New Constructions of Resilient and Correlation Immune Boolean Functions Achieving Upper Bound on Nonlinearity

Recently, weight divisibility results on resilient and correlation immune Boolean functions have received a lot of attention. These results have direct consequences towards the upper bound on nonlinearity of resilient and correlation immune Boolean functions of certain order. Now the clear requirement in the design of resilient Boolean functions (which optimizes Siegenthalerʹs inequality) is to provide results which attain the upper bound on nonlinearity. Here we construct a 7-variable, 2-resilient Boolean function with nonlinearity 56. This solves the maximum nonlinearity issue for 7-variable functions with any order of resiliency. Using this 7-variable function, we also construct a 10-variable, 4-resilient Boolean function with nonlinearity 480. Construction of these two functions was posed as important open questions in Crypto 2000. Also, we provide methods to generate an infinite sequence of Boolean functions on n = 7 + 3i variables (i ≥ 0) with order of resiliency m = 2 + 2i, algebraic degree 4 + i and nonlinearity 2n-1 - 2m+1, which were not known earlier. We conclude with constructions of some unbalanced correlation immune functions of 5 and 6 variables which attain the upper bound on nonlinearity.