The nonlinearity of a Boolean function, which is defined as its distance from the set of affine functions, is an important measuring index in cryptographic applications. The distribution of nonlinearities over all the Boolean functions is equivalent to the weight distribution of first order Reed-Muller codes and is very difficult to determine. As the first step towards solving this problem, the distribution of Boolean functions with nonlinearity ::; 2 n -2 is presented in this paper. It is

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... that the number of Boolean functions with nonlinearity t is exactly ( 2; ) •2"+1 for t < 2"-2 and 2Ml [( 2~~2 ) -(2" -1) ( ;:=: ) + ( 2";-1 )1 for t = 2n-2. Preliminaries A function f: GFn(2)-+GF( 2 ) is called a Boolean function of n variables. f(x) is called an affine function if there exist ao, al, ... , an EGF(2) such that f(x) = ao EB alXl EB ... EB anx n , where x = (Xl, ... , Xn) EGFn(2) and EB means modulo 2