Orthogonal arrays, resilient functions, error-correcting codes, and linear programming bounds

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Orthogonal arrays (OAs) are basic combinatorial structures, which appear under various disguises in cryptology and the theory of algorithms. Among their applications are universal hashing, authentication codes, resilient and correlation-immune functions, derandomization of algorithms, and perfect local randomizers. In this paper, we give new explicit bounds on the size of orthogonal arrays using Delsarte's linear programming method. Specifically, we prove that the minimum number of rows in a binary orthogonal array of length n and strength t is at least 2n - (n2n-1/t + 1) and also at least 2n - (2n-2(n + 1)/[t+1/2]). We also prove that these bounds are as powerful as the linear programming bound itself for many parametric situations. An (n, m, t)-resilient function is a function f : {0, 1}n → {0, 1}m 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 bits are chosen independently at random. A basic problem is to maximize t given m and n, i.e., to determine the largest value of t such that an (n, m, t)-resilient function exists. In this paper, we obtain upper and lower bounds for the optimal values of t where 1 ≤ n ≤ 25 and 1 ≤ m ≤ n. The upper bounds are derived from Delsarte's linear programming bound, and the lower bounds come from constructions based on error-correcting codes. We also obtain new explicit upper bounds for the optimal values of t. It was proved by Chor et al. in [Proc. 26th IEEE Symp. on Foundations of Computer Science, 1985, pp. 396-407] that an (n, 2, t)-resilient function exists if and only if t < [2n/3]. This result was generalized by Friedman [Proc. 33rd IEEE Symp. on Foundations of Computer Science, 1992, pp. 314-319], who proved a bound for general m. We also prove some new bounds, and complete the determination of the optimal resiliency of resilient functions with m = 3 and most of the cases for m = 4. Several other infinite classes of (optimal) resilient functions are also constructed using the theory of anticodes.

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SIAM Journal on Discrete Mathematics