On algebraic immunity of Maiorana-McFarland like functions and applications of algebraic attacks to some stream cipher schemes

This paper addresses the resistance of the Maiorana-McFarland like functions against recently introduced algebraic attacks. In [4] it was proved that for any Boolean function f in n variables there will exist low degree multipliers, that is a function g such that deg(fg) = dn 2 e. We show that the functions in Maiorana-McFarland class admit nontrivial low degree multipliers whose degree is always strictly less than dn 2 e. This implies that this class does not posses the optimized algebraic immunity. Furthermore, we consider a class of additive stream ciphers known as nonlinear combining generator, see [12, 16]. This scheme uses a nonlinear resilient function F : GF (2)n → GF (2)m, whose n binary inputs are some linear combinations of secret key bits generated by n linear feedback shift registers (LFSRs). In connection to this design we investigate two main classes of functions used in this scheme. We show that increasing the number of outputs m results in the existence of annihilators of very low degree (quadratic or cubic) which in turn implies that running algebraic attacks on such ciphers becomes computationally feasible even for the key sizes of the order of 512 bits. The length of the ciphertext needed is in general less then few kilobytes. Keyword : Algebraic attacks, Annihilators, Stream ciphers, Nonlinear combiner, Boolean function, Resiliency, Algebraic Degree.