Secure and Fast Digital Signatures using BCH Codes

Since the introduction of public key cryptography in the 70’s [1], many cryptosystems have been proposed and many cryptographic schemes have been broken. The most used cryptosystems rely on number theory problem like the factorization problem [3] and the discrete logarithm over suitable group [2]. The McEliece cryptosystem [5] and the Neiderreiter variante [6] rely on coding theory, they are ones of the few cryptosystems, which are very secure and which are not broken although they do not rely on number theory. These cryptosystems present many advantages: they are very fast for both encryption and decryption and the best attacks complexity are exponential in the length of the code. These cryptosystems have the drawback to have a large public key which is a generator matrix or a check parity matrix of a long code. Another drawback related to the belief that we can not deduce a digital signature from these public key cryptosystems. In 2001, Courtois, Finiasz and Sendrier [15] introduced the first signature scheme based on McEliece cryptosystem. Firstly, they have presented a scheme based on McEliece cryptosystem using generator matrix. With the proposed secure parameters, this scheme is impractical. Secondly, they have introduced a short practical signature based on the Neiderreiter variant using the parity check matrix as key. This scheme has the drawback to have a slow signature algorithm. In this paper, we introduce new performant digital signature schemes based on coding theory similar to those based on McEliece and Niederreiter cryptosystems. The idea of our schemes consists in considering a chained BCH code. The resulting code will be a secret code which will be scrambled and permuted to obtain the public code.