Generalization of Boolean Functions Properties to Functions Defined over GF(p)

Abstract

Problem statement: Traditionally, cryptographic applications designed on hardware have always tried to take advantage of the simplicity of implementation functions over GF(p), p = 2, to reduce costs and improve performance. On the contrast, functions defined over GF(p); p > 2, possess far better cryptographic properties than GF(2) functions. Approach: We generalize some of the previous results on cryptographic Boolean functions to functions defined over GF(p); p > 2. Results: We generalize Siegenthaler's construction to functions defined over finite field. We characterize the linear structures of functions over GF(p) in terms of their Walsh transform values. We then investigate the relation between the autocorrelation coefficients of functions over GF(p) and their Walsh spectrum. We also derive an upper bound for the dimension of the linear space of the functions defined over GF(p). Finally, we present a method to construct a bent function from semi-bent functions. Conclusion: Functions defined over GF(p) can achieve better cryptographic bounds than GF(2) functions. In this paper we gave a generalization of several of the GF(2) cryptographic properties to functions defined over GF(p), where p is an odd prime.