Construction of Perfect Nonlinear and Maximally Nonlinear Multiple-Output Boolean Functions Satisfying Higher Order Strict Avalanche Criteria

Abstract

We consider the problem of constructing perfect nonlinear multiple-output Boolean functions satisfying higher order strict avalanche criteria (SAC). Our first construction is an infinite family of 2-output perfect nonlinear functions satisfying higher order SAC. This construction is achieved using the theory of bilinear forms and symplectic matrices. Next we build on a known connection between 1-factorization of a complete graph and SAC to construct more examples of 2- and 3-output perfect nonlinear functions. In certain cases, the constructed S-boxes have optimal tradeoff between the following parameters: numbers of input and output variables, nonlinearity, and order of SAC. In case the number of input variables is odd, we modify the construction for perfect nonlinear S-boxes to obtain a construction for maximally nonlinear S-boxes satisfying higher order SAC. Our constructions present the first examples of perfect nonlinear and maximally nonlinear multiple-output S-boxes satisfying higher order SAC. Finally, we present a simple method for improving the degree of the constructed functions with a small tradeoff in nonlinearity and the SAC property. This yields functions which have possible applications in the design of block ciphers.