A Family of Cryptographically Significant Boolean Functions Based on the Hidden Weighted Bit Function

Based on the hidden weighted bit function, we propose a family of cryptographically significant Boolean functions. We investigate its algebraic degree and use Schur polynomials to study its algebraic immunity. For a subclass of this family, we deduce a lower bound on its nonlinearity. Moreover, we give an infinite class of balanced functions with very good cryptographic properties: optimum algebraic degree, optimum algebraic immunity, high nonlinearity (higher than the Carlet-Feng function and the function proposed by [25]) and a good behavior against fast algebraic attacks. These functions seem to have the best cryptographic properties among all currently known functions.

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