Cryptographic properties of the hidden weighted bit function

Abstract

The hidden weighted bit function (HWBF), introduced by R. Bryant in IEEE Trans. Comp. 40 and revisited by D. Knuth in Vol. 4 of The Art of Computer Programming, is a function that seems to be the simplest one with exponential Binary Decision Diagram (BDD) size. This property is interesting from a cryptographic viewpoint since BDD-based attacks are receiving more attention in the cryptographic community. But, to be usable in stream ciphers, the functions must also satisfy all the other main criteria. In this paper, we investigate the cryptographic properties of the HWBF and prove that it is balanced, with optimum algebraic degree and satisfies the strict avalanche criterion. We calculate its exact nonlinearity and give a lower bound on its algebraic immunity. Moreover, we investigate its normality and its resistance against fast algebraic attacks. The HWBF is simple, can be implemented efficiently, has a high BDD size and rather good cryptographic properties, if we take into account that its number of variables can be much larger than for other functions with the same implementation efficiency. Therefore, the HWBF is a good candidate for being used in real ciphers. Indeed, contrary to the case of symmetric functions, which allow such fast implementation but also offer to the attacker some specific possibilities due to their symmetry, its structure is not suspected to be related to such dedicated attacks.