Vectorial Boolean functions play an important role in cryptography, sequences and coding theory. Both affine equivalence and EA-equivalence are well known equivalence relations between vectorial Boolean functions. In this paper, we give an exact formula for the number of affine equivalence classes, and an asymptotic formula for the number of EA-equivalence classes of vectorial Boolean functions.

In recent years, algebraic attacks and fast algebraic attacks have received a lot of attention in the cryptographic community. There are three Boolean functions achieving optimal algebraic immunity based on primitive element of F2n. The support of Boolean functions in [1]-[3] have the same parameter s, which makes us have a large number of Boolean functions with good properties. However, we prove that the Boolean functions are affine equivalence when s takes different values.

Since 2008, three different classes of Boolean functions with optimal algebraic immunity have been proposed by Carlet and Feng [2], Wang et al.[8] and Chen et al.[3]. We call them C-F functions, W-P-K-X functions and C-T-Q functions for short. In this paper, we propose three affine equivalent classes of Boolean functions containing C-F functions, W-P-K-X functions and C-T-Q functions as a subclass, respectively. Based on the affine equivalence relation, we construct more classes of Boolean functions with optimal algebraic immunity. Moreover, we deduce a new lower bound on the nonlinearity of C-F functions, which is better than all the known ones.

A function F:F2n F2n is almost perfect nonlinear (APN) if, for every a 0, b in F2n, the equation F(x)+F(x+a)=b has at most two solutions in F2n. When used as an S-box in a block cipher, it contributes optimally to the resistance to differential cryptanalysis. The function F is almost bent (AB) if the minimum Hamming distance between all its component functions v F, v∈F2n {0} (where "" denotes any inner product in F2n ) and all affine Boolean functions on F2n takes the maximal value 2n-1-2. AB functions exist for n odd only and contribute optimally to the resistance to the linear cryptanalysis. Every AB function is APN, and in the n odd case, any quadratic APN function is AB. The APN and AB properties are preserved by affine equivalence: F F' if F'=A1 F A2, where A1,A2 are affine permutations. More generally, they are preserved by CCZ-equivalence, that is, affine equivalence of the graphs of F: {(x,F(xv)) | x∈ F2n} and of F'. Until recently, the only known constructions of APN and AB functions were CCZ-equivalent to power functions F(x)=xd over finite fields (F2n being identified with F2n and an inner product being x y=tr(xy) where tr is the trace function). Several recent infinite classes of APN functions have been proved CCZ-inequivalent to power functions. In this paper, we describe the state of the art in the domain and we also present original results. We indicate what are the most important open problems and make some new observations about them. Many results presented are from joint works with Lilya Budaghyan, Gregor Leander and Alexander Pott.