Square code attack on a modified Sidelnikov cryptosystem

Abstract

This paper presents a cryptanalysis of a modified version of the Sidelnikov cryptosystem which is based on Reed-Muller codes. This modified scheme consists in inserting random columns in the secret generating matrix or parity check matrix. The cryptanalysis relies on the computation of the squares of the public code. The particular nature of Reed-Muller which are defined by means of multivariate binary polynomials, permits to predicate the value of dimension of the square codes and then to fully recover in polynomial time the secret positions of the random columns. Our work shows that the insertion of random columns in the Sidelnikov scheme does not bring any security improvement.