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Section: New Results

Solving Systems in Finite Fields, Applications in Cryptology and Algebraic Number Theory

On the Complexity of Solving Quadratic Boolean Systems

A fundamental problem in computer science is to find all the common zeroes of $m$ quadratic polynomials in $n$ unknowns over ${𝔽}_2$. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity bound was reached by an exhaustive search in $4{log}_{2}n{2}^n$ operations. We give an algorithm that reduces the problem to a combination of exhaustive search and sparse linear algebra. This algorithm has several variants depending on the method used for the linear algebra step. Under precise algebraic assumptions on the input system, we show in [4] , that the deterministic variant of our algorithm has complexity bounded by $O\left(2^{0.841n}\right)$ when $m=n$, while a probabilistic variant of the Las Vegas type has expected complexity $O\left(2^{0.792n}\right)$. Experiments on random systems show that the algebraic assumptions are satisfied with probability very close to 1. We also give a rough estimate for the actual threshold between our method and exhaustive search, which is as low as 200, and thus very relevant for cryptographic applications.

Decomposing polynomial sets into simple sets over finite fields: The positive-dimensional case

Our work in [19] presents an algorithm for decomposing any positive-dimensional polynomial set into simple sets over an arbitrary finite field. The algorithm is based on some relationship established between simple sets and radical ideals, reducing the decomposition problem to the problem of computing the radicals of certain ideals. In addition to direct application of the algorithms of Matsumoto and Kemper, the algorithm of Fortuna and others is optimized and improved for the computation of radicals of special ideals. Preliminary experiments with an implementation of the algorithm in Maple and Singular are carried out to show the effectiveness and efficiency of the algorithm.

Using Symmetries in the Index Calculus for Elliptic Curves Discrete Logarithm

In 2004, an algorithm is introduced to solve the DLP for elliptic curves defined over a non prime finite field ${𝔽}_{q^n}$. One of the main steps of this algorithm requires decomposing points of the curve $E\left({𝔽}_{q^n}\right)$ with respect to a factor base, this problem is denoted PDP. In [11] , we apply this algorithm to the case of Edwards curves, the well-known family of elliptic curves that allow faster arithmetic as shown by Bernstein and Lange. More precisely, we show how to take advantage of some symmetries of twisted Edwards and twisted Jacobi intersections curves to gain an exponential factor $2^{\omega (n-1)}$ to solve the corresponding PDP where $\omega$ is the exponent in the complexity of multiplying two dense matrices. Practical experiments supporting the theoretical result are also given. For instance, the complexity of solving the ECDLP for twisted Edwards curves defined over ${𝔽}_{q^5}$, with $q\approx 2^{64}$, is supposed to be $\sim$ $2^{160}$ operations in $E\left({𝔽}_{q^5}\right)$ using generic algorithms compared to $2^{130}$ operations (multiplication of two 32-bits words) with our method. For these parameters the PDP is intractable with the original algorithm. The main tool to achieve these results relies on the use of the symmetries and the quasi-homogeneous structure induced by these symmetries during the polynomial system solving step. Also, we use a recent work on a new algorithm for the change of ordering of Gröbner basis which provides a better heuristic complexity of the total solving process.

A Distinguisher for High Rate McEliece Cryptosystems [12]

The Goppa Code Distinguishing (GD) problem consists in distinguishing the matrix of a Goppa code from a random matrix. The hardness of this problem is an assumption to prove the security of code-based cryptographic primitives such as McEliece's cryptosystem. Up to now, it is widely believed that the GD problem is a hard decision problem. We present in [12] the first method allowing to distinguish alternant and Goppa codes over any field. Our technique can solve the GD problem in polynomial-time provided that the codes have sufficiently large rates. The key ingredient is an algebraic characterization of the key-recovery problem. The idea is to consider the rank of a linear system which is obtained by linearizing a particular polynomial system describing a key-recovery attack. Experimentally it appears that this dimension depends on the type of code. Explicit formulas derived from extensive experimentations for the rank are provided for "generic" random, alternant, and Goppa codes over any alphabet. Finally, we give theoretical explanations of these formulas in the case of random codes, alternant codes over any field of characteristic two and binary Goppa codes.

Cryptanalysis of HFE, multi-HFE and variants for odd and even characteristic [6]

We investigate in this paper the security of HFE and Multi-HFE schemes as well as their minus and embedding variants. Multi-HFE is a generalization of the well-known HFE schemes. The idea is to use a multivariate quadratic system instead of a univariate polynomial in HFE over an extension field as a private key. According to the authors, this should make the classical direct algebraic (message-recovery) attack proposed by Faugère and Joux on HFE no longer efficient against Multi-HFE. We consider here the hardness of the key-recovery in Multi-HFE and its variants, but also in HFE (both for odd and even characteristic). We first improve and generalize the basic key recovery proposed by Kipnis and Shamir on HFE. To do so, we express this attack as matrix/vector operations. In one hand, this permits to improve the basic Kipnis-Shamir (KS) attack on HFE. On the other hand, this allows to generalize the attack on Multi-HFE. Due to its structure, we prove that a Multi-HFE scheme has much more equivalent keys than a basic HFE. This induces a structural weakness which can be exploited to adapt the KS attack against classical modifiers of multivariate schemes such as minus and embedding. Along the way, we discovered that the KS attack as initially described cannot be applied against HFE in characteristic 2. We have then strongly revised KS in characteristic 2 to make it work. In all cases, the cost of our attacks is related to the complexity of solving MinRank. Thanks to recent complexity results on this problem, we prove that our attack is polynomial in the degree of the extension field for all possible practical settings used in HFE and Multi-HFE. This makes then Multi-HFE less secure than basic HFE for equally-sized keys. As a proof of concept, we have been able to practically break the most conservative proposed parameters of multi-HFE in few days (256 bits security broken in 9 days).

Cryptanalysis of a Public-Key Encryption Scheme Based on New Multivariate Quadratic Assumptions [24]

In [24] , we investigate the security of a public-key encryption scheme introduced by Huang, Liu and Yang (HLY) at PKC'12. This new scheme can be provably reduced to the hardness of solving a set of quadratic equations whose coefficients of highest degree are chosen according to a discrete Gaussian distributions. The other terms being chosen uniformly at random. Such a problem is a variant of the classical problem of solving a system of non-linear equations (PoSSo), which is known to be hard for random systems. The main hypothesis of Huang, Liu and Yang is that their variant is not easier than solving PoSSo for random instances. In this paper, we disprove this hypothesis. To this end, we exploit the fact that the new problem proposed by Huang, Liu and Yang reduces to an easy instance of the Learning With Errors (LWE) problem. The main contribution of this paper is to show that security and efficiency are essentially incompatible for the HLY proposal. That is, one cannot find parameters which yield a secure and a practical scheme. For instance, we estimate that a public-key of at least 1.03 GB is required to achieve 80-bit security against known attacks. As a proof of concept, we present practical attacks against all the parameters proposed Huang, Liu and Yang. We have been able to recover the private-key in roughly one day for the first challenge (i.e. Case 1) proposed by HLY and in roughly three days for the second challenge (i.e. Case 2).

On the Complexity of the BKW Algorithm on LWE [3]

In [3] , we present a study of the complexity of the Blum-Kalai-Wasserman (BKW) algorithm when applied to the Learning with Errors (LWE) problem, by providing refined estimates for the data and computational effort requirements for solving concrete instances of the LWE problem. We apply this refined analysis to suggested parameters for various LWE-based cryptographic schemes from the literature and compare with alternative approaches based on lattice reduction. As a result, we provide new upper bounds for the concrete hardness of these LWE-based schemes. Rather surprisingly, it appears that BKW algorithm outperforms known estimates for lattice reduction algorithms starting in dimension $n\approx 250$ when LWE is reduced to SIS. However, this assumes access to an unbounded number of LWE samples.

Combined Attack on CRT-RSA. Why Public Verification Must Not Be Public?

In [25] we introduce a new Combined Attack on a CRT-RSA implementation resistant against Side-Channel Analysis and Fault Injection attacks. Such implementations prevent the attacker from obtaining the signature when a fault has been induced during the computation. Indeed, such a value would allow the attacker to recover the RSA private key by computing the gcd of the public modulus and the faulty signature. The principle of our attack is to inject a fault during the signature computation and to perform a Side-Channel Analysis targeting a sensitive value processed during the Fault Injection countermeasure execution. The resulting information is then used to factorize the public modulus, leading to the disclosure of the whole RSA private key. After presenting a detailed account of our attack, we explain how its complexity can be significantly reduced by using Coppersmith's techniques based on lattice reduction. We also provide simulations that confirm the efficiency of our attack as well as two different countermeasures having a very small impact on the performance of the algorithm. As it performs a Side-Channel Analysis during a Fault Injection countermeasure to retrieve the secret value, this article recalls the need for Fault Injection and Side-Channel Analysis countermeasures as monolithic implementations.

Polynomial root finding over local rings and application to error correcting codes

Guruswami and Sudan designed a polynomial-time list-decoding algorithm. Their method divides into two steps. First it computes a polynomial $Q$ in ${𝔽}_q\left[x\right]\left[y\right]$ such that the possible transmitted messages are roots of $Q$ in ${𝔽}_q\left[x\right]$. In the second step one needs to determine all such roots of $Q$. Several techniques have been investigated to solve both steps of the problem.

The Guruswami and Sudan algorithm has been adapted to other families of codes such as algebraic-geometric codes and alternant codes over fields. Extensions over certain types of finite rings have further been studied for Reed-Solomon codes, for alternant codes, and for algebraic-geometric codes. In all these cases, the two main steps of the Guruswami and Sudan algorithm are roughly preserved, but to the best of our knowledge, the second step has never been studied into deep details from the complexity point of view. In [5] , we investigate root-finding for polynomials over Galois rings, which are often used within these error correcting codes, and that are defined as non-ramified extension of $\mathbb{Z}/p^n\mathbb{Z}$. We study the cost of our algorithms, discuss their practical performances, and apply our results to the Guruswami and Sudan list decoding algorithm over Galois rings.