On an Almost-Universal Hash Function Family with Applications to Authentication and Secrecy Codes

摘要

Universal hashing, discovered by Carter and Wegman in 1979, has many important applications in computer science. MMH*, which was shown to be Delta-universal by Halevi and Krawczyk in 1997, is a well-known universal hash function family. We introduce a variant of MMH*, that we call GRDH, where we use an arbitrary integer n 1 instead of prime p and let the keys x = x(1), ... , x(k) is an element of Z(n)(k) satisfy the conditions gcd(x(i), n) = t(i) (1 = i = k), where t(1), ... , t(k) are given positive divisors of n. Then via connecting the universal hashing problem to the number of solutions of restricted linear congruences, we prove that the family GRDH is an epsilon-almost-Delta-universal family of hash functions for some epsilon 1 if and only if n is odd and gcd(x(i), n) = t(i) = 1 (1 = i = k). Furthermore, if these conditions are satisfied then GRDH is 1/p-1-almost-Delta-universal, where p is the smallest prime divisor of n. Finally, as an application of our results, we propose an authentication code with secrecy scheme which strongly generalizes the scheme studied by Alomair et al.