Abstract

Informally, a permutation family is k-wise independent if any k distinct points are mapped to any k distinct points equally likely; a permutation family is k-rankwise indepen-dent if any k distinct points are mapped to any order equally likely; a permutation family is k-restricted rain-wise indepen-dent if any element in at most k distinct points is mapped to be the minimum among those images equally likely (note that n-restricted rain-wise independence is referred to sim-ply as rain-wise independence). These permutation families have theoretical applications, e.g., derandomization, univer-sal hashing, search engines, etc. In this paper, we precisely investigate the size of those permutation families and show that (1) for lower bounds, IlCll> n- 1 and IlCll __ _ (n/2) tk/4J for families C of k-restricted rain-wise and k-rankwise inde-pendent permutations on {0, 1,..., n- 1}, respective!~,; (2) for upper bounds, IlCll __ _ n c1+¢2/tg,)) ~ and IlCll-- n °ok / ~"~) for families C of k-restricted rain-wise and k-rankwise in-dependent permutations on {0, 1,...,n- 1}, respectively; (3) for optimal upper bounds, IlCll = Icm(n, n- 1,..., 1) = e n-°(n) for families C of rain-wise independent permutations on {0, 1,..., n- 1}, which achieves the known lower bound.

Citations