Construction of Nonlinear Boolean Functions with Important Cryptographic Properties

This paper addresses the problem of obtaining new construction methods for cryptographically significant Boolean functions. We show that for each positive integer m, there are infinitely many integers n (both odd and even), such that it is possible to construct n-variable, m-resilient functions having nonlinearity greater than 2n-1 -2[n/2]. Also we obtain better results than all published works on the construction of n-variable, m-resilient functions, including cases where the constructed functions have the maximum possible algebraic degree n - m - 1. Next we modify the Patterson-Wiedemann functions to construct balanced Boolean functions on n-variables having nonlinearity strictly greater than 2n-1 - 2n-1/2 for all odd n ≥ 15. In addition, we consider the properties strict avalanche criteria and propagation characteristics which are important for design of S-boxes in block ciphers and construct such functions with very high nonlinearity and algebraic degree.