Design And Analysis For Diffusion Structure Of Block Ciphers

The design and analysis of diffusion structure for block cipher is one of the focuses in the cryptography nowadays. So this paper studies the topic above, and our contributions mainly include the following four parts:1. This paper deals with those MDS matrices which are Hadamard matrices and Cauchy matrices simultaneously, called Cauchy–Hadamard matrices. Then the structure, the method of constructing and the count value are presented. This paper proves that a Cauchy-Hadamard type MDS matrix can be transformed into an involution Cauchy-Hadamard type MDS, and then proposes the method of construct an involution Cauchy-Hadamard type MDS matrix from a Cauchy-Hadamard type MDS matrix.2. This paper investigates the properties of Cauchy matrices, we provide the necessary and sufficient condition of the same Cauchy matrix generated by different ordered-arrays, and prove the count value formula of Cauchy matrices over GF (2ⁿ). Then by take consider of implementation, this paper proposes definitions of simplest Cauchy matrices and optimization simplest Cauchy matrices, and provides necessary and sufficient condition of simplest Cauchy matrices. We prove that a cyclic-shift matrix could never be a Cauchy matrix. This paper provides the structure and count value of simplest Cauchy matrices, and points that any simplest Cauchy matrix can be transformed into an optimization simplest Cauchy matrix, proves that a simplest Cauchy matrix can be transformed into an involution simplest Cauchy matrix, and the method of transforming is given.3. This paper provides count value formulas of characteristic vectors for quasi-involution MDS matrices and multi-bit cyclic-shift transform.4. This paper studies the design of 0-1 matrices over GF(2ⁿ) with largest branch number whose order are 8 and 16. This paper proves that the branch number of matrices over GF(2ⁿ) are equal to that over extension field GF(2^mn), base on this verdict, the result of upper bound of branch number of binary matrices proposed by Choy et.al. is corrected. We construct a kind of invertible 0-1 matrices with order 8 and branch number 5, and we propose a kind of matrices with equal differential branch number and linear branch number, from which we searched out a large number of 0-1 matrices and involution 0-1 matrices with order 16 and branch number 8.