c i ) โ‰  ฯƒ i ( c i โ€ฒ ) } | = d โข ( ฯƒ โข ( c ) , ฯƒ โข ( c โ€ฒ ) ) .

The proof for response codes relies on the same argument. โˆŽ

Remark these results imply that, if ๐’ž is linear, then the minimum distance of R โข ( ฯƒ โข ( ๐’ž ) ) is the minimum weight of R โข ( ๐’ž ) .

Definition 3.8.

Let ฮต โˆˆ [ 0 , 1 ] and ( ๐’ฌ , V ) be a verification structure for a code ๐’ž โŠ† ๐”ฝ q n . We say r โˆˆ โ„› ๐’ฌ is ฮต-close to ( ๐’ฌ , V ) if

wt ( V ( r ) ) : = | { u โˆˆ ๐’ฌ , V ( u , r u ) โ‰  0 } | โ‰ค ฮต N .

Let now c โˆˆ ๐’ž and ฮฒ โˆˆ [ 0 , 1 ] . We say that r โˆˆ โ„› ๐’ฌ is a ฮฒ-liar for ( ๐’ฌ , V , c ) if

| { u โˆˆ ๐’ฌ , V ( u , r u ) = 0 and r u โ‰  R ( u , c ) } | โ‰ค ฮฒ N .

Bounded-distance error-and-erasure decoder

Let ๐’œ โŠ† ๐”ฝ q n be any code of minimum distance d, and let a โˆˆ ๐’œ be corrupted with b errors and e erasures, resulting in a word r โ€ฒ โˆˆ ( ๐”ฝ q โˆช { โŠฅ } ) n . Then it is well known that, as long as 2 โข b + e < d , it is possible to retrieve a from r โ€ฒ thanks to a so-called bounded-distance error-and-erasure decoding algorithm. This is precisely the decoding algorithm that we employ in Figure 3 on the code ๐’œ = R โข ( ๐’ž ) .

Our framework allows us to reformulate the extraction success in terms of a probability to decode corrupted codewords. More precisely:

Proposition 3.9.

Let ฯƒ โˆˆ S โข ( F q ) n , m โˆˆ F q k , and denote by d the minimum distance of R โข ( C ) of length N. Let also r โˆˆ R Q be the response word, output of a proving algorithm P taking w = ฯƒ โข ( C โข ( m ) ) as input. Finally, assume that r is ฮต-close to ( Q , V ฯƒ ) and a ฮฒ-liar for ( Q , V ฯƒ , w ) , with ( ฮต + 2 โข ฮฒ ) โข N < d . Then Extract โข ( r , ฯƒ ) = m , where Extract โข ( r , ฯƒ ) is defined in Figure 3.

Proof.

Recall that r โ€ฒ โˆˆ ( โ„› โˆช { โŠฅ } ) ๐’ฌ represents the word we get from r after step (ii) of the algorithm given in Figure 3. Let us now translate our assumptions on r in coding-theoretic terminology:

  • โ€ข

    r is ฮต-close to ( ๐’ฌ , V ฯƒ ) means that there are at most ฮต โข N challenges u โˆˆ ๐’ฌ for which we know that the coordinate r u โ€ฒ is not authentic. This justifies that we assign erasure symbols to these coordinates.

  • โ€ข

    r is a ฮฒ-liar for ( ๐’ฌ , V , c ) means that there are at most ฮฒ โข N other corrupted values r u โ€ฒ , but we cannot identify them. Therefore, we can assimilate these coordinates to errors.

To sum up, we see r โ€ฒ as a corruption of R โข ( ๐’ž โข ( m ) ) with at most ฮต โข N erasures and at most ฮฒ โข N errors, where N = | ๐’ฌ | . Since we assume that ( ฮต + 2 โข ฮฒ ) โข N < d , we know from the previous discussion that the decoding succeeds to retrieve m. โˆŽ

3.3.2 Bounding the extraction failure

According to Definition 2.3, our ๐–ฏ๐—ˆ๐–ฑ scheme is weakly ( ฮต , ฯ„ ) -sound if, for every polynomial-time algorithm ๐–ฏ outputting a response word r ( w ) from a file w, we have

Pr ฯƒ , m โก ( decoding r ( w ) โข into โข m โข fails wt ( V ฯƒ ( r ( w ) ) ) โ‰ค ฮต N | m โ† R ๐”ฝ q k ฯƒ โ† R ๐”– โข ( ๐”ฝ q ) n w = ฯƒ โข ( ๐’ž โข ( m ) ) ) โ‰ค ฯ„ .

Using Proposition 3.9, the security analysis of our ๐–ฏ๐—ˆ๐–ฑ scheme reduces to measuring the ability of the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ to produce a response word r which is ฮต-close to ( ๐’ฌ , V ฯƒ ) and a ฮฒ-liar for ( ๐’ฌ , V ฯƒ , w ) , with ( ฮต + 2 โข ฮฒ ) โข N โ‰ฅ d .

For fixed r โˆˆ โ„› ๐’ฌ , ฯƒ โˆˆ ๐”– โข ( ๐”ฝ q ) n and w = ฯƒ โข ( ๐’ž โข ( m ) ) the authentic file given to the prover, we define three subsets of ๐’ฌ :

  • โ€ข

    ๐’Ÿ ( r , w ) : = { u โˆˆ ๐’ฌ , r u โ‰  R ( w ) u } and D ( r , w ) : = | ๐’Ÿ ( r , w ) | = wt ( r - R ( w ) ) . This represents challenges u on which the response word r differs from the authentic one R โข ( w ) .

  • โ€ข

    โ„ฐ ( r , ฯƒ ) : = { u โˆˆ ๐’ฌ , V ฯƒ ( u , r u ) โ‰  0 } and E ( r , ฯƒ ) : = | โ„ฐ ( r , ฯƒ ) | = wt ( V ฯƒ ( r ) ) . These are challenges u on which the associated coordinate r u is not accepted by the verification map (it corresponds to erasures in the decoding process).

  • โ€ข

    โ„ฌ ( r , ฯƒ , w ) : = { u โˆˆ ๐’ฌ , r u โ‰  R ( w ) u and V ฯƒ ( u , r u ) = 0 } and B ( r , ฯƒ , m ) : = | โ„ฌ ( r , ฯƒ , m ) | . These are the challenges u on which the associated coordinate r u is accepted by the verification map, but differs from the authentic response s u (it corresponds to errors in the decoding process).

One can easily check that, for every ฯƒ, the sets โ„ฐ โข ( r , ฯƒ ) and โ„ฌ โข ( r , ฯƒ , w ) define a partition of ๐’Ÿ โข ( r , w ) . The probability of extraction failure can thus be written as

(3.1) Pr โก ( 2 โข D โข ( r , w ) - E โข ( r , ฯƒ ) โ‰ฅ d min โข ( R โข ( ๐’ž ) ) E โข ( r , ฯƒ ) โ‰ค ฮต โข N | m โ† R ๐”ฝ q k ฯƒ โ† R ๐”– โข ( ๐”ฝ q ) n w = ฯƒ โข ( ๐’ž โข ( m ) ) ) .

For w โˆˆ ๐”ฝ q n , let us define the set of admissible permutations and messages

ฮฆ w : = { ( ฯƒ , m ) โˆˆ ๐”– ( ๐”ฝ q ) n ร— ๐”ฝ q k , w = ฯƒ ( ๐’ž ( m ) ) } ,

so that equation (3.1) rewrites

Pr โก ( 2 โข D โข ( r , w ) - E โข ( r , ฯƒ ) โ‰ฅ d min โข ( R โข ( ๐’ž ) ) E โข ( r , ฯƒ ) โ‰ค ฮต โข N | ( ฯƒ , m ) โ† R ฮฆ w ) .

Later on, we will use the notation Pr ฮฆ w to refer to the fact that ( ฯƒ , m ) is uniformly drawn from ฮฆ w . Similarly we will use notation ๐”ผ ฮฆ w for the expectancy and Var ฮฆ w for the variance.

Given r โˆˆ โ„› ๐’ฌ , we also define

ฮฑ ( r , w ) : = max u โˆˆ ๐’Ÿ โข ( r , w ) Pr ฮฆ w ( V ฯƒ ( u , r u ) = 0 )

and ฮฑ : = max ( r , w ) ฮฑ ( r , w ) , where ( r , w ) are such that D โข ( r , w ) โ‰  0 . The parameter ฮฑ โˆˆ ( 0 , 1 ) is called the bias of the verification structure ( ๐’ฌ , V ) for ๐’ž . It corresponds to the maximum probability that a response is accepted but not authentic.

Lemma 3.10.

For all r โˆˆ R Q and w โˆˆ F q n , we have

๐”ผ ฮฆ w โข ( E โข ( r , ฯƒ ) ) โ‰ฅ ( 1 - ฮฑ ) โข D โข ( r , w ) .

Proof.

A simple computation shows

๐”ผ ฮฆ w โข ( E โข ( r , ฯƒ ) ) = ๐”ผ ฮฆ w โข ( โˆ‘ u โˆˆ ๐’Ÿ โข ( r , w ) ๐Ÿ™ V ฯƒ โข ( u , r u ) โ‰  0 ) = โˆ‘ u โˆˆ ๐’Ÿ โข ( r , w ) Pr ฮฆ w โก ( V ฯƒ โข ( u , r u ) โ‰  0 ) โ‰ฅ โˆ‘ u โˆˆ ๐’Ÿ โข ( r , w ) ( 1 - ฮฑ ) โ‰ฅ ( 1 - ฮฑ ) โข D โข ( r , w ) . โˆŽ

Lemma 3.10 essentially means that, if an adversary to our ๐–ฏ๐—ˆ๐–ฑ scheme wants its response word to be (in average) ฮต-close to the verification structure, then he should modify at most D โข ( r , w ) โ‰ค ฮต โข N 1 - ฮฑ responses. Below, we take advantage of this result, and we measure the probability of an extraction failure.

First, for ฮด , ฮต โˆˆ ( 0 , 1 ) , let

p โข ( r , w ; ฮต , ฮด ) : = Pr ฮฆ w ( 2 D ( r , w ) - E ( r , ฯƒ ) โ‰ฅ ฮด N and E ( r , ฯƒ ) โ‰ค ฮต N ) = Pr ฮฆ w โก ( E โข ( r , ฯƒ ) โ‰ค min โก { ฮต โข N , 2 โข D โข ( r , w ) - ฮด โข N } ) .

The probability p โข ( r , w ; ฮต , ฮด ) represents the probability that the extraction fails for a response code of relative distance ฮด and an adversarial response word r associated to w, which is ฮต-close to the verification structure. Let us bound p โข ( r , w ; ฮต , ฮด ) .

Proposition 3.11.

Let ฮด , ฮต โˆˆ ( 0 , 1 ) such that ฮด โข 1 - ฮฑ 1 + ฮฑ > ฮต . Let also r โˆˆ R Q and w โˆˆ F q n . Then we have

p โข ( r , w ; ฮต , ฮด ) โ‰ค Var ฮฆ w โข ( E โข ( r , ฯƒ ) ) ( 1 + ฮฑ 2 โข ( ฮด โข 1 - ฮฑ 1 + ฮฑ - ฮต ) ) 2 โข N 2 .

Proof.

We distinguish three cases.

(i) 2 โข D โข ( r , w ) - ฮด โข N < 0 . The event E โข ( r , ฯƒ ) โ‰ค min โก { ฮต โข N , 2 โข D โข ( r , w ) - ฮด โข N } never occurs since E โข ( r , ฯƒ ) โ‰ฅ 0 . Hence p โข ( r , w ; ฮต , ฮด ) = 0 .

(ii) ฮต โข N โ‰ค 2 โข D โข ( r , w ) - ฮด โข N . The inequality E โข ( r , ฯƒ ) โ‰ค ฮต โข N implies

E โข ( r , ฯƒ ) - ๐”ผ ฮฆ w โข ( E ) โ‰ค ฮต โข N - ( 1 - ฮฑ ) โข D โข ( r , w ) โ‰ค ฮต โข N - ( 1 - ฮฑ ) โข ฮต + ฮด 2 โข N โ‰ค - 1 + ฮฑ 2 โข ( ฮด โข 1 - ฮฑ 1 + ฮฑ - ฮต ) โข N .

Hence, using Chebychevโ€™s inequality,

p โข ( r , w ; ฮต , ฮด ) = Pr ฮฆ w โก ( E โข ( r , ฯƒ ) โ‰ค ฮต โข N ) โ‰ค Pr ฮฆ w โก ( | E โข ( r , ฯƒ ) - ๐”ผ ฮฆ w โข ( E ) | โ‰ฅ 1 + ฮฑ 2 โข ( ฮด โข 1 - ฮฑ 1 + ฮฑ - ฮต ) โข N ) โ‰ค Var ฮฆ w โข ( E โข ( r , ฯƒ ) ) ( 1 + ฮฑ 2 โข ( ฮด โข 1 - ฮฑ 1 + ฮฑ - ฮต ) ) 2 โข N 2 .

(iii) 0 โ‰ค 2 โข D โข ( r , w ) - ฮด โข N < ฮต โข N . In this case, E โข ( r , ฯƒ ) โ‰ค 2 โข D โข ( r , w ) - ฮด โข N implies

E โข ( r , ฯƒ ) - ๐”ผ ฮฆ w โข ( E ) โ‰ค ( 1 + ฮฑ ) โข D โข ( r , w ) - ฮด โข N โ‰ค ( 1 + ฮฑ ) โข ฮต + ฮด 2 โข N - ฮด โข N โ‰ค - 1 + ฮฑ 2 โข ( ฮด โข 1 - ฮฑ 1 + ฮฑ - ฮต ) โข N .

Therefore, similarly to the previous case, we obtain the claimed result. โˆŽ

For any u โˆˆ ๐’Ÿ โข ( r , w ) , denote by X u the { 0 , 1 } -random variable โ€œ ๐Ÿ™ V ฯƒ โข ( u , r u ) = 0 โ€ when ฯƒ is uniformly drawn from ฮฆ w . It holds that E โข ( r , ฯƒ ) = โˆ‘ u โˆˆ ๐’Ÿ โข ( r , w ) ( 1 - X u ) .

Recall that two real random variables Y , Z are uncorrelated if ๐”ผ โข ( Y โข Z ) = ๐”ผ โข ( Y ) โข ๐”ผ โข ( Z ) . For instance, two independent random variables are uncorrelated.

Lemma 3.12.

Let r โˆˆ R Q and w โˆˆ F q n . If the random variables { X u } u โˆˆ D โข ( r , w ) are pairwise uncorrelated, then

Var ฮฆ w โข ( E โข ( r , ฯƒ ) ) โ‰ค D โข ( r , w ) .

Proof.

By assumption, { X u } u โˆˆ ๐’Ÿ โข ( r , w ) are pairwise uncorrelated; hence

Var ฮฆ w โข ( E โข ( r , ฯƒ ) ) = โˆ‘ u โˆˆ ๐’Ÿ โข ( r , w ) Var ฮฆ w โข ( 1 - X u ) .

The trivial bound Var ฮฆ w โข ( 1 - X u ) โ‰ค 1 gives the result. โˆŽ

As a corollary of Proposition 3.11 and Lemma 3.12, under the same hypothesis and assuming ฮด โข 1 - ฮฑ 1 + ฮฑ > ฮต , we get

p โข ( r , w ; ฮต , ฮด ) โ‰ค 4 N โข ( ( 1 - ฮฑ ) โข ฮด - ( 1 + ฮฑ ) โข ฮต ) 2

since D โข ( r , w ) โ‰ค N . Moreover, if lim N โ†’ โˆž โก ฮด > 0 and lim N โ†’ โˆž โก ฮฑ = 0 , then p โข ( r , w ; ฮต , ฮด ) = ๐’ช โข ( 1 / N ) .

Therefore, we end up with the following theorem.

Theorem 3.13.

Let ( Q , V ) be a verification structure for C with bias ฮฑ. Let N = | Q | , and let ฮด = d min โข ( R โข ( C ) ) / N be the relative distance of the associated response code. Finally, assume that, for any r โˆˆ R Q and any w โˆˆ F q n , the variables { X u } u โˆˆ D โข ( r , w ) are pairwise uncorrelated. Then, for any ฮต < ฮด โข 1 - ฮฑ 1 + ฮฑ , the PoR scheme associated to C and ( Q , V ) is ( ฮต , ฯ„ ) -sound, where

ฯ„ = 4 N โข ( ( 1 - ฮฑ ) โข ฮด - ( 1 + ฮฑ ) โข ฮต ) 2 .

For asymptotically small ฮฑ, a code ๐’ž equipped with a verification structure satisfying the conditions of Theorem 3.13 thus gives an ( ฮต , ฯ„ ) -sound ๐–ฏ๐—ˆ๐–ฑ scheme for every ฮต < ( 1 + o โข ( 1 ) ) โข ฮด and ฯ„ = ๐’ช โข ( 1 / N ) .

According to Theorem 3.13, we thus need to look for (sequences of) codes ๐’ž and associated verification structures ( ๐’ฌ , V ) such that

  1. (i)

    the response code R โข ( ๐’ž ) admits a good relative distance ฮด = d min โข ( R โข ( ๐’ž ) ) / N ,

  2. (ii)

    the bias ฮฑ is small,

  3. (iii)

    random variables { X u } u โˆˆ ๐’Ÿ โข ( r , w ) are pairwise uncorrelated.

Sections 3.4 and 3.5 characterise conditions under which the last two points are fulfilled. Then, in Section 5, we discuss which response codes can achieve good relative distance.

3.4 Estimating ฮฑ

In this section, we prove that, assuming ฮฆ w approximates the uniform distribution over ๐”– โข ( ๐”ฝ q ) n in a sense that we make precise later, the bias ฮฑ can be bounded according to parameters of the verification structure.

Let us fix r โˆˆ โ„› ๐’ฌ , w โˆˆ ๐”ฝ q n and u โˆˆ ๐’ฌ . We recall that ฮฑ is defined by

ฮฑ = max r , w โก max u โˆˆ ๐’Ÿ โข ( r , w ) โก Pr ฮฆ w โก ( V ฯƒ โข ( u , r u ) = 0 ) ,

where randomness comes from ฯƒ โ† R ฮฆ w = { ( ฯƒ , m ) โˆˆ ๐”– ( ๐”ฝ q ) n ร— ๐”ฝ q k , w = ฯƒ ( ๐’ž ( m ) ) } . We notice that this is equivalent to write ฯƒ โ† R { ฯƒ โˆˆ ๐”– ( ๐”ฝ q ) n , ฯƒ - 1 ( w ) โˆˆ ๐’ž } .

For convenience, we will view r u โˆˆ โ„› = ๐”ฝ q โ„“ as a vector indexed by u = ( u 1 , โ€ฆ , u โ„“ ) , so that we can easily denote by r u โข [ u j ] โˆˆ ๐”ฝ q its j-th coordinate, 1 โ‰ค j โ‰ค โ„“ . We define the code K u : = ker V ( u , โ‹… ) โŠ† ๐”ฝ q โ„“ , and up to re-indexing coordinates, ๐’ž | u โŠ† K u . This allows us to write that, for every ฯƒ, we have V ฯƒ โข ( u , r u ) = 0 if and only if ฯƒ u - 1 โข ( r u ) โˆˆ K u . Finally, we denote by Z u : = { i โˆˆ u , r u [ i ] โ‰  R ( w ) u [ i ] } the set of coordinates of r u that are not authentic.

Let Y u โข ( ฯƒ ) represent the event โ€œ ฯƒ u - 1 ( r u ) โˆˆ K u โˆฃ supp ( ฯƒ u - 1 ( r u ) ) = Z u โ€. Informally, the reason why we consider an event Y u โข ( ฯƒ ) conditioned by supp โก ( ฯƒ u - 1 โข ( r u ) ) = Z u is that the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ is free to choose any support Z u on which he can modify the original file. More formally, this constraint will help us to bound the probability Pr ฮฆ w โก ( V ฯƒ โข ( u , r u ) = 0 ) in Lemma 3.14. We say that ฮฆ w is sufficiently uniform if, for every u โˆˆ ๐’ฌ , we have

ฮณ u : = Pr โก [ Y u โข ( ฯƒ ) โˆฃ ฯƒ โ† R ฮฆ w ] - Pr โก [ Y u โข ( ฯƒ ) โˆฃ ฯƒ โ† R ๐”– โข ( ๐”ฝ q ) n ] Pr โก [ Y u โข ( ฯƒ ) โˆฃ ฯƒ โ† R ๐”– โข ( ๐”ฝ q ) n ] = o ( 1 )

when the file size n โข log โก q โ†’ โˆž . In other words, ฮฆ w is sufficiently uniform if it is a good approximation of the whole set of n-tuples of permutations, when considering the probability that Y u โข ( ฯƒ ) happens.

Lemma 3.14.

Let r, w, u and Z u be defined as above. Let also A u = | { x โˆˆ K u , supp ( x ) = Z u } | . Then

Pr ฮฆ w โก ( V ฯƒ โข ( u , r u ) = 0 ) โ‰ค ( 1 + ฮณ u ) โข A u ( q - 1 ) | Z u | .

Proof.

For every ฯƒ such that ( ฯƒ , m ) โˆˆ ฮฆ w , we know that ฯƒ u - 1 โข ( R โข ( w ) u ) โˆˆ K u , and we recall that V ฯƒ โข ( u , r u ) = 0 if and only if ฯƒ u - 1 โข ( r u ) โˆˆ K u . Since K u is linear, and up to considering ฯƒ u - 1 โข ( R โข ( w ) u - r u ) instead, we can assume without loss of generality that ฯƒ u - 1 โข ( r u ) โข [ i ] = 0 for every i โˆˆ u โˆ– Z u . In other words, we assume that supp โก ( ฯƒ u - 1 โข ( r u ) ) = Z u .

Remark that

Pr ฯƒ โ† R ๐”– โข ( ๐”ฝ q ) n [ ฯƒ u - 1 ( r u ) โˆˆ K u โˆฃ supp ( ฯƒ u - 1 ( r u ) ) = Z u ] = Pr x โ† R ๐”ฝ q โ„“ โก [ x โˆˆ K u โˆฃ supp โก ( x ) = Z u ] = Pr x โ† R ๐”ฝ q โ„“ โก [ x โˆˆ K u โˆฃ supp โก ( x ) = Z u ] = A u ( q - 1 ) | Z u |

since A u counts the number of codewords in K u whose support is Z u .

Therefore, we get

Pr ฮฆ w โก ( V ฯƒ โข ( u , r u ) = 0 ) โ‰ค Pr ฮฆ w โก [ V ฯƒ โข ( u , r u ) = 0 โˆฃ supp โก ( ฯƒ u - 1 โข ( r u ) ) = Z u ] = ( 1 + ฮณ u ) โข Pr ๐”– โข ( ๐”ฝ q ) n โก [ V ฯƒ โข ( u , r u ) = 0 โˆฃ supp โก ( ฯƒ u - 1 โข ( r u ) ) = Z u ] = ( 1 + ฮณ u ) โข Pr x โ† R ๐”ฝ q โ„“ โก [ x โˆˆ K u โˆฃ supp โก ( x ) = Z u ] = ( 1 + ฮณ u ) โข A u ( q - 1 ) | Z u | . โˆŽ

Lemma 3.15.

Let S u be the F q -vector space ใ€ˆ { x โˆˆ K u , supp ( x ) = Z u } ใ€‰ , and assume that S u โ‰  { 0 } . We have

A u โ‰ค q | Z u | - d min โข ( S u ) + 1 .

Proof.

We prove that, if A u > q e for some integer e โ‰ฅ 0 , then d min โข ( S u ) โ‰ค | Z u | - e , which clearly induces our result. If A u > q e , then dim โก S u > e since | S u | โ‰ฅ A u . The Singleton bound then provides

d min โข ( S u ) โ‰ค | Z u | - dim โก S u + 1 โ‰ค | Z u | - e . โˆŽ

Finally, we get the following upper bound on ฮฑ.

Proposition 3.16.

Let ฮ” = min โก { d min โข ( K u ) , u โˆˆ Q } . Then

ฮฑ โ‰ค ( 1 + ฮณ ) โข ( 1 + 1 q - 1 ) โ„“ โข q - ฮ” + 1 ,

where ฮณ u = max โก ฮณ u .

Proof.

Remark that S u , defined in previous lemma, is a subcode of K u shortened on u โˆ– Z u . Hence

d min โข ( K u ) โ‰ค d min โข ( S u ) ,

and we can apply previous results and obtain the desired bound

ฮฑ โ‰ค max u , r โก ( 1 + ฮณ u ) โข ( q q - 1 ) | Z u | โข q - d min โข ( K u ) + 1 โ‰ค ( 1 + ฮณ ) โข ( 1 + 1 q - 1 ) โ„“ โข q - ฮ” + 1 ,

where ฮณ = max u โก ฮณ u . โˆŽ

If every ฮฆ w is sufficiently uniform, then, by definition, we have ฮณ = o โข ( 1 ) when the file size n โข log โก q โ†’ โˆž . This assumption is significant since we desire to have a small bias ฮฑ, which is deeply linked to the soundness of ๐–ฏ๐—ˆ๐–ฑ s (see Theorem 3.13). In Appendix A, we present experimental estimates of ฮฑ, validating that the assumption that ฮฆ w is sufficiently uniform.

3.5 Pairwise uncorrelation of { X u } u โˆˆ ๐’Ÿ

This section is devoted to proving that variables { X u } u โˆˆ ๐’Ÿ โข ( r , w ) are pairwise uncorrelated if the supports of challenges u โˆˆ ๐’Ÿ โข ( r , w ) have small pairwise intersection. For this purpose, let us recall that, for fixed r โˆˆ โ„› ๐’ฌ , w and u โˆˆ ๐’Ÿ โข ( r , w ) , the random variable X u represents ๐Ÿ™ V ฯƒ โข ( u , r u ) = 0 when ฯƒ is uniformly picked in ฮฆ w .

We first state a technical lemma that will be useful to prove Proposition 3.18 below. For clarity, we denote by d โŠฅ โข ( ๐’ž ) the minimum distance of the dual code ๐’ž โŠฅ of a linear code ๐’ž .

Lemma 3.17.

Let C โŠ† F q n be a linear code and T โŠ‚ [ 1 , n ] , | T | = t , where t < d โŠฅ โข ( C ) . For a โˆˆ F q T , we define

๐’ฑ a = { c โˆˆ ๐’ž , c | T = a } โ€ƒ ๐‘Ž๐‘›๐‘‘ โ€ƒ N a = | ๐’ฑ a | .

Then

  1. (i)

    ๐’ฑ 0 = { v โˆˆ ๐’ž , v | T = 0 } is a linear subcode of ๐’ž ;

  2. (ii)

    for every non-zero a โˆˆ ๐”ฝ q T , there exists a non-zero c ( a ) โˆˆ ๐’ž such that ๐’ฑ a = ๐’ฑ 0 + { c ( a ) } ;

  3. (iii)

    for every a โˆˆ ๐”ฝ q T , N a = q k - t , where k = dim โก ๐’ž .

Proof.

(i) The fact that ๐’ฑ 0 = { v โˆˆ ๐”ฝ q X , v | T = 0 } is actually the well-known definition of the shortening of a code. It is easy to prove that it defines a linear code.

(ii) Let a โˆˆ ๐”ฝ q T be non-zero, and let us first prove that there exists c ( a ) โˆˆ ๐’ž such that c | T ( a ) = a . If it were not the case, then, by definition, we would have ๐’ž | T โ‰  ๐”ฝ q t . But this is impossible since ๐’ž โŠฅ contains no non-zero codeword of weight less that t. It is then easy to check that ๐’ฑ a = ๐’ฑ 0 + { c ( a ) } .

(iii) First notice that ๐’ฑ a โˆฉ ๐’ฑ b = โˆ… if a โ‰  b . Since

๐’ž = โ‹ƒ a โˆˆ ๐”ฝ q t ๐’ฑ a ,

we get the expected result. โˆŽ

Proposition 3.18.

If max โก { | u โˆฉ v | , u โ‰  v โˆˆ Q } < min โก { d โŠฅ โข ( C | u ) , u โˆˆ Q } , then the random variables { X u } u โˆˆ Q are pairwise uncorrelated.

Proof.

Recall that K u : = ker V ( u , โ‹… ) and that, by definition of a verification structure, we have ๐’ž | u โŠ† K u . For u โ‰  v โˆˆ ๐’ฌ , let us prove that ๐”ผ โข ( X u โข X v ) = ๐”ผ โข ( X u ) โข ๐”ผ โข ( X v ) . First,

๐”ผ โข ( X u โข X v ) = Pr โก ( V ฯƒ โข ( u , r u ) = 0 โข and โข V ฯƒ โข ( v , r v ) = 0 ) = Pr โก ( ฯƒ - 1 โข ( r u ) | u โˆˆ K u โข and โข ฯƒ - 1 โข ( r v ) | v โˆˆ K v ) .

Denote t = | u โˆฉ v | , and let ( ๐š , ๐› ) โˆˆ ( ๐”ฝ q t ) 2 . We denote by Z โข ( ฯƒ , ๐š , ๐› ) the event

ฯƒ - 1 โข ( r u ) | u โˆฉ v = ๐š โ€ƒ and โ€ƒ ฯƒ - 1 โข ( r v ) | u โˆฉ v = ๐› .

We first notice that { ฯƒ | u โˆฉ v - 1 , ฯƒ โˆˆ ฮฆ w } = ๐”– ( ๐”ฝ q ) t . Indeed, we can here use an argument similar to the proof of Lemma 3.17: the constraint ฯƒ - 1 โข ( w ) โˆˆ ๐’ž is ineffective on ฯƒ | u โˆฉ v - 1 since | u โˆฉ v | โ‰ค t < d โŠฅ โข ( ๐’ž | z ) for every z โˆˆ ๐’ฌ . Therefore, for every ( ๐š , ๐› ) โˆˆ ( ๐”ฝ q t ) 2 , we have

Pr โก ( Z โข ( ฯƒ , ๐š , ๐› ) ) = q - 2 โข t ,

and it follows that

๐”ผ โข ( X u โข X v ) = 1 q 2 โข t โข โˆ‘ ๐š , ๐› โˆˆ ( ๐”ฝ q t ) 2 Pr โก ( ฯƒ - 1 โข ( r u ) | u โˆˆ K u โข and โข ฯƒ - 1 โข ( r v ) | v โˆˆ K v โˆฃ Z โข ( ฯƒ , ๐š , ๐› ) ) .

Recall now that t < min โก { d โŠฅ โข ( ๐’ž | u ) , u โˆˆ ๐’ฌ } โ‰ค min โก { d โŠฅ โข ( K u ) , u โˆˆ ๐’ฌ } . Hence, for fixed ๐š and ๐› , the variables ฯƒ - 1 ( r u ) | u โˆˆ K u โˆฃ Z ( ฯƒ , ๐š , ๐› ) and ฯƒ - 1 ( r v ) | v โˆˆ K v โˆฃ Z ( ฯƒ , ๐š , ๐› ) are independent (once again, it is a consequence of the structure results of Lemma 3.17). Therefore,

๐”ผ โข ( X u โข X v ) = 1 q 2 โข t โข โˆ‘ ๐š , ๐› โˆˆ ( ๐”ฝ q t ) 2 Pr โก ( ฯƒ - 1 โข ( r u ) | u โˆˆ K u โˆฃ Z โข ( ฯƒ , ๐š , ๐› ) ) โข Pr โก ( ฯƒ - 1 โข ( r v ) | v โˆˆ K v โˆฃ Z โข ( ฯƒ , ๐š , ๐› ) ) .

Then

๐”ผ โข ( X u โข X v ) = 1 q 2 โข t โข โˆ‘ ๐š , ๐› โˆˆ ( ๐”ฝ q t ) 2 Pr โก ( ฯƒ - 1 โข ( r u ) | u โˆˆ K u โˆฃ ฯƒ - 1 โข ( r u ) | u โˆฉ v = ๐š ) โข Pr โก ( ฯƒ - 1 โข ( r v ) | v โˆˆ K v โˆฃ ฯƒ - 1 โข ( r v ) | u โˆฉ v = ๐› ) ,

and we conclude since

๐”ผ โข ( X u ) = q - t โข โˆ‘ ๐š โˆˆ ๐”ฝ q t Pr โก ( ฯƒ - 1 โข ( r u ) | u โˆˆ K u โˆฃ ฯƒ - 1 โข ( r u ) | u โˆฉ v = ๐š ) . โˆŽ

4 Performance

4.1 Efficient scrambling of the encoded file

In the ๐–ฏ๐—ˆ๐–ฑ scheme we propose, the storage cost of an n-tuple of permutations in ๐”– โข ( ๐”ฝ q ) n is excessive since it is superlinear in the original file size. In this subsection, we propose a storage-efficient way to scramble the codeword c โˆˆ ๐’ž produced by the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ .

Precisely, we want to define a family of maps ( ฯƒ ( ฮบ ) ) ฮบ , where ฯƒ ( ฮบ ) : ๐’ž โ†’ ๐”ฝ q n , c โ†ฆ w โˆˆ ๐”ฝ q n , with the following requirements:

  • โ€ข

    For every ฮบ, the map ฯƒ ( ฮบ ) is efficiently computable and requires a low storage.

  • โ€ข

    For every ฮบ and every c โˆˆ ๐’ž , if w = ฯƒ ( ฮบ ) โข ( c ) , then, for every i โˆˆ [ 1 , n ] , the local inverse map w i โ†ฆ c i is efficiently computable.

  • โ€ข

    If ฮบ is randomly generated but unknown, then, given the knowledge of w = ฯƒ ( ฮบ ) โข ( c ) and ๐’ž , it is hard to produce a response word r โˆˆ โ„› ๐’ฌ such that, for many u โˆˆ ๐’ฌ , both V ฯƒ ( ฮบ ) โข ( u , r u ) = 0 and r u โ‰  w | u hold. To be more specific and in light of the security analysis of Section 3.3, we require that it is hard to distinguish ฯƒ ( ฮบ ) โข ( c ) from a random ( z 1 , โ€ฆ , z n ) โˆˆ ๐”ฝ q n , where symbols z i are picked independently and uniformly at random.

We here propose to derive ฯƒ ( ฮบ ) from a suitable block cipher, yielding the explicit construction given below. Of course, other proposals can be envisioned.

The construction

Let IV denote a random initialisation vector for AES in CTR mode ( IV could be a nonce concatenated with a random value). Vector IV is kept secret by the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ , as well as a randomly chosen key ฮบ for the cipher. Let also f be a permutation polynomial over ๐”ฝ q of degree d > 1 . For instance, one could choose f โข ( x ) = x d with gcd โก ( d , q - 1 ) = 1 . Notice that polynomial f can be made public.

Let s = โŒŠ 256 โŒˆ log 2 โก q โŒ‰ โŒ‹ be the number of ๐”ฝ q -symbols one can store in a 256-bit word[2]. Up to appending a few random bits to c, we assume that s โˆฃ n , and we define t = n / s . Let us fix a partition of [ 1 , n ] into s-tuples i = ( i 1 , โ€ฆ , i s ) ; it can be, for instance, ( 1 , โ€ฆ , s ) , ( s + 1 , โ€ฆ , 2 โข t ) , โ€ฆ , ( ( t - 1 ) โข s + 1 , โ€ฆ , n ) . Notice that this partition does not need to be chosen at random. Given c = ( c 1 , โ€ฆ , c n ) โˆˆ ๐’ž and i an element of the above partition, we now define

b i = ( f โข ( c i 1 ) โข โˆฃ โ€ฆ โˆฃ โข f โข ( c i s ) ) โŠ• AES ฮบ โข ( IV โŠ• i ) โˆˆ { 0 , 1 } 256 .

If log 2 โก q โˆค 256 , trailing zeroes can be added to evaluations of f. Finally, the pseudo-random permutation ฯƒ is defined by

ฯƒ ( c ) : = ( b 1 , โ€ฆ , b t ) .

Design rationale

AES is a natural choice when one needs a (secret-)keyed pseudo-random permutation. Also notice that, with this construction, one only needs to store the key ฮบ and the vector IV since the other objects (the polynomial f, the partition) are made public. Hence our objectives in terms of storage are met.

We now point out the necessity to use i as a part of the input of the AES cipher. Assume that we do not. Then the local permutation ฯƒ j , 1 โ‰ค j โ‰ค n , would not depend on j. As a consequence, for a certain class of codes, the local verification map r u โ†ฆ V ฯƒ โข ( u , r u ) would not depend on u, and a malicious ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ would then be able to produce accepted answers while storing only a small piece of the file w (e.g., w | u for only one u โˆˆ ๐’ฌ ).

Another mandatory feature is the non-linearity of the permutation polynomial f. Indeed, assume, for instance, that f = id . Then, given the knowledge of w = ฯƒ โข ( c ) , it would be very easy for a malicious ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ to produce a word w โ€ฒ โ‰  w such that r โ€ฒ = R โข ( w โ€ฒ ) is always accepted by the ๐–ต๐–พ๐—‹๐—‚๐–ฟ๐—‚๐–พ๐—‹ . Simply, the ๐–ฏ๐—‹๐—ˆ๐—๐–พ๐—‹ defines w โ€ฒ = w + c โ€ฒ , where c โ€ฒ is any non-zero codeword of ๐’ž . Hence one sees that the polynomial f must be non-linear in order to prevent such kind of attacks.

4.2 Parameters

We here consider a ๐–ฏ๐—ˆ๐–ฑ built upon a code ๐’ž โŠ† ๐”ฝ q n with verification structure ( ๐’ฌ , V ) satisfying โ„› = ๐”ฝ q โ„“ and V โข ( โ„› ) = ๐”ฝ q s . We also assume that we use an n-tuple of pseudo-random permutations as described in the previous subsection.

Communication complexity

At each verification step, the client sends an โ„“ -tuple of coordinates ( u 1 , โ€ฆ , u โ„“ ) , u i โˆˆ [ 1 , n ] . The server then answers with corresponding symbols w u i โˆˆ ๐”ฝ q . Therefore, the upload communication cost is โ„“ โข log 2 โก n bits, while the download communication cost is โ„“ โข log 2 โก q , thus a total of โ„“ โข ( log 2 โก n + log 2 โก q ) bits.

Computation complexity

In the initialisation phase, following the encryption described in Section 4.1, the client essentially has

  • โ€ข

    to compute the codeword c โˆˆ ๐’ž associated to its message,

  • โ€ข

    to make n evaluations of the permutation polynomial f over ๐”ฝ q ,

  • โ€ข

    to compute t = n โข log 2 โก q 256 AES ciphertexts to produce the word w to be sent to the server.

Given a generator matrix of ๐’ž , the codeword c can be computed in ๐’ช โข ( k โข n ) operations over ๐”ฝ q with a matrix-vector product. Notice that quasi-linear-time encoding algorithms exist for some classes of codes. Besides, if a monomial or a sparse permutation polynomial is used, then the cost of each evaluation is ๐’ช โข ( ( log 2 โก q ) 3 ) . If we denote by c the bitcost of an AES encryption, we get a total bitcost of ๐’ช โข ( n โข k โข ( log 2 โก q ) 2 + n โข ( log 2 โก q ) 3 + c โข n โข log 2 โก q ) for the initialisation phase. Recall this is a worst-case scenario in which the encoding process is inefficient.

At each verification step, an honest server only needs to read โ„“ symbols from the file it stores. Hence its computation complexity is ๐’ช โข ( โ„“ ) . The client has to compute a matrix-vector product over ๐”ฝ q , where the matrix has size s ร— โ„“ and the vector has size โ„“ , thus a computation cost of ๐’ช โข ( โ„“ โข s ) operations over ๐”ฝ q .

Storage needs

The client stores 2 ร— 256 bits for secret material ฮบ and IV to use in AES. The server storage overhead exactly corresponds to the redundancy of the linear code ๐’ž , that is, ( n - dim โก ๐’ž ) โข log 2 โก q bits.

Other features

Our ๐–ฏ๐—ˆ๐–ฑ scheme is unbounded-use since every challenge reveals nothing about the secret data held by the client. It does not feature dynamic updates of files. Though, we must emphasise that the file w the client produces can be split among several servers, and the verification step remains possible even if the servers do not communicate with each other. Indeed, computing a response to a challenge does not require mixing distinct symbols w i of the uploaded file. Therefore, our scheme is well suited for the storage of large static distributed databases. Parameters of the ๐–ฏ๐—ˆ๐–ฑ schemes we propose are reported in Figure 4.

Figure 4

Summary of parameters of our ๐–ฏ๐—ˆ๐–ฑ construction for an original file of size k โข log 2 โก q bits and a code ๐’ž of dimension k over ๐”ฝ q equipped with a verification structure ( ๐’ฌ , V ) such that | u | = โ„“ and rank โก V โข ( u , โ‹… ) โ‰ค s for all u โˆˆ ๐’ฌ .

5 Instantiations

In this section, we present several instantiations of our ๐–ฏ๐—ˆ๐–ฑ construction. We first recall basics and notation from coding theory.

The code Rep โข ( โ„“ ) โŠ† ๐”ฝ q โ„“ denotes the repetition code ใ€ˆ ( 1 , โ€ฆ , 1 ) ใ€‰ . We recall that Rep โข ( โ„“ ) โŠฅ is the parity code Par ( โ„“ ) : = { c โˆˆ ๐”ฝ q โ„“ , โˆ‘ i = 1 โ„“ c i = 0 } . Let ๐’ž , ๐’ž โ€ฒ be two linear codes over ๐”ฝ q of respective parameters [ n , k , d ] and [ n , k โ€ฒ , d โ€ฒ ] . Their tensor product ๐’ž โŠ— ๐’ž โ€ฒ is the ๐”ฝ q -linear code generated by words

( c i c j โ€ฒ : 1 โ‰ค i โ‰ค n , โ€‰1 โ‰ค j โ‰ค n โ€ฒ ) โˆˆ ๐”ฝ q n โข n โ€ฒ .

It has dimension k โข k โ€ฒ and minimum distance d โข d โ€ฒ . We also denote by

๐’ž โŠ— s : = ๐’ž โŠ— โ€ฆ โŠ— ๐’ž โŸ s โข  times โŠ† ๐”ฝ q n s

the s-fold tensor product of ๐’ž with itself.

5.1 Tensor-product codes

The upcoming subsection illustrates our construction with a non practical but simple instance. The next ones lead to practical ๐–ฏ๐—ˆ๐–ฑ instances.

5.1.1 A simple but non-practical instance

Let n = N โข โ„“ and ๐’ฌ = { u i = { i โ„“ + 1 , i โ„“ + 2 , โ€ฆ , ( i + 1 ) โ„“ } , i โˆˆ [ 0 , N - 1 ] } . The set ๐’ฌ defines a partition of [ 1 , n ] . We define the code

๐’ž = { c โˆˆ ๐”ฝ q n , โˆ‘ j โˆˆ u c j = 0 for all u โˆˆ ๐’ฌ } โŠ† ๐”ฝ q n .

In other words, ๐’ž = Par โข ( โ„“ ) โŠ— ๐”ฝ q N , and a parity-check matrix H for ๐’ž is given by

H = ( 1 โ‹ฏ 1 0 โ‹ฏ โ‹ฏ โ‹ฏ โ‹ฏ โ‹ฏ 0 0 โ‹ฏ 0 1 โ‹ฏ 1 โ‹ฑ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฑ โ‹ฑ 0 0 โ‹ฏ โ‹ฏ โ‹ฏ โ‹ฏ โ‹ฏ 0 1 โ‹ฏ 1 ) .

The verification map V : ๐’ฌ ร— ๐”ฝ q โ„“ โ†’ ๐”ฝ q is defined by V ( u , b ) : = โˆ‘ j = 1 โ„“ b u j for all ( u , b ) โˆˆ ๐’ฌ ร— ๐”ฝ q โ„“ . By construction (see the fundamental Example 3.2), the pair ( ๐’ฌ , V ) defines a verification structure for ๐’ž .

Lemma 5.1.

Let C = Par โข ( โ„“ ) โŠ— F q N as above. Then the response code R โข ( C ) has minimum distance 1.

Proof.

We see that the restriction map R sends the codeword ( 1 , - 1 , 0 , 0 , โ€ฆ , 0 ) โˆˆ ๐’ž to a word of weight 1. Besides, R is injective, so d min โข ( R โข ( ๐’ž ) ) > 0 . โˆŽ

Since ฮด = d min โข ( R โข ( ๐’ž ) ) / N = 1 / N โ†’ 0 when N goes to infinity, an attempt to build a ๐–ฏ๐—ˆ๐–ฑ scheme from ๐’ž cannot be practical.

5.1.2 Higher order tensor-product codes

Let ๐’œ โŠ† ๐”ฝ q โ„“ be a non-degenerate [ โ„“ , k ๐’œ , d ๐’œ ] q -linear code, and define ๐’ž = ๐’œ โŠ— s โŠ† ๐”ฝ q n , where n = โ„“ s . Notice that it will be more convenient to see coordinates of words w โˆˆ ๐”ฝ q n as elements of [ 1 , โ„“ ] s .

For ๐š โˆˆ [ 1 , โ„“ ] s and 1 โ‰ค i โ‰ค s , we define L i , ๐š โŠ‚ [ 1 , โ„“ ] s , the โ€œi-th axis-parallel line with basis ๐š โ€, as

L i , ๐š : = { ๐ฑ โˆˆ [ 1 , โ„“ ] s such that x j = a j for all j โ‰  i } .

By definition of ๐’ž , a word c lies in ๐’ž if and only if, for every L = L i , ๐š , the restriction c | L โˆˆ ๐’œ . This means that we can define

  • โ€ข

    a set of queries ๐’ฌ = { L i , ๐š , i โˆˆ [ 1 , s ] , ๐š โˆˆ [ 1 , โ„“ ] s } ,

  • โ€ข

    a verification map

    V : ๐’ฌ ร— โ„› โ†’ ๐”ฝ q โ„“ - k ๐’œ ,
    ( L , r ) โ†ฆ H โข r ,

    where H is a parity-check matrix for ๐’œ whose columns are ordered according to the line L.

By the previous discussion, it is clear that c โˆˆ ๐’ž implies that V โข ( L , c | L ) = 0 for every L โˆˆ ๐’ฌ (in fact, these two assertions are equivalent). Hence ( ๐’ฌ , V ) defines a verification structure for ๐’ž , and we have N = | ๐’ฌ | = s โข โ„“ s - 1 .

Lemma 5.2.

Let C = A โŠ— s as above. Then R โข ( C ) has minimum distance s โ‹… d A s - 1 .

Proof.

Let us first prove that the minimum distance of R โข ( ๐’ž ) is larger than s โ‹… d ๐’œ s - 1 . Let r = R โข ( c ) โˆˆ R โข ( ๐’ž ) , and assume r โ‰  0 . Then there exists L โˆˆ ๐’ฌ such that 0 โ‰  r L = c | L โˆˆ ๐’œ . Therefore, c ๐ฑ โ‰  0 for some ๐ฑ โˆˆ L โŠ‚ [ 1 , โ„“ ] s . Consider the set

S i , ๐ฑ = { ๐ฒ โˆˆ [ 1 , โ„“ ] s , y i = x i } .

Very informally, the set S i , ๐ฑ corresponds to the hyperplane passing through ๐ฑ and โ€œorthogonalโ€ to the i-th axis. By definition of ๐’ž = ๐’œ โŠ— s , we know that c | S i , ๐ฑ โˆˆ ๐’œ โŠ— ( s - 1 ) โˆ– { 0 } for every 1 โ‰ค i โ‰ค s . Let

U i = supp โก ( c | S i , ๐ฑ ) = { ๐ฎ ( i , 1 ) , โ€ฆ , ๐ฎ ( i , t i ) }

with t i โ‰ฅ d min โข ( ๐’œ โŠ— ( s - 1 ) ) = ( d ๐’œ ) s - 1 . Every ๐ฎ ( i , j ) โˆˆ U i defines a line L i , ๐ฎ ( i , j ) on which c | L i , ๐ฎ ( i , j ) is a non-zero codeword of ๐’œ . Equivalently, r is non-zero on index L i , ๐ฎ ( i , j ) โˆˆ ๐’ฌ . Therefore,

wt ( r ) = | { L โˆˆ ๐’ฌ , r L โ‰  0 } | โ‰ฅ | โ‹ƒ i = 1 s { L i , ๐ฎ ( i , j ) , โ€‰1 โ‰ค j โ‰ค t i } | โ‰ฅ โˆ‘ i = 1 s t i โ‰ฅ s ( d ๐’œ ) s - 1 .

Let us now build a word r โˆˆ R โข ( ๐’ž ) of weight s โข ( d ๐’œ ) s - 1 . Let w โˆˆ ๐’œ โˆ– { 0 } be a minimum-weight codeword of ๐’œ , and define W : = supp ( w ) โŠ† A . Define c = w โŠ— s โˆˆ ๐’ž ; then supp โก ( c ) = W s . Let finally r = R โข ( c ) . We see that r L i , ๐ฑ โ‰  0 if and only if ๐ฑ โˆˆ W s . Hence we get

wt ( r ) = | { L โˆˆ ๐’ฌ , r L โ‰  0 } | = | โ‹ƒ i = 1 s { L i , ๐ฑ , ๐ฑ โˆˆ W s } | = s โ‹… d ๐’œ s - 1

since each line L i , ๐ฑ is counted d ๐’œ times when ๐ฑ runs over W s . โˆŽ

Proposition 5.3.

Let ฮด > 0 , and let A be an [ โ„“ , โ„“ โข ( 1 - ฮด ) + 1 , โ„“ โข ฮด ] q MDS code. Define C = A โŠ— s and ( Q , V ) as above. If every ฮฆ w is sufficiently uniform, then the PoR scheme associated to C and ( Q , V ) is ( ฮต , ฯ„ ) -sound for ฯ„ = O โข ( 1 ( ฮด โข โ„“ ) s โข s ) and every ฮต < ฮต 0 , where ฮต 0 = ( 1 + O โข ( q - ฮด โข โ„“ + 1 ) ) โข ฮด s when โ„“ โ†’ โˆž .

Proof.

First, the relative distance of R โข ( ๐’ž ) is ฮด s according to Lemma 5.2. Then the random variables { X u } u โˆˆ ๐’Ÿ are pairwise uncorrelated because the inequality

max u โ‰  v โˆˆ ๐’ฌ 2 โก | u โˆฉ v | = 1 < โ„“ โข ( 1 - ฮด ) + 2 = min u โˆˆ ๐’ฌ โก d min โข ( ( ๐’ž | u ) โŠฅ )

allows us to apply Proposition 3.18. Besides, if every ฮฆ w is sufficiently uniform, then the bias ฮฑ satisfies ฮฑ = ๐’ช โข ( q - ฮด โข โ„“ + 1 ) and hence 1 - ฮฑ 1 + ฮฑ = 1 + ๐’ช โข ( q - ฮด โข โ„“ + 1 ) . Therefore, we can use Theorem 3.13, and we get the desired result. โˆŽ

Parameters

We mainly focus on the download communication complexity in the verification step and on the server storage overhead since these are the most crucial parameters which depend on the family of codes ๐’ž we use. Besides, we consider that it is more relevant to analyse the ratio between these quantities and the file size than their absolute values.

Here, for an initial file of size | F | = ( ( 1 - ฮด ) โข q + 1 ) s โข log 2 โก q bits, we get

  • โ€ข

    a redundancy rate

    n โข log 2 โก q | F | = ( q ( 1 - ฮด ) โข q + 1 ) s โ‰ค 1 ( 1 - ฮด ) s ,

  • โ€ข

    a communication complexity rate

    โ„“ โข log 2 โก q | F | = q ( ( 1 - ฮด ) โข q + 1 ) s โ‰ค 1 ( 1 - ฮด ) s โข q 1 - s .

Example 5.4.

In Table 3, we present various parameters of ๐–ฏ๐—ˆ๐–ฑ instances admitting 0.10 โ‰ค ฮต 0 โ‰ค 0.16 , for files of size approaching 10 4 , 10 6 and 10 9 bits. Here ๐’œ is a [ q , ( 1 - ฮด ) โข q + 1 , ฮด โข q ] q MDS code (e.g., a Reedโ€“Solomon code), and ๐’ž = ๐’œ โŠ— s .

Table 3

Parameters of ๐–ฏ๐—ˆ๐–ฑ instances admitting 0.10 โ‰ค ฮต 0 โ‰ค 0.16 .

q ฮด โข q s File size (bits) Comm. rate Redundancy rate ฮต 0
16 10 4 9,604 6.664 ร— 10 - 3 27.3 0.153
25 13 3 10,985 1.138 ร— 10 - 2 7.112 0.141
64 24 2 10,086 3.807 ร— 10 - 2 2.437 0.141
32 21 5 1,244,160 1.286 ร— 10 - 4 134.8 0.122
47 28 4 960,000 2.938 ร— 10 - 4 30.5 0.126
101 47 3 1,164,625 6.071 ร— 10 - 4 6.193 0.101
512 180 2 998,001 4.617 ร— 10 - 3 2.364 0.124
128 85 5 1,154,413,568 7.762 ร— 10 - 7 208.3 0.129
256 150 4 1,048,636,808 1.953 ร— 10 - 6 32.77 0.118
1,024 550 3 1,071,718,750 9.555 ร— 10 - 6 10.02 0.155
12,167 3,900 2 957,037,536 1.78 ร— 10 - 4 2.166 0.103
16,384 5,500 2 1,658,765,150 1.383 ร— 10 - 4 2.266 0.113

The previous example shows that, while the communication rate is reasonable for these ๐–ฏ๐—ˆ๐–ฑ instances over large files, the storage needs remain large.

5.2 Reedโ€“Muller and related codes

Low-degree Reedโ€“Muller codes are known to admit many distinct low-weight parity-check equations, whose supports correspond to affine subspaces of the ambient space. Therefore, they seem naturally adapted to our construction. Let us first consider the plane (or bivariate) Reedโ€“Muller code case.

5.2.1 The plane Reedโ€“Muller code RM q โข ( 2 , q - 2 )

Let ๐’ž be the Reedโ€“Muller code

๐’ž = RM q ( 2 , q - 2 ) : = { ( f ( x , y ) ) ( x , y ) โˆˆ ๐”ฝ q 2 , f โˆˆ ๐”ฝ q [ X , Y ] , deg f โ‰ค q - 2 } .

It is well known that ๐’ž has length q 2 and dimension ( q - 1 ) โข ( q - 2 ) / 2 . Besides, for every line

L = { ๐ฑ = ( a t + b , c t + d ) , t โˆˆ ๐”ฝ q } โŠ‚ ๐”ฝ q 2

and every c โˆˆ ๐’ž , we can check that โˆ‘ ๐ฑ โˆˆ L c ๐ฑ = 0 . Indeed, let f โˆˆ ๐”ฝ q โข [ X , Y ] , deg โก f = a โ‰ค q - 2 . The restriction of f on an affine line L can be interpolated as a univariate polynomial f | L of degree at most a. Our claim follows since โˆ‘ z โˆˆ ๐”ฝ q z i = 0 for every i โ‰ค q - 2 .

Therefore, we can define ๐’ฌ as the set of affine lines L of ๐”ฝ q 2 and V โข ( L , r ) = โˆ‘ j = 1 โ„“ r j โˆˆ ๐”ฝ q . From the previous discussion, we see that ( ๐’ฌ , V ) is a verification structure for ๐’ž . Also notice there are q โข ( q + 1 ) distinct affine lines in ๐”ฝ q 2 ; hence N = q โข ( q + 1 ) .

Lemma 5.5.

Let C = RM q โข ( 2 , q - 2 ) , equipped with its verification structure defined as above. Then the response code R โข ( C ) has minimum distance q 2 + 2 .

Proof.

Any non-zero codeword c โˆˆ ๐’ž consists in the evaluation of a non-zero polynomial f โข ( X , Y ) โˆˆ ๐”ฝ q โข [ X , Y ] of degree at most q - 2 . Denote by L 1 , โ€ฆ , L a โŠ‚ ๐”ฝ q 2 the affine lines on which f vanishes, i.e., f โข ( P ) = 0 for every P โˆˆ L i , 1 โ‰ค i โ‰ค a . We claim that a โ‰ค q - 2 . Indeed, since f has total degree less than q - 1 , it also vanishes on closed lines L 1 ยฏ , โ€ฆ , L a ยฏ , considered as affine lines in ๐”ฝ q ยฏ 2 , where ๐”ฝ q ยฏ denotes the algebraic closure of ๐”ฝ q . Denote by g i โˆˆ ๐”ฝ q โข [ X , Y ] the monic polynomial of degree 1 which defines L i ยฏ . From Hilbertโ€™s Nullstellensatz, there exists r > 0 such that ( โˆ i = 1 a g i ) โˆฃ f r . Since the g i โ€™s have degree 1 and are distinct, we get a โ‰ค deg โก f โ‰ค q - 2 . Hence the affine lines different from L 1 , โ€ฆ , L a correspond to non-zero coordinates of R โข ( c ) . There are q โข ( q + 1 ) - a โ‰ฅ q 2 + 2 such lines, so d min โข ( R โข ( ๐’ž ) ) โ‰ฅ q 2 + 2 .

Now we claim there exists a word r โˆˆ R โข ( ๐’ž ) of weight N - q + 2 = q 2 + 2 . Let L ( 0 ) and L ( 1 ) be two distinct parallel affine lines, respectively defined by X = 0 and X = 1 . We build the word c which is -1 on coordinates corresponding to points in L ( 0 ) , 1 on those corresponding to points in L ( 1 ) and 0 elsewhere. One can check that c โˆˆ ๐’ž ; indeed, c corresponds to the evaluation of โˆ z โˆˆ ๐”ฝ q โˆ– { 0 , 1 } ( z - X ) . Now, if we want to compute wt โก ( R โข ( c ) ) , we only need to count the number of lines which do not intersect L ( 0 ) nor L ( 1 ) . Clearly, there are only q - 2 such lines. Hence wt โก ( R โข ( c ) ) = q โข ( q + 1 ) - ( q - 2 ) , and this concludes the proof. โˆŽ

Proposition 5.6.

Let C = RM โข ( 2 , q - 2 ) , and let ( Q , V ) be its associated verification structure. If every ฮฆ w is sufficiently uniform, then the PoR scheme associated to C and ( Q , V ) is ( ฮต , ฯ„ ) -sound for ฮต = 1 - o โข ( 1 ) and ฯ„ = O โข ( 1 ( 1 - ฮต ) โข q 2 ) , when q โ†’ โˆž .

Proof.

One can check that the random variables { X u } u โˆˆ ๐’Ÿ are pairwise uncorrelated since

max u โ‰  v โˆˆ ๐’ฌ 2 โก | u โˆฉ v | = 1 < โ„“ โข ( 1 - ฮด ) + 2 = min u โˆˆ ๐’ฌ โก d min โข ( ( ๐’ž | u ) โŠฅ ) .

Besides, the relative distance of R โข ( ๐’ž ) is q 2 + 2 q โข ( q + 1 ) โ†’ 1 according to Lemma 5.5. If every ฮฆ w is sufficiently uniform, the bias ฮฑ satisfies ฮฑ โˆˆ ๐’ช โข ( 1 / q ) and hence 1 - ฮฑ 1 + ฮฑ = 1 + ๐’ช โข ( 1 / q ) . Therefore, we can use Theorem 3.13, and we get the desired result. โˆŽ

Parameters

For an initial file of size | F | = 1 2 โข ( q - 1 ) โข ( q - 2 ) โข log 2 โก q bits, we get

  • โ€ข

    a redundancy rate

    q 2 โข log 2 โก q | F | = 2 ( 1 - 1 / q ) โข ( 1 - 2 / q ) โ†’ 2 ,

  • โ€ข

    a communication complexity rate

    q โข log 2 โก q | F | = 2 q โข 1 ( 1 - 1 / q ) โข ( 1 - 2 / q ) = ๐’ช โข ( 1 / q ) .

5.2.2 Storage improvements via lifted codes

The redundancy rate of Reedโ€“Muller codes presented above stays stuck above 2. Affine lifted codes, introduced by Guo, Kopparty and Sudan [5], allow to break this barrier while keeping the same verification structure. Generically, they are defined as follows:

Lift ( m , d ) : = { ( f ( ๐ ) ) ๐ โˆˆ ๐”ฝ q m , f โˆˆ ๐”ฝ q [ X 1 , โ€ฆ , X m ] for every affine line L โŠ‚ ๐”ฝ q m , ( f ( ๐ ) ) ๐ โˆˆ L โˆˆ RS q ( d + 1 ) } .

We refer to [5] for more details about the construction. Here we focus on Lift โข ( 2 , q - 2 ) since it can be compared to RM โข ( 2 , q - 2 ) . Indeed, one sees that

(5.1) RM โข ( 2 , q - 2 ) โŠ† Lift โข ( 2 , q - 2 ) ,

and equation (5.1) turns into a proper inclusion as long as q is not a prime. Besides, by definition of lifted codes, Lift โข ( 2 ,