We give a series of constructions of correlation-immune function over finite fields. We prove that F/sub 2/ and F/sub 3/ are the only finite fields F/sub q/ with the property that every (n-1)th correlation-immune function in n>2 variables over F/sub q/ is linear. We also show that by choosing larger finite fields one can alleviate the tradeoff between the length of the linear equivalent and the order of correlation immunity. This is useful for the design of various cryptosystems.

Through the use of mutually orthogonal frequency squares, a generalization of the original Golomb-Posner code is constructed. The minimum distance and weight distribution of the generalized code when it is constructed from a complete set of orthogonal squares over a finite field are also determined.>

A necessary and sufficient condition is established for nonsingularity of a feedback shift-register of degree at most two over a finite field. Several relations between permutation polynomials and nonsingular feedback shift-registers over finite fields are also studied. Four theorems are given and discussed.>