Nonlinear Vector Resilient Functions

An (n, m, k)-resilient function is a function f : F2 → F2 such that every possible output m-tuple is equally likely to occur when the values of k arbitrary inputs are fixed by an adversary and the remaining n − k input bits are chosen independently at random. In this paper we propose a new method to generate a (n + D + 1, m, d − 1)-resilient function for any non-negative integer D whenever a [n, m, d] linear code exists. This function has algebraic degree D and nonlinearity at least 2 − 2b √ 2n+D+1c + 2. If we apply this method to the simplex code, we can get a (t(2 − 1) + D + 1, m, t2 − 1)-resilient function with algebraic degree D for any positive integers m, t and D. Note that if we increase the input size by D in the proposed construction, we can get a resilient function with the same parameter except algebraic degree increased by D.