The notion of k-wise independent permutations has several applications. From the practical point of view, it often suffices to consider almost (i.e., ε-approximate) k-wise independent permutation families rather than k-wise independent permutation families, however, we know little about how to construct families of ε-approximate k-wise independent permutations of small size. For any n > 0, let Sn be the set of all permutations on {0,1,..., n - 1}. In this paper, we investigate the size of families of ε-approximate k-wise independent permutations and show that (1) for any constant ε 0, if a family Sn of permutations is ε-approximate k-wise independent, then || n(n - 1) (n - k + 1) if ε< 1; || {n(n - 1) (n - k + 1)}/(1 +ε) otherwise; (2) for any constant 0< ε 1, there exists a family Sn of ε-approximate k-wise independent permutations such that || = ; (3) for any constant ε> 0 and any n = pm - 1 with p prime, it is possible to construct a polynomial time samplable family Sn of ε-approximate pairwise independent permutations such that || = O(n(n - 1)/ε); (4) for any constant ε> 0 and any n = pm with p prime, it is possible to construct a polynomial time samplable family Sn of ε-approximate 3-wise independent permutations such that || = O(n(n - 1)(n - 2)/ε). Our results are derived by combinatorial arguments, i.e., probabilistic methods and linear algebra methods.

We show that the permanent of an m n rectangular matrix can be computed with O(n 2m 3m) multiplications and additions. Asymptotically, this is better than straightforward extensions of the best known algorithms for the permanent of a square matrix when m/n log3 2 and n .