Title: The cryptologic characteristics of circulant matrices

Authors: Haiqing Han; Siru Zhu; Qin Li; Yanping He; Xiao Wang; Yuxia Wang

Addresses: School of Computer, Wuhan University, Wuhan 430072, Hubei, China; School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, Hubei, China ' Air Force Early Warning Academy of PLA, Wuhan City 430072, Hubei Province, China ' Normal Department, Hubei Polytechnic University, Huangshi 435003, Hubei Province, China ' School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, Hubei, China ' School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, Hubei, China ' School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, Hubei, China

Abstract: A 4 × 4 invertible circulant matrix on GF(28) can represent the Mixcolumn operation of AES, which plays an important role as a confusion operation. Starting from the analysis of the Mixcolumns operation of AES, we have mainly research the properties of circulant matrix over finite field, and present a novel algorithm that generates the 4 × 4 circulant inverse matrices and the 4 × 4 circulant matrices with the maximal branch number in this paper. At last, some characteristics of the orthormorphic or symmetrical circulant matrices to arrive at the maximal branch number have been discussed in this paper. An algorithm for generating the 4 × 4 orthormorphic circulant matrix with the maximal branch number is also obtained on the finite field. Furthermore, the conclusion is gained that the symmetrical matrix with maximal branch number does not exist.

Keywords: circulant matrix; branch number; orthormorphic matrix; cryptologic characteristic; symmetrical permutation.

DOI: 10.1504/IJICA.2021.119302

International Journal of Innovative Computing and Applications, 2021 Vol.12 No.5/6, pp.248 - 254

Received: 22 Dec 2019
Accepted: 02 Apr 2020

Published online: 26 Nov 2021 *

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