Three Characterizations of Non-binary Correlation-Immune and Resilient Functions

A function f (X 1 ; X 2 ; : : : ; X n) is said to be t th-order correlation-immune if the ran-ally, if all possible outputs are equally likely, then f is called a t ?resilient function. In this paper, we provide three diierent characterizations of t th-order correlation immune functions and resilient functions where the random variable is over GF (q). The rst is in terms of the structure of a certain associated matrix. The second characterization involves Fourier transforms. The third characterization establishes the equivalence of resilient functions and large sets of orthogonal arrays. 1 Deenitions Let GF(q) denote the Galois Field with q elements, where q = p a is a prime power. Let f : GF (q)] n ?! GF (q)] m be a function and let fX 1 ; X 2 ; : : : ; X n g be the set of random input variables assuming values from GF(q) with independent equiprobable distributions (that is, every possible input vector occurs with equal probability 1=q n). Such a function is said to have balanced input. Let fZ 1 ; Z 2 ; : : : ; Z m g denote the set of output variables of the function f. The function f is said to be correlation-immune with respect to T f1; 2; : : : ; ng (or equivalently independent of T) if the probability distribution of the random variable