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2011•Computer Engineering and ScienceRequires access

The Nonlinearity of Complementary Symmetric Boolean Functions

Peizhong Lu

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Abstract

Complementary symmetric Boolean functions are a special class of symmetric Boolean functions.A high proportion of symmetric Boolean functions with optimum algebraic immunity are complementary symmetric Boolean functions.Especially for the case of 2m variables,it reaches a high proportion of 2/3.By the relationship between the nonlinearity and the Walsh spectrum of the Boolean functions,and that between the Walsh spectrum of the Boolean functions and the Krawtchouk polynomial,the nonlinearity of complementary symmetric Boolean functions is determined.As a result,the nonlinearity of all complementary symmetric Boolean functions with n variables is2n-1-12[nn/2]

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What this paper is about

Complementary symmetric Boolean functions are a special class of symmetric Boolean functions.A high proportion of symmetric Boolean functions with optimum algebraic immunity are complementary symmetric Boolean functions.Especially for the case of 2m variables,it reaches a high proportion of 2/3.By the relationship between the nonlinearity and the Walsh spectrum of the Boolean functions,and that between the Walsh spectrum of the Boolean functions and the Krawtchouk polynomial,the nonlinearity of complementary symmetric Boolean functions is determined.As a result,the nonlinearity of all complementary symmetric Boolean functions with n variables is2n-1-12[nn/2]

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Available abstract

Complementary symmetric Boolean functions are a special class of symmetric Boolean functions.A high proportion of symmetric Boolean functions with optimum algebraic immunity are complementary symmetric Boolean functions.Especially for the case of 2m variables,it reaches a high proportion of 2/3.By the relationship between the nonlinearity and the Walsh spectrum of the Boolean functions,and that between the Walsh spectrum of the Boolean functions and the Krawtchouk polynomial,the nonlinearity of complementary symmetric Boolean functions is determined.As a result,the nonlinearity of all complementary symmetric Boolean functions with n variables is2n-1-12[nn/2]

Key concepts: Boolean function, Parity function, Two-element Boolean algebra, Boolean expression, Boolean network, Complete Boolean algebra, Symmetric function, Mathematics

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