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The preservation of good cryptographic properties of MDS matrix under direct exponent transformation

In this paper, some new results on the preservation of many good cryptographic properties of MDS matrices under direct exponent transformation are presented. These good cryptographic properties include MDS, involutory, symmetric, recursive (exponent of a companion matrix ), the number of 1 0 s and distinct elements in a matrix, circulant and circulant-like | Journal of Computer Science and Cybernetics, V.31, N.4 (2015), 291–303 DOI: 10.15625/1813-9663/31/4/7059 THE PRESERVATION OF GOOD CRYPTOGRAPHIC PROPERTIES OF MDS MATRIX UNDER DIRECT EXPONENT TRANSFORMATION TRAN THI LUONG1 , NGUYEN NGOC CUONG2 , LUONG THE DUNG1,∗ 1 Academy of Cryptography Techniques, Hanoi, Vietnam; ttluong@bcy.gov.vn; ∗ ltdung@bcy.gov.vn 2 Vietnam Government Information Security Commission, Hanoi, Vietnam; nguyenngoccuong189@gmail.com Abstract. Maximum Distance Separable (MDS) code has been studied for a long time in the coding theory and has been applied widely in cryptography. The methods for transforming an MDS into other ones have been proposed by many authors in the literature. These methods are called MDS matrix transformations in order to generate different MDS matrices (dynamic MDS matrices) from an existing one. In this paper, some new results on the preservation of many good cryptographic properties of MDS matrices under direct exponent transformation are presented. These good cryptographic properties include MDS, involutory, symmetric, recursive (exponent of a companion matrix ), the number of 1 s and distinct elements in a matrix, circulant and circulant-like. In addition, these results are shown to have important applications in constructing dynamic diffusion layers for block ciphers. The strength of the ciphers against developing cryptanalytic techniques can be enhanced by the dynamic MDS diffusion layers. Keywords. MDS matrix, dynamic MDS matrix, direct exponent matrix, cryptographic properties. 1. INTRODUCTION Claude Shannon, in his paper of “Communication Theory of Secrecy Systems” [1] defined confusion and diffusion as two mandatory properties, required for the design of block ciphers. Confusion is to make the relationship of statistical independence between ciphertext string and plaintext string more complicated while diffusion is associated with dependency of output bits on input bits. As we know, MDS matrices were first .