The Properties And Construction Of Bent Functions

Abstract: In the cryptosystems,the design of keystream generators of nonlinear combine functions is crucial to the safety of cryptosystems.The designer of cryptography designed various kinds of nonlinear combine functions to withstand attack,for example,the Bent function is put forward by Rothaus to withstand difference attack. Bent function has important applications not only in cryptography,but also in other fields such as, dyadic theory and coding theory.The dissertation mainly discusses the properties and construction of Bent function, the main results are as follows.1. The dissertation gives some necessary and sufficient conditions for an nboolean function in n variables to be a Bent function when n is an even number by use of number theory and relations between boolean auto-correlation spectrum and cyclic spectrum.2. Decomposition formula of Walsh spectrum of a class of boolean functions are given by use of identity of boolean function,by which some recursive construction of Bent function are given.3. A necessary and sufficient condition for a f(x + t(y)) + g(y) to be a bent function is disscussed,on which basis a necessary and sufficient condition for a function F(x,y) = fix + t(y))q(y) + g(y) to be a bent function is given when q(y) is not a balanced-function by use of cyclic walsh spectrum and decomposition formula of walsh spectrum.Meanwhile it is proved that an n-Bent function of algebraic degree 2 is linearly equivalent to x₁,x₂ + ... + x_n-1x_n and some constructions of Bent function are given.4. A necessary and sufficient condition for a boolean function F(x₁ ,...x_m,y₁,..., y_n)=t(y)·x +g(y)to be a Bent function is studied ,which conclusiongeneralized the construction of Bent function by Rotuaus,a enumeration of construction of F(x₁ ,...x_m,y₁,...,y_n) = t(y)·x + g(y) are given in the case, n - m = 2, the lower bound for the enumeration of Bent function is greatly improved.5. The relations between partially-Bent function and Bent function are given by Li shi qu,therefore the research into the properties construction and...