About The Nature Of The Logic Functions In Cryptography

Using the theories of probability, algebra and spectral theory comprehensively, we investigate some related characteristics of logic functions in cryptography: Firstly, we introduce m order generalized s - correlation immunity of Boolean vector functions and prove that the higher order generalized ε - correlation immunity can guarantee the lower order generalized ε- correlation immunity. Then by applying decomposition formula of joint distribution of Boolean random vectors, we give a spectrum criterion of m order generalized e - correlation immunity of Boolean vector functions. Furthermore, we show that the algebraic degree of m order generalized e - correlation immune Boolean vector functions is not restricted by the correlation immune orders. Secondly, we analyze the similarities and differences about the cryptographic properties of two Boolean functions in the sense of linear equivalence, and obtain a sufficient and necessary condition about a Boolean function linearly equivalent to some m order correlation-immune Boolean function. We also obtain a sufficient and necessary condition about a Boolean function linearly equivalent to some Boolean function satisfying k order propagation criterion. Moreover, as an example of application, for a given Boolean function which is not correlation-immured and does not satisfy the propagation criterion, we construct some correlation-immune Boolean function satisfying the propagation criterionwhich is linearly equivalent to the former one. Lastly, combining the p-adic decomposition of the variables in Z_p^r , and the thought ofprobability theory, we present the decompose properties and equivalent description of p^r - valued r.v. Then we show the p - adic decomposition of both the p^r - valued logic functions and the variables. And then, we give a linear combining lemma of k order correlation-immunity of p^r -valued logic functions in the sense of p - adic decomposition. Based on this, we also provide a spectrum criterion of k order correlation-immunity of p^r - valued logic functions by using Chrestenson transformation of p - valued logic functions directly. In the end, we give a method of constructing some correlation-immunelogic functions in 2 n variables over Z4 and Z9.