Group theory in cryptography

This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.Comment: 25 pages References updated, and a few extra references added. Minor typographical changes. To appear in Proceedings of Groups St Andrews 2009 in Bath, U

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

Cryptanalysis of public-key cryptosystems that use subcodes of algebraic geometry codes

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

Expanded Gabidulin Codes and Their Application to Cryptography

This paper presents a new family of linear codes, namely the expanded Gabidulin codes. Exploiting the existing fast decoder of Gabidulin codes, we propose an efficient algorithm to decode these new codes when the noise vector satisfies a certain condition. Furthermore, these new codes enjoy an excellent error-correcting capability because of the optimality of their parent Gabidulin codes. Based on different masking techniques, we give two encryption schemes by using expanded Gabidulin codes in the McEliece setting. According to our analysis, both of these two cryptosystems can resist the existing structural attacks. Our proposals have an obvious advantage in public-key representation without using the cyclic or quasi-cyclic structure compared to some other code-based cryptosystems