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.Comment: To appear in Designs, Codes and Cryptography Journa

New algorithms for decoding in the rank metric and an attack on the LRPC cryptosystem

We consider the decoding problem or the problem of finding low weight codewords for rank metric codes. We show how additional information about the codeword we want to find under the form of certain linear combinations of the entries of the codeword leads to algorithms with a better complexity. This is then used together with a folding technique for attacking a McEliece scheme based on LRPC codes. It leads to a feasible attack on one of the parameters suggested in \cite{GMRZ13}.Comment: A shortened version of this paper will be published in the proceedings of the IEEE International Symposium on Information Theory 2015 (ISIT 2015

Two attacks on rank metric code-based schemes: RankSign and an Identity-Based-Encryption scheme

RankSign [GRSZ14a] is a code-based signature scheme proposed to the NIST competition for quantum-safe cryptography [AGHRZ17] and, moreover, is a fundamental building block of a new Identity-Based-Encryption (IBE) [GHPT17a]. This signature scheme is based on the rank metric and enjoys remarkably small key sizes, about 10KBytes for an intended level of security of 128 bits. Unfortunately we will show that all the parameters proposed for this scheme in [AGHRZ17] can be broken by an algebraic attack that exploits the fact that the augmented LRPC codes used in this scheme have very low weight codewords. Therefore, without RankSign the IBE cannot be instantiated at this time. As a second contribution we will show that the problem is deeper than finding a new signature in rank-based cryptography, we also found an attack on the generic problem upon which its security reduction relies. However, contrarily to the RankSign scheme, it seems that the parameters of the IBE scheme could be chosen in order to avoid our attack. Finally, we have also shown that if one replaces the rank metric in the [GHPT17a] IBE scheme by the Hamming metric, then a devastating attack can be found

An algebraic approach to the Rank Support Learning problem

Rank-metric code-based cryptography relies on the hardness of decoding a random linear code in the rank metric. The Rank Support Learning problem (RSL) is a variant where an attacker has access to N decoding instances whose errors have the same support and wants to solve one of them. This problem is for instance used in the Durandal signature scheme. In this paper, we propose an algebraic attack on RSL which clearly outperforms the previous attacks to solve this problem. We build upon Bardet et al., Asiacrypt 2020, where similar techniques are used to solve MinRank and RD. However, our analysis is simpler and overall our attack relies on very elementary assumptions compared to standard Gr{\"o}bner bases attacks. In particular, our results show that key recovery attacks on Durandal are more efficient than was previously thought