On Cryptographic Functions And Their Constructions

Cryptographic Boolean functions play an important role in both stream ciphers and block ciphers. In this dissertation, some important problems on cryptographic Boolean functions are investigated. The main results are as follows:1) A class of plateaued functions has been get by way of using the Maiorana- McFarland construction. A variety of desirable criteria for functions with cryptographic application could be satisfied: balancedness, high nonlinearity, correlation immunity of reasonably high order, strict avalanche criterion, non- existence of non-zero linear structures and good global avalanche characteristics etc.2) The notion of multi-output plateaued functions is introduced and some methods to construct this kind of cryptographic functions are provided. An effective method for finding a set of [ n , k ] disjoint linear codes is presented. When n≥2k, we could find a set of [ n , k ] disjoint linear codes with cardinality 2 n ? k + ?? ( n ?k )/k??; When n < 2k, there does not exist a set a disjoint linear codes with cardinality at least 2. The method on constructing a set of [ n , k ,≥? d/2 ?] disjoint linear codes is also be considered. We show how, thanks to our method, a (9,2,1) multi- output plateaued functions with highly nonlinearity could be constructed.3) We derive several results towards a better understanding the characterization of separable Boolean functions. Some properties of separable plateaued functions are also given. Two indicators related to the inseparability of cryptographic functions are introduced.4) We derive an upper bound on the algebraic immunity of a k-normal Boolean function. An effective algorithm to check whether a given bent function is normal or not is present.5) A large class of resilient functions is constructed via concatenating nonlinear resilient functions. We show that how this technique can be used to construct resilient functions which could achieve Siegenthalor bound. The nonlinearity of such functions is also considered. By way of concatenating 2d plateaued functions on F2 n ?d, which possess some properties, a resilient function of n variables with highly nonlinearity could be constructed.6) We show how to calculate periods of product polynomials via combinatorial techniques on factorization. A general formula for the period of a product polynomial is present. Then, the result is applied to compute the period of a convolution sequence.