Highly Nonlinear Balanced Boolean Functions with Very Good Autocorrelation Property

Abstract

Constructing highly nonlinear balanced Boolean functions with very good autocorrelation property is an interesting open question. In this direction we use the measure Δ_f, the highest magnitude of all autocorrelation coefficients for a function f. We provide balanced functions f with currently best known nonlinearity and Δ_f values together. Our results for 15-variable functions disprove the conjecture proposed by Zhang and Zheng (1995) for different ranges of nonlinearity, where our constructions are based on modifications of Patterson-Wiedemann (1983) functions. Also we propose a simple bent based construction technique to get functions with very good Δ_f values for odd number of variables. This construction has a root in Kerdock Codes. Moreover, our construction on even number of variables is a recursive one and we conjecture (similar to Dobbertin's conjecture (1994) with respect to nonlinearity) that this provides the minimum possible value of Δ_f for a balanced function f on even number of variables.