Excluding Some Weak Keys in the McEliece Cryptosystem

Abstract

A Goppa code which has a non trivial automorphism group is a weak key for the McEliece cryptosystem. A quasicyclic code clearly has a non trivial automorphism group. Hence any quasicyclic code is a weak key. Some new classes of quasicyclic irreducible Goppa codes have recently been established and it is conjectured that, in the binary case, these new classes contain all binary quasicyclic ireducible Goppa codes. Using simple numerical conditions on the parameters of a Goppa code, we show that if we adopt the parameters as suggested by McEliece himself for choosing Goppa codes in the implementation of his cryptosystem, then some such codes are weak keys. We suggest other parameters which we claim would reduce the probability of choosing a weak key.