Large Families Of Boolean Functions Constructed By Using Pseudorandom Subsets In Finite Fields And The Cryptography Properties

Boolean functions play an important role in stream ciphers,block cipher and hash functions.In a series of papers many Boolean functions with good cryptography properties were constructed using algebraic and number theoretic methods.In this paper we construct large families of Boolean functions by using the pseudorandom subsets in finite fields,and study the cryptography properties.Our results are as follows:Firstly,a certain construction of Boolean functions in finite fields of odd characteristic is studied.We construct large family of Boolean functions by using the pseudorandom subsets in finite fields of odd characteristic,which gen-eralizes the construction given by Lange and Winterhof.Using upper bounds on certain character sums over finite fields,we study the cryptography properties:maximum Fourier coefficient,nonlinearity,average sensitivity,sparsity,and the lower bound on the nonlinearity is superior to the result given by Lange and Winterhof.Secondly,certain constructions of Boolean functions in finite fields of char-acteristic two are studied.We construct two families of Boolean functions by using the pseudorandom subsets in finite fields of characteristic two,which are generalizations and improvements on the results obtained by predecessors.Us-ing upper bounds on certain character sums over finite fields and Krawtchouk polynomial,we study the cryptography properties of these Boolean functions:maximum Fourier coefficient,nonlinearity,sparsity.The first family of Boolean functions includes the functions given by Brandstatter,Carlet and Wang et al.The second family of Boolean functions is generalizations and improvements on the results obtained by Dalai et al.