Construction of Boolean functions with excellent cryptographic criteria using bivariate polynomial representation

A class of (Formula presented.) -variable Boolean functions with excellent cryptographic criteria is proposed in this paper, using bivariate polynomial representation (BPR). By comparing known Boolean functions created by the ‘BPR-method’, three conjectures on the relationship between cryptographic criteria and parameter settings are given as guidelines to the research. Then on the basis of certain combinatorial facts and computer experiments, we prove that our functions possess the optimal algebraic immunity k, and validate that, at least for (Formula presented.) , the functions preserve almost perfect immunity against fast algebraic attacks. In addition, we show the functions to be 1-resilient with the maximum algebraic degree of (Formula presented.) and give a proof of the lower bound for nonlinearity by means of Gauss sum. Our functions demonstrate great performance in meeting the desired cryptographic criteria for use in the filter model of pseudorandom generators.