WDC algorithm and enumeration of Bent functions with 6 variables
In the research of Boolean functions,there is an important problem:given a subset A={(λi,ai)|λi∈Fn2,ai∈Z,i=0,1,2,…,m-1 },find all the Boolean functions f in n variables,such thatf(λi)=ai for all i=0,1,2,…,m-1,where f(λi)is the Walsh spectral value of the Boolean function f at the address λi.In general,this problem is quite difficult.But if the given set of addres-ses is a vector subspace of Fn2,there is a simple solution.This paper,gives an algorithm called WDC Algorithm,that can solve this problem efficiently in this special case.The WDC Algorithm contains the following three parts:① constructs all the Boolean functions that satisfying all these conditions;② finds the number of such Boolean functions;and ③ gives a necessary and sufficient condition of the spectral distribution on the subspace to ensure the existence of such Boolean functions.On the other hand,the Bent function is the Boolean function that has the maximal nonlinearity,thus possess-ing excellent cryptographic properties.This paper uses WDC algorithm to find all 6-variable Bent functions by means of computer searching,the total number of such functions is 5 425 430 528.