Improving the Lower Bound on the Higher Order Nonlinearity of Boolean Functions With Prescribed Algebraic Immunity

The recent algebraic attacks have received a lot of attention in cryptographic literature. The algebraic immunity of a Boolean function quantifies its resistance to the standard algebraic attacks of the pseudorandom generators using it as a nonlinear filtering or combining function. Very few results have been found concerning its relation with the other cryptographic parameters or with the rth-order nonlinearity. As recalled by Carlet at CRYPTO´06, many papers have illustrated the importance of the r th-order nonlinearity profile (which includes the first-order nonlinearity). The role of this parameter relatively to the currently known attacks has been also shown for block ciphers. Recently, two lower bounds involving the algebraic immunity on the rth-order nonlinearity have been shown by Carlet . None of them improves upon the other one in all situations. In this paper, we prove a new lower bound on the rth-order nonlinearity profile of Boolean functions, given their algebraic immunity, that improves significantly upon one of these lower bounds for all orders and upon the other one for low orders.