Correlation-immune and resilient functions over a finite alphabet and their applications in cryptography

We extend the notions of correlation-immune functions and resilient functio ns to functions over any finite alphabet. A previous result due to Gopalakr ishnan and Stinson is generalized as we give an orthogonal array characteri zation, a Fourier transform and a matrix characterization for correlation-i mmune and resilient functions over any finite alphabet endowed with the str ucture of an Abelian group. We then point out the existence of a tradeoff b etween the degree of the algebraic normal form and the correlation-immunity order of any function defined on a finite field and we construct some infi nite families of t-resilient functions with optimal nonlinearity which are particularly well-suited for combining linear feedback shift registers. We also point out the link between correlation-immune functions and some crypt ographic objects as perfect local randomizers and multipermutations.