Nonlinearity Criteria for Cryptographic Functions

Nonlinearity criteria for Boolean functions are classified in view of their suitability for cryptographic design. The classification is set up in terms of the largest transformation group leaving a criterion invariant. In this respect two criteria turn out to be of special interest, the distance to linear structures and the distance to affine functions, which are shown to be invariant under all affine transformations. With regard to these criteria an optimum class of functions is considered. These functions simultaneously have maximum distance to affine functions and maximum distance to linear structures, as well as minimum correlation to affine functions. The functions with these properties are proved to coincide with certain functions known in combinatorial theory, where they are called bent functions. They are shown to have practical applications for block ciphers as well as stream ciphers. In particular they give rise to a new solution of the correlation problem.