Constructions And Applications To Coding Theory Of Nonlinear Cryptographic Functions

Nonlinear cryptographic functions have an important role in cryptography. To resist some known attacks, the functions should satisfy corresponding cryptographic prop-erties. Cryptographic functions are defined differently according to different crypto-graphic properties, such as PN functions, APN functions, bent functions, almost bent functions, Boolean functions with optimal algebraic immunity and so on.Boolean functions play a very important role in stream cipher design and analy-sis. This paper mainly concentrate on a cryptographic property of Boolean functions: algebraic immunity which is proposed to measure the ability of resisting algebraic at-tacks. In2003, Courtois, Meier et al succeeded in implementing algebraic attacks by establishing the equation system with low algebraic degree. In2010, Rizomiliotis pro-posed a necessary and sufficient condition on Boolean functions with optimal algebraic immunity. Based on this condition, some classes of balanced Boolean functions with optimal algebraic immunity are proposed. We prove that there always exist Boolean functions with optimal algebraic degree in our constructions. A lower bound on non-linearity of Boolean functions in this paper is proposed, and it is the best bound on Carlet-Feng functions so far. Experimental data shows that the nonlinearity of Boolean functions in this paper can achieve even more than that of Carlet-Feng functions. In addition, for small values of the number of variables, experimental data also shows that Boolean functions in this paper have a good behavior against fast algebraic attacks. Re-cent study shows that there does not exist an n-variable Boolean function with optimal resistance against fast algebraic attacks for most values of n.BCH codes were independently proposed by Hocquenghem in1959and Bose and Ray-Chaudhuri in1960, which are mostly studied in coding theory, particularly, in error-correcting codes. To determine the minimum distance a binary triple-error-correcting BCH code C1,d1,d2with three zeros α,αd1and αd2of length2n-1and the weight distribution of its dual code, where n≥5is odd and α is a primitive element of the finite field F2n, it is usually to combine Walsh transform and the theory of binary quadratic forms, which will restrict the weight of exponents d1,d2. Recently, Hollmann and Xiang proposed the modular add-with-carry algorithms and divisibility of weights, which can effectively break the shackles of the quadratic theory. In2006, the relation between the minimum distance of codes and APN functions was proposed by Carlet et al. In this paper, we restrict the exponents d1, d2to APN exponents. A new class of binary triple-error-correcting BCH code C1,3,13is proposed and the weight distribution is determined by the theory introduced by Hollmann and Xiang.