New constructions of resilient and correlation immune Boolean functions achieving upper bounds on nonlinearity (2001) [6 citations — 1 self]
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Abstract:
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 benchmark in the design of resilient Boolean functions (which optimizes Siegenthaler 's 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 were justied as important open questions in Crypto 2000. Also we provide methods to generate an innite sequence of Boolean functions on n = 7 + 3i variables (i 0) with order of resiliency m = 2 + 2i, algebraic degree 4 + i and nonlinearity 2 n 1 2 m+1, which were not known earlier. We conclude with a few interesting construction results on unbalanced correlation immune functions of 5 and 6 variables.
