Highly Nonlinear Resilient Functions Optimizing Siegenthaler's Inequality

Siegenthaler proved that an n input 1 output, m-resilient (balanced mth order correlation immune) Boolean function with algebraic degree d satisfies the inequality : m + d ≤ n - 1. We provide a new construction method using a small set of recursive operations for a large class of highly nonlinear, resilient Boolean functions optimizing Siegenthaler's inequality m + d = n - 1. Comparisons to previous constructions show that better nonlinearity can be obtained by our method. In particular, we show that as n increases, for almost all m, the nonlinearity obtained by our method is better than that provided by Seberry et al in Eurocrypt'93. For small values of n, the functions constructed by our method is better than or at least comparable to those constructed using the methods provided in papers by Filiol et al and Millan et al in Eurocrypt'98. Our technique can be used to construct functions on large number of input variables with simple hardware implementation.