Code-Based Cryptosystems Using Generalized Concatenated Codes

Public-key cryptosystems nowadays are mostly based on number theoretic problems like factorization (RSA) and the discrete logarithm problem (Elgamal). However, such systems can be broken with quantum computers by applying Shor’s algorithms [1] for solving both problems, factorization and discrete logarithm, in polynomial time. Hence there is a need for post-quantum cryptography, i.e., methods resisting quantum computers. Code-based cryptography, introduced by McEliece in 1978 [2], is one of these candidates. In the original work, the McEliece cryptosystem uses Goppa codes. Ongoing research work is investigating other classes of codes for use in this cryptosystem. Code-based cryptosystems based on Ordinary Concatenated (OC) codes were suggested by Nicolas Sendrier in [3]. OC codes are characterized by a lower decoding complexity than non-concatenated codes. However, in order to reach the same level of security as the original cryptosystem, systems based on OC codes require larger key sizes than the ones based on Goppa codes. Generalized Concatenated (GC) codes also have the advantage of low decoding complexity at the cost of possessing larger key sizes. As explained in [4], comparing a GC and an OC code with the same number of codewords, a GC code has a larger minimum distance. On the other hand, when they both have the same minimum distance, a GC code has more codewords. In [3, 5], it is shown that the structure of a randomly permuted OC code could be discovered. A cryptosystem using OC codes, can then be attacked through obtaining the structure of the inner and outer codes from the public generator matrix. The attack consists of three main steps. The first step is based on identifying the positions of the inner code blocks. The second step orders the positions of the elements of the inner code blocks with respect to each other. Finally, in the third step, a generator matrix for an equivalent inner code is obtained. Moreover, a generator matrix of a π-equivalent outer code is also obtained, where π symbolizes the Frobenius field automorphism and also any power of π results in a field automorphism. After obtaining the structures of the inner and outer codes, already known attacks could be applied to each of them in order to break the whole system. In this work, code-based cryptosystems using GC codes are analyzed in light of Sendrier’s attack [3, 5]. If a GC code could be converted to an OC code, the attack