Codes, graphs, and schemes from nonlinear functions

We consider functions on binary vector spaces which are far from linear functions in di erent senses. We compare three existing notions: almost perfect nonlinear (APN) functions, almost bent (AB) functions, and crooked (CR) functions. Such functions are of importance in cryptography because of their resistance to linear and di erential attacks on certain cryptosystems. We give a new combinatorial characterization of almost bent functions in terms of the number of solutions to a certain system of equations, and a characterization of crooked functions in terms of the Fourier transform. We also show how these functions can be used to construct several combinatorial structures; such as semi-biplanes, di erence sets, distance regular graphs, symmetric association schemes, and uniformly packed (BCH and Preparata) codes. 1 Almost perfect nonlinear, almost bent, and crooked functions We consider functions on binary vector spaces which are far from linear functions in di erent senses. We compare three existing notions: almost perfect nonlinear (APN) functions, almost bent (AB) functions, and crooked (CR) functions. Such functions are of importance in cryptography because of their resistance to linear and di erential attacks on certain cryptosystems (cf. [8], [9], [10, p. 1037]). Furthermore they are of interest in the study of linear feedback shift register sequences with low crosscorrelation (cf. [17, pp. 1795-1810]). Also in the construction of certain combinatorial structures they have proven to be useful; we will give an overview and update on this in Section 2. Furthermore we give a new combinatorial characterization of almost bent functions in terms of the number of solutions to a certain system of equations (similar to such a characterization of APN functions), and a new characterization of crooked functions in terms of the Fourier transform. First we introduce some notation which will be used throughout the paper. Let V be an n-dimensional space over the eld GF (2); and let N = 2n = jV j. By h ; i we shall denote the standard inner product on V . By jX j we denote the size of a nite set X . Let f : V ! V be any function. For 0 6= a 2 V , we denote by Ha(f), or simply Ha, the set Ha = Ha(f) = ff(x) + f(x+ a) j x 2 V g: The Fourier transform (also called Walsh transform) f : V V ! IR of f is de ned by the formula f (a; b) = X