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A study of correlation-immune, resilient and related cryptographic functions
Abstract
Orthogonal arrays (OAs) are basic combinatorial structures, which appear under various disguises in cryptology and the theory of algorithms. Among their applications are resilient and correlation-immune functions, derandomization of algorithms, random pattern testing of VLSI chips, authentication codes, universal hash functions, threshold schemes, and perfect local randomisers. This dissertation is a study of correlation-immune and resilient functions from a combinatorial point of view, emphasizing their connections to orthogonal arrays. An (n,m,t) resilient function is a function from n variables to m variables such that every possible output m-tuple is equally likely to occur when the values of t arbitrary inputs are fixed by an opponent and the remaining $n-t$ input variables are chosen independently at random. We provide three characterizations of non-binary correlation-immune and resilient functions; one in terms of the structure of certain associated matrix, one using Fourier transforms, and one based on orthogonal arrays. New bounds on orthogonal arrays and resilient functions are developed using Delsarte's linear programming technique. Several methods of construction of resilient functions are presented. Some of these methods of construction use linear and non-linear codes; the rest are constructions of new resilient functions from old. A table of bounds for resilient functions is also presented. New explicit bounds on orthogonal arrays and resilient functions are derived as corollaries of the linear programming technique and these bounds are shown to be as powerful as the linear programming bound itself for many parametric situations. The duality between the bounds on codes and the bounds on orthogonal arrays is studied. Also, several classes of optimal resilient functions are constructed. The constructions involve MDS codes, perfect codes and the theory of anticodes. A conjecture of Chor et al concerning symmetric resilient functions is disproved by construction of an infinite class of counterexamples. A short proof of a recent result by Lloyd concerning the non-existence of certain cryptographic functions is also presented.
Subject Area
Computer science|Mathematics
Recommended Citation
Gopalakrishnan, K, "A study of correlation-immune, resilient and related cryptographic functions" (1994). ETD collection for University of Nebraska-Lincoln. AAI9510970.
https://digitalcommons.unl.edu/dissertations/AAI9510970