arXiv subjects

Philippe Gaborit

Publications and source records attributed to Philippe Gaborit.

At least 37 records · Page 2Linked to original sources

arXiv2022

LRPC codes with multiple syndromes: near ideal-size KEMs without ideals

We introduce a new rank-based key encapsulation mechanism (KEM) with public key and ciphertext sizes around 3.5 Kbytes each, for 128 bits of security, without using ideal structures. Such structures allow to compress objects, but give reductions to specific problems whose security is potentially weaker than for unstructured problems. To the best of our knowledge, our scheme improves in size all the existing unstructured post-quantum lattice or code-based algorithms such as FrodoKEM or Classic McEliece. Our technique, whose efficiency relies on properties of rank metric, is to build upon existing Low Rank Parity Check (LRPC) code-based KEMs and to send multiple syndromes in one ciphertext, allowing to reduce the parameters and still obtain an acceptable decoding failure rate. Our system relies on the hardness of the Rank Support Learning problem, a well-known variant of the Rank Syndrome Decoding problem. The gain on parameters is enough to significantly close the gap between ideal and non-ideal constructions. It enables to choose an error weight close to the rank Gilbert-Varshamov bound, which is a relatively harder zone for algebraic attacks. We also give a version of our KEM that keeps an ideal structure and permits to roughly divide the bandwidth by two compared to previous versions of LRPC KEMs submitted to the NIST with a Decoding Failure Rate (DFR) of $2^{-128}$.

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arXiv2022

Quasi-Cyclic Stern Proof of Knowledge

The ongoing NIST standardization process has shown that Proof of Knowledge (PoK) based signatures have become an important type of possible post-quantum signatures. Regarding code-based cryptography, the original approach for PoK based signatures is the Stern protocol which allows to prove the knowledge of a small weight vector solving a given instance of the Syndrome Decoding (SD) problem over F2. It features a soundness error equal to 2/3. This protocol was improved a few years later by Véron who proposed a variation of the scheme based on the General Syndrome Decoding (GSD) problem which leads to better results in term of communication. A few years later, the AGS protocol introduced a variation of the Véron protocol based on Quasi-Cyclic (QC) matrices. The AGS protocol permits to obtain an asymptotic soundness error of 1/2 and an improvement in term of communications. In the present paper, we introduce the Quasi-Cyclic Stern PoK which constitutes an adaptation of the AGS scheme in a SD context, as well as several new optimizations for code-based PoK. Our main optimization on the size of the signature can't be applied to GSD based protocols such as AGS and therefore motivated the design of our new protocol. In addition, we also provide a special soundness proof that is compatible with the use of the Fiat-Shamir transform for 5-round protocols. This approach is valid for our protocol but also for the AGS protocol which was lacking such a proof. We compare our results with existing signatures including the recent code-based signatures based on PoK leveraging the MPC in the head paradigm. In practice, our new protocol is as fast as AGS while reducing its associated signature length by 20%. As a consequence, it constitutes an interesting trade-off between signature length and execution time for the design of a code-based signature relying only on the difficulty of the SD problem.

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arXiv2022

Code-based Signatures from New Proofs of Knowledge for the Syndrome Decoding Problem

In this paper, we study code-based signatures constructed from Proof of Knowledge (PoK). This line of work can be traced back to Stern who introduces the first efficient PoK for the syndrome decoding problem in 1993. Afterward, different variations were proposed in order to reduce signature's size. In practice, obtaining a smaller signature size relies on the interaction of two main considerations: (i) the underlying protocol and its soundness error and (ii) the type of optimizations which are compatible with a given protocol. Over the years, different variations were proposed to improve the Stern scheme such as the Veron scheme (with public key a noisy codeword rather than a syndrome), the AGS scheme which is a 5-pass protocol with cheating probability asymptotically equal to 1/2 and more recently the FJR approach which permits to decrease the cheating probability to 1/N but induces a performance overhead. Overall the length of the signature depends on a trade-off between: the scheme in itself, the possible optimizations and the cost of the implementation. The recent approaches which increase the cost of the implementation opens the door to many different type of trade-offs. In this paper we propose three new schemes and different trade-offs, which are all interesting in themselves, since depending on potential future optimizations a scheme may eventually become more efficient than another. All the schemes we propose use a trusted helper: a first scheme permits to get a 1/2 cheating probability, a second scheme permits to decrease the cheating probability in 1/N but with a different approach than the recent FJR scheme and at last a third scheme propose a Veron-like adaptation of the FJR scheme in which the public key is a noisy codeword rather than a syndrome. We provide an extensive comparison table which lists various trade-offs between our schemes and previous ones.

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arXiv2021

On the hardness of code equivalence problems in rank metric

In the recent years, the notion of rank metric in the context of coding theory has known many interesting developments in terms of applications such as space time coding, network coding or public key cryptography. These applications raised the interest of the community for theoretical properties of this type of codes, such as the hardness of decoding in rank metric. Among classical problems associated to codes for a given metric, the notion of code equivalence (to decide if two codes are isometric) has always been of the greatest interest, for its cryptographic applications or its deep connexions to the graph isomorphism problem. In this article, we discuss the hardness of the code equivalence problem in rank metric for $\mathbb{F}_{q^m}$-linear and general rank metric codes. In the $\mathbb{F}_{q^m}$-linear case, we reduce the underlying problem to another one called {\em Matrix Codes Right Equivalence Problem}. We prove the latter problem to be either in $\mathcal{P}$ or in $\mathcal{ZPP}$ depending of the ground field size. This is obtained by designing an algorithm whose principal routines are linear algebra and factoring polynomials over finite fields. It turns out that the most difficult instances involve codes with non trivial {\em stabilizer algebras}. The resolution of the latter case will involve tools related to finite dimensional algebras and Wedderburn--Artin theory. It is interesting to note that 30 years ago, an important trend in theoretical computer science consisted to design algorithms making effective major results of this theory. These algorithmic results turn out to be particularly useful in the present article. Finally, for general matrix codes, we prove that the equivalence problem (both left and right) is at least as hard as the well--studied {\em Monomial Equivalence Problem} for codes endowed with the Hamming metric.

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arXiv2021

Improvements of Algebraic Attacks for solving the Rank Decoding and MinRank problems

Rank Decoding (RD) is the main underlying problem in rank-based cryptography. Based on this problem and quasi-cyclic versions of it, very efficient schemes have been proposed recently, such as those in the ROLLO and RQC submissions, which have reached the second round of the NIST Post-Quantum competition. Two main approaches have been studied to solve RD: combinatorial ones and algebraic ones. While the former has been studied extensively, a better understanding of the latter was recently obtained by Bardet et al. (EUROCRYPT20) where it appeared that algebraic attacks can often be more efficient than combinatorial ones for cryptographic parameters. This paper gives substantial improvements upon this attack in terms both of complexity and of the assumptions required by the cryptanalysis. We present attacks for ROLLO-I-128, 192, and 256 with bit complexity respectively in 70, 86, and 158, to be compared to 117, 144, and 197 for the aforementionned previous attack. Moreover, unlike this previous attack, ours does not need generic Gröbner basis algorithms since it only requires to solve a linear system. For a case called overdetermined, this modeling allows us to avoid Gröbner basis computations by going directly to solving a linear system. For the other case, called underdetermined, we also improve the results from the previous attack by combining the Ourivski-Johansson modeling together with a new modeling for a generic MinRank instance; the latter modeling allows us to refine the analysis of MinRank's complexity given in the paper by Verbel et al. (PQC19). Finally, since the proposed parameters of ROLLO and RQC are completely broken by our new attack, we give examples of new parameters for ROLLO and RQC that make them resistant to our attacks. These new parameters show that these systems remain attractive, with a loss of only about 50\% in terms of key size for ROLLO-I.

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arXiv2020

HQC-RMRS, an instantiation of the HQC encryption framework with a more efficient auxiliary error-correcting code

The HQC encryption framework is a general code-based encryption scheme for which decryption returns a noisy version of the plaintext. Any instantiation of the scheme will therefore use an error-correcting procedure relying on a fixed auxiliary code. Unlike the McEliece encryption framework whose security is directly related to how well one can hide the structure of an error-correcting code, the security reduction of the HQC encryption framework is independent of the nature of the auxiliary decoding procedure which is publicly available. What is expected from it is that the decoding algorithm is both efficient and has a decoding failure rate which can be easily modelized and analyzed. The original error-correction procedure proposed for the HQC framework was to use tensor products of BCH codes and repetition codes. In this paper we consider another code family for removing the error vector deriving from the general framework: the concatenation of Reed-Muller and Reed-Solomon codes. We denote this instantiation of the HQC framework by HQC-RMRS. These codes yield better decoding results than the BCH and repetition codes: overall we gain roughly 17\% in the size of the key and the ciphertext, while keeping a simple modelization of the decoding error rate. The paper also presents a simplified and more precise analysis of the distribution of the error vector output by the HQC protocol.

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arXiv2020

An Algebraic Attack on Rank Metric Code-Based Cryptosystems

The Rank metric decoding problem is the main problem considered in cryptography based on codes in the rank metric. Very efficient schemes based on this problem or quasi-cyclic versions of it have been proposed recently, such as those in the submissions ROLLO and RQC currently at the second round of the NIST Post-Quantum Cryptography Standardization Process. While combinatorial attacks on this problem have been extensively studied and seem now well understood, the situation is not as satisfactory for algebraic attacks, for which previous work essentially suggested that they were ineffective for cryptographic parameters. In this paper, starting from Ourivski and Johansson's algebraic modelling of the problem into a system of polynomial equations, we show how to augment this system with easily computed equations so that the augmented system is solved much faster via Groebner bases. This happens because the augmented system has solving degree $r$, $r+1$ or $r+2$ depending on the parameters, where $r$ is the rank weight, which we show by extending results from Verbel et al. (PQCrypto 2019) on systems arising from the MinRank problem; with target rank $r$, Verbel et al. lower the solving degree to $r+2$, and even less for some favorable instances that they call superdetermined. We give complexity bounds for this approach as well as practical timings of an implementation using Magma. This improves upon the previously known complexity estimates for both Groebner basis and (non-quantum) combinatorial approaches, and for example leads to an attack in 200 bits on ROLLO-I-256 whose claimed security was 256 bits.

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arXiv2019

Improved Veron Identification and Signature Schemes in the Rank Metric

It is notably challenging to design an efficient and secure signature scheme based on error-correcting codes. An approach to build such signature schemes is to derive it from an identification protocol through the Fiat-Shamir transform. All such protocols based on codes must be run several rounds, since each run of the protocol allows a cheating probability of either 2/3 or 1/2. The resulting signature size is proportional to the number of rounds, thus making the 1/2 cheating probability version more attractive. We present a signature scheme based on double circulant codes in the rank metric, derived from an identification protocol with cheating probability of 2/3. We reduced this probability to 1/2 to obtain the smallest signature among signature schemes based on the Fiat-Shamir paradigm, around 22 KBytes for 128 bit security level. Furthermore, among all code-based signature schemes, our proposal has the lowest value of signature plus public key size, and the smallest secret and public key sizes. We provide a security proof in the Random Oracle Model, implementation performances, and a comparison with the parameters of the most important code-based signature schemes.

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arXiv2019

Low Rank Parity Check Codes: New Decoding Algorithms and Applications to Cryptography

We introduce a new family of rank metric codes: Low Rank Parity Check codes (LRPC), for which we propose an efficient probabilistic decoding algorithm. This family of codes can be seen as the equivalent of classical LDPC codes for the rank metric. We then use these codes to design cryptosystems à la McEliece: more precisely we propose two schemes for key encapsulation mechanism (KEM) and public key encryption (PKE). Unlike rank metric codes used in previous encryption algorithms -notably Gabidulin codes - LRPC codes have a very weak algebraic structure. Our cryptosystems can be seen as an equivalent of the NTRU cryptosystem (and also to the more recent MDPC \cite{MTSB12} cryptosystem) in a rank metric context. The present paper is an extended version of the article introducing LRPC codes, with important new contributions. We have improved the decoder thanks to a new approach which allows for decoding of errors of higher rank weight, namely up to $\frac{2}{3}(n-k)$ when the previous decoding algorithm only decodes up to $\frac{n-k}{2}$ errors. Our codes therefore outperform the classical Gabidulin code decoder which deals with weights up to $\frac{n-k}{2}$. This comes at the expense of probabilistic decoding, but the decoding error probability can be made arbitrarily small. The new approach can also be used to decrease the decoding error probability of previous schemes, which is especially useful for cryptography. Finally, we introduce ideal rank codes, which generalize double-circulant rank codes and allow us to avoid known structural attacks based on folding. To conclude, we propose different parameter sizes for our schemes and we obtain a public key of 3337 bits for key exchange and 5893 bits for public key encryption, both for 128 bits of security.

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arXiv2017

RankSign: an efficient signature algorithm based on the rank metric

In this paper we propose a new approach to code-based signatures that makes use in particular of rank metric codes. When the classical approach consists in finding the unique preimage of a syndrome through a decoding algorithm, we propose to introduce the notion of mixed decoding of erasures and errors for building signature schemes. In that case the difficult problem becomes, as is the case in lattice-based cryptography, finding a preimage of weight above the Gilbert-Varshamov bound (case where many solutions occur) rather than finding a unique preimage of weight below the Gilbert-Varshamov bound. The paper describes RankSign: a new signature algorithm for the rank metric based on a new mixed algorithm for decoding erasures and errors for the recently introduced Low Rank Parity Check (LRPC) codes. We explain how it is possible (depending on choices of parameters) to obtain a full decoding algorithm which is able to find a preimage of reasonable rank weight for any random syndrome with a very strong probability. We study the semantic security of our signature algorithm and show how it is possible to reduce the unforgeability to direct attacks on the public matrix, so that no information leaks through signatures. Finally, we give several examples of parameters for our scheme, some of which with public key of size $11,520$ bits and signature of size $1728$ bits. Moreover the scheme can be very fast for small base fields.

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arXiv2017

Polynomial-Time Key Recovery Attack on the Faure-Loidreau Scheme based on Gabidulin Codes

Encryption schemes based on the rank metric lead to small public key sizes of order of few thousands bytes which represents a very attractive feature compared to Hamming metric-based encryption schemes where public key sizes are of order of hundreds of thousands bytes even with additional structures like the cyclicity. The main tool for building public key encryption schemes in rank metric is the McEliece encryption setting used with the family of Gabidulin codes. Since the original scheme proposed in 1991 by Gabidulin, Paramonov and Tretjakov, many systems have been proposed based on different masking techniques for Gabidulin codes. Nevertheless, over the years all these systems were attacked essentially by the use of an attack proposed by Overbeck. In 2005 Faure and Loidreau designed a rank-metric encryption scheme which was not in the McEliece setting. The scheme is very efficient, with small public keys of size a few kiloBytes and with security closely related to the linearized polynomial reconstruction problem which corresponds to the decoding problem of Gabidulin codes. The structure of the scheme differs considerably from the classical McEliece setting and until our work, the scheme had never been attacked. We show in this article that this scheme like other schemes based on Gabidulin codes, is also vulnerable to a polynomial-time attack that recovers the private key by applying Overbeck's attack on an appropriate public code. As an example we break concrete proposed $80$ bits security parameters in a few seconds.

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arXiv2016

Efficient Encryption from Random Quasi-Cyclic Codes

We propose a framework for constructing efficient code-based encryption schemes from codes that do not hide any structure in their public matrix. The framework is in the spirit of the schemes first proposed by Alekhnovich in 2003 and based on the difficulty of decoding random linear codes from random errors of low weight. We depart somewhat from Aleknovich's approach and propose an encryption scheme based on the difficulty of decoding random quasi-cyclic codes. We propose two new cryptosystems instantiated within our framework: the Hamming Quasi-Cyclic cryptosystem (HQC), based on the Hamming metric, and the Rank Quasi-Cyclic cryptosystem (RQC), based on the rank metric. We give a security proof, which reduces the IND-CPA security of our systems to a decisional version of the well known problem of decoding random families of quasi-cyclic codes for the Hamming and rank metrics (the respective QCSD and RQCSD problems). We also provide an analysis of the decryption failure probability of our scheme in the Hamming metric case: for the rank metric there is no decryption failure. Our schemes benefit from a very fast decryption algorithm together with small key sizes of only a few thousand bits. The cryptosystems are very efficient for low encryption rates and are very well suited to key exchange and authentication. Asymptotically, for λthe security parameter, the public key sizes are respectively in $O(λ^{2})$ for HQC and in $O(λ^{4/3})$ for RQC. Practical parameter compares well to systems based on ring-LPN or the recent MDPC system.

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arXiv2016

RankSynd a PRNG Based on Rank Metric

In this paper, we consider a pseudo-random generator based on the difficulty of the syndrome decoding problem for rank metric codes. We also study the resistance of this problem against a quantum computer. Our results show that with rank metric it is possible to obtain fast PRNG with small public data, without considering additional structure for public matrices like quasi-cyclicity for Hamming distance.

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arXiv2014

Distinguisher-Based Attacks on Public-Key Cryptosystems Using Reed-Solomon Codes

Because of their interesting algebraic properties, several authors promote the use of generalized Reed-Solomon codes in cryptography. Niederreiter was the first to suggest an instantiation of his cryptosystem with them but Sidelnikov and Shestakov showed that this choice is insecure. Wieschebrink proposed a variant of the McEliece cryptosystem which consists in concatenating a few random columns to a generator matrix of a secretly chosen generalized Reed-Solomon code. More recently, new schemes appeared which are the homomorphic encryption scheme proposed by Bogdanov and Lee, and a variation of the McEliece cryptosystem proposed by Baldi et \textit{al.} which hides the generalized Reed-Solomon code by means of matrices of very low rank. In this work, we show how to mount key-recovery attacks against these public-key encryption schemes. We use the concept of distinguisher which aims at detecting a behavior different from the one that one would expect from a random code. All the distinguishers we have built are based on the notion of component-wise product of codes. It results in a powerful tool that is able to recover the secret structure of codes when they are derived from generalized Reed-Solomon codes. Lastly, we give an alternative to Sidelnikov and Shestakov attack by building a filtration which enables to completely recover the support and the non-zero scalars defining the secret generalized Reed-Solomon code.

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arXiv2013

On the complexity of the Rank Syndrome Decoding problem

In this paper we propose two new generic attacks on the Rank Syndrome Decoding (RSD) problem Let $C$ be a random $[n,k]$ rank code over $GF(q^m)$ and let $y=x+e$ be a received word such that $x \in C$ and the $Rank(e)=r$. The first attack is combinatorial and permits to recover an error $e$ of rank weight $r$ in $min(O((n-k)^3m^3q^{r\lfloor\frac{km}{n}\rfloor}, O((n-k)^3m^3q^{(r-1)\lfloor\frac{(k+1)m}{n}\rfloor}))$ operations on $GF(q)$. This attack dramatically improves on previous attack by introducing the length $n$ of the code in the exponent of the complexity, which was not the case in previous generic attacks. which can be considered The second attack is based on a algebraic attacks: based on the theory of $q$-polynomials introduced by Ore we propose a new algebraic setting for the RSD problem that permits to consider equations and unknowns in the extension field $GF(q^m)$ rather than in $GF(q)$ as it is usually the case. We consider two approaches to solve the problem in this new setting. Linearization technics show that if $n \ge (k+1)(r+1)-1$ the RSD problem can be solved in polynomial time, more generally we prove that if $\lceil \frac{(r+1)(k+1)-(n+1)}{r} \rceil \le k$, the problem can be solved with an average complexity $O(r^3k^3q^{r\lceil \frac{(r+1)(k+1)-(n+1)}{r} \rceil})$. We also consider solving with \grob bases for which which we discuss theoretical complexity, we also consider consider hybrid solving with \grob bases on practical parameters. As an example of application we use our new attacks on all proposed recent cryptosystems which reparation the GPT cryptosystem, we break all examples of published proposed parameters, some parameters are broken in less than 1 s in certain cases.

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arXiv2012

A new class of codes for Boolean masking of cryptographic computations

We introduce a new class of rate one-half binary codes: {\bf complementary information set codes.} A binary linear code of length $2n$ and dimension $n$ is called a complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune Boolean functions of use in the security of hardware implementations of cryptographic primitives. Such codes permit to improve the cost of masking cryptographic algorithms against side channel attacks. In this paper we investigate this new class of codes: we give optimal or best known CIS codes of length $<132.$ We derive general constructions based on cyclic codes and on double circulant codes. We derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all be classified in small lengths $\le 12$ by the building up construction. Some nonlinear permutations are constructed by using $\Z_4$-codes, based on the notion of dual distance of an unrestricted code.

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arXiv2011

A new zero-knowledge code based identification scheme with reduced communication

In this paper we present a new 5-pass identification scheme with asymptotic cheating probability 1/2 based on the syndrome decoding problem. Our protocol is related to the Stern identification scheme but has a reduced communication cost compared to previous code-based zero-knowledge schemes, moreover our scheme permits to obtain a very low size of public key and secret key. The contribution of this paper is twofold, first we propose a variation on the Stern authentication scheme which permits to decrease asymptotically the cheating probability to 1/2 rather than 2/3 (and very close to 1/2 in practice) but with less communication. Our solution is based on deriving new challenges from the secret key through cyclic shifts of the initial public key syndrome; a new proof of soundness for this case is given Secondly we propose a new way to deal with hashed commitments in zero-knowledge schemes based on Stern's scheme, so that in terms of communication, on the average, only one hash value is sent rather than two or three. Overall our new scheme has the good features of having a zero-knowledge security proof based on well known hard problem of coding theory, a small size of secret and public key (a few hundred bits), a small calculation complexity, for an overall communication cost of 19kb for authentication (for a $2^{16}$ security) and a signature of size of 93kb (11.5kB) (for security $2^{80}$), an improvement of 40% compared to previous schemes based on coding theory.

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arXiv2011

Classification of extremal and $s$-extremal binary self-dual codes of length 38

In this paper we classify all extremal and $s$-extremal binary self-dual codes of length 38. There are exactly 2744 extremal $[38,19,8]$ self-dual codes, two $s$-extremal $[38,19,6]$ codes, and 1730 $s$-extremal $[38,19,8]$ codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self-dual codes of length 36, and from a generalization of this recursive algorithm for the shadow. The classification of $s$-extremal $[38,19,6]$ codes permits to achieve the classification of all $s$-extremal codes with d=6.

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