*09:17* [Pub][ePrint]
McEliece in the world of Escher, by Danilo Gligoroski and Simona Samardjiska and H{\\aa}kon Jacobsen and Sergey Bezzateev
We present a new family of linear binary codes of length $n$ and dimension $k$ accompanied with a fast list decoding algorithm that can correct up to $\\frac{n}{2}$ errors in a bounded channel with an error density $\\rho$. The decisional problem of decoding random codes using these generalized error sets is NP-complete. Next we use the properties of these codes to design both an encryption scheme and a signature scheme. Although in the open literature there have been several proposals how to produce digital signatures from the McEliece public key scheme, as far as we know, this is the first public key scheme based on codes where signatures are produced in a straightforward manner from the decryption procedure of the scheme. The security analysis of our scheme have two main parts:1. An extensive list of attacks using the Information Set Decoding techniques adopted for our codes; 2. An analysis of the cost of a distinguishing attack based on rank attacks on the generator matrix of the code or on its dual code. Based on this security analysis we suggest some concrete parameters for the security levels in the range of $2^{80} - 2^{128}$.

*09:17* [Pub][ePrint]
Folding Alternant and Goppa Codes with Non-Trivial Automorphism Groups, by Jean-Charles Faugère and Ayoub Otmani and Ludovic Perret and Frédéric de Portzamparc and Jean-Pierre Tillich
The main practical limitation of the McEliece public-key encryption scheme is probably the size of its key. A famous trend to overcome this issue is to focus on subclasses of alternant/Goppa codes with a non trivial automorphism group. Such codes display then symmetries allowing compact parity-check or generator matrices. Forinstance, a key-reduction is obtained by taking quasi-cyclic (QC) or quasi-dyadic (QD) alternant/Goppa codes.

We show that the use of such symmetric alternant/Goppa codes in cryptography introduces a fundamental weakness. It is indeed possible to reduce the key-recovery on the original symmetric public-code to the key-recovery on a (much) smaller code that has not anymore symmetries. This result is obtained thanks to a new operation on codes called folding that exploits the knowledge of the automorphism group. This operation consists in adding the coordinates of codewords which belong to the same orbit under the action of the automorphism group. The advantage is twofold:

the reduction factor can be as large as the size of the orbits, and it preserves a fundamental property: folding the dual of an alternant (resp. Goppa) code provides the dual of an alternant (resp. Goppa) code. A key point is to show that all the existing constructions of alternant/Goppa codes with symmetries follow a common principal of taking codes whose support is globally invariant under the action of affine transformations (by building upon prior works of T. Berger and A. D¨ur). This enables not only to present a unified view but also to generalize the construction of QC, QD and even quasi-monoidic (QM) Goppa codes. All in all, our results can be harnessed to boost up any key-recovery attack on McEliece systems based on symmetric alternant or Goppa codes, and in particular algebraic attacks.