Construction methods for mds matrices using companion and permutation matrices for lightweight cryptography

In this work, we propose a new construction method of MDS-matrices of dimension k = 4,6 by means of summation of a power r of the companion matrix of a certain polynomial and a fixed permutation matrix over the finite field GF(2⁸). The method is represented by the expression S^ + P for a polynomial f (x) = = x^k + f_k-1x^k-1 + ... + f₁x + f₀, where Sf is the companion matrix of the polynomial f (x), P is a permutation matrix, r = 3k/2, and the coefficients f_i £ {0,1, a, a^-1, a², a³}. For its effective implementation, it is proposed to apply Sf as a linear feedback shift register with characteristic polynomial f (x) and P as a Feistel network with k entrances. The XOR-count metric is used to show the effectiveness of the proposed method in algorithms that require low implementation cost.

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