Julien Schrek

Philippe Gaborit, Olivier Ruatta, Julien Schrek et al.

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.

Philippe Gaborit, Olivier Ruatta, Julien Schrek

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.