We consider functions on binary vector spaces which are far from linear functions in different 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 differential 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, difference sets, distance regular graphs, symmetric association schemes, and uniformly packed (BCH and Preparata) codes.
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Paper provided by Tilburg University, Faculty of Economics and Business Administration in its series Research Memorandum with number
790.