arXiv subjects
Publications and source records attributed to Alain Couvreur.
We present a polynomial time structural attack against the McEliece system based on Wild Goppa codes from a quadratic finite field extension. This attack uses the fact that such codes can be distinguished from random codes to compute some filtration, that is to say a family of nested subcodes which will reveal their secret algebraic description.
The BBCRS scheme is a variant of the McEliece public-key encryption scheme where the hiding phase is performed by taking the inverse of a matrix which is of the form $\mathbf{T} +\mathbf{R}$ where $\mathbf{T}$ is a sparse matrix with average row/column weight equal to a very small quantity $m$, usually $m < 2$, and $\mathbf{R}$ is a matrix of small rank $z\geqslant 1$. The rationale of this new transformation is the reintroduction of families of codes, like generalized Reed-Solomon codes, that are famously known for representing insecure choices. We present a key-recovery attack when $z = 1$ and $m$ is chosen between $1$ and $1 + R + O( \frac{1}{\sqrt{n}} )$ where $R$ denotes the code rate. This attack has complexity $O(n^6)$ and breaks all the parameters suggested in the literature.
We give a polynomial time attack on the McEliece public key cryptosystem based on subcodes of algebraic geometry (AG) codes. The proposed attack reposes on the distinguishability of such codes from random codes using the Schur product. Wieschebrink treated the genus zero case a few years ago but his approach cannot be extent straightforwardly to other genera. We address this problem by introducing and using a new notion, which we call the t-closure of a code.
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.
We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison and Shokrollahi. It is based on the Cayley graph of Fn together with a set of generators regarded as the columns of the parity-check matrix of a classical code. We give a general lower bound on the minimum distance of the Quantum code in $\mathcal{O}(dn^2)$ where d is the minimum distance of the classical code. When the classical code is the $[n, 1, n]$ repetition code, we are able to compute the exact parameters of the associated Quantum code which are $[[2^n, 2^{\frac{n+1}{2}}, 2^{\frac{n-1}{2}}]]$.
For a given support $L \in \mathbb{F}_{q^m}^n$ and a polynomial $g\in \mathbb{F}_{q^m}[x]$ with no roots in $\mathbb{F}_{q^m}$, we prove equality between the $q$-ary Goppa codes $Γ_q(L,N(g)) = Γ_q(L,N(g)/g)$ where $N(g)$ denotes the norm of $g$, that is $g^{q^{m-1}+\cdots +q+1}.$ In particular, for $m=2$, that is, for a quadratic extension, we get $Γ_q(L,g^q) = Γ_q(L,g^{q+1})$. If $g$ has roots in $\mathbb{F}_{q^m}$, then we do not necessarily have equality and we prove that the difference of the dimensions of the two codes is bounded above by the number of distinct roots of $g$ in $\mathbb{F}_{q^m}$. These identities provide numerous code equivalences and improved designed parameters for some families of classical Goppa codes.
In this article, we present a new construction of codes from algebraic curves. Given a curve over a non-prime finite field, the obtained codes are defined over a subfield. We call them Cartier Codes since their construction involves the Cartier operator. This new class of codes can be regarded as a natural geometric generalisation of classical Goppa codes. In particular, we prove that a well-known property satisfied by classical Goppa codes extends naturally to Cartier codes. We prove general lower bounds for the dimension and the minimum distance of these codes and compare our construction with a classical one: the subfield subcodes of Algebraic Geometry codes. We prove that every Cartier code is contained in a subfield subcode of an Algebraic Geometry code and that the two constructions have similar asymptotic performances. We also show that some known results on subfield subcodes of Algebraic Geometry codes can be proved nicely by using properties of the Cartier operator and that some known bounds on the dimension of subfield subcodes of Algebraic Geometry codes can be improved thanks to Cartier codes and the Cartier operator.
We give the parameters of any evaluation code on a smooth quadric surface. For hyperbolic quadrics the approach uses elementary results on product codes and the parameters of codes on elliptic quadrics are obtained by detecting a BCH structure of these codes and using the BCH bound. The elliptic quadric is a twist of the surface P^1 x P^1 and we detect a similar BCH structure on twists of the Segre embedding of a product of any d copies of the projective line.
In this article, new regular incidence structures are presented. They arise from sets of conics in the affine plane blown-up at its rational points. The LDPC codes given by these incidence matrices are studied. These sparse incidence matrices turn out to be redundant, which means that their number of rows exceeds their rank. Such a feature is absent from random LDPC codes and is in general interesting for the efficiency of iterative decoding. The performance of some codes under iterative decoding is tested. Some of them turn out to perform better than regular Gallager codes having similar rate and row weight.
In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound "à la Weil" of Aubry and Perret together with the bound of Homma and Kim for plane curves. The parameters of several codes from rational surfaces are computed. Among them, the codes defined by the evaluation of forms of degree 3 on an elliptic quadric are studied. As far as we know, such codes have never been treated before. Two other rational surfaces are studied and very good codes are found on them. In particular, a [57,12,34] code over $\mathbf{F}_7$ and a [91,18,53] code over $\mathbf{F}_9$ are discovered, these codes beat the best known codes up to now.
We study the list-decoding problem of alternant codes, with the notable case of classical Goppa codes. The major consideration here is to take into account the size of the alphabet, which shows great influence on the list-decoding radius. This amounts to compare the \emph{generic} Johnson bound to the \emph{$q$-ary} Johnson bound. This difference is important when $q$ is very small. Essentially, the most favourable case is $q=2$, for which the decoding radius is greatly improved, notably when the relative minimum distance gets close to 1/2. Even though the announced result, which is the list-decoding radius of binary Goppa codes, is new, it can be rather easily made up from previous sources (V. Guruswami, R. M. Roth and I. Tal, R .M. Roth), which may be a little bit unknown, and in which the case of binary Goppa codes has apparently not been thought at. Only D. J. Bernstein treats the case of binary Goppa codes in a preprint. References are given in the introduction. We propose an autonomous treatment and also a complexity analysis of the studied algorithm, which is quadratic in the blocklength $n$, when decoding at some distance of the relative maximum decoding radius, and in $O(n^7)$ when reaching the maximum radius.
In this paper, we study residues of differential 2-forms on a smooth algebraic surface over an arbitrary field and give several statements about sums of residues. Afterwards, using these results we construct algebraic-geometric codes which are an extension to surfaces of the well-known differential codes on curves. We also study some properties of these codes and extend to them some known properties for codes on curves.