Codes, graphs, and schemes from nonlinear functions

Authors:
E. R. van Dam;D. Fon-Der-Flaass
Affiliations:
Department of Econometrics and O.R., Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands;Institute of Mathematics, Novosibirsk 90, 630090, Russia
Venue:
European Journal of Combinatorics
Year:
2003

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Abstract

We consider functions on binary vector spaces which are far from linear functions in different senses. We compare three existing notions: almost perfect nonlinear 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 AB functions in terms of the number of solutions to a certain system of equations, and a characterization of CF 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.