Constructing Families of <I>ε</I>-Approximate <I>k</I>-Wise Independent Permutations

Toshiya ITOH; Yoshinori TAKEI; Jun TARUI

Constructing Families of ε-Approximate k-Wise Independent Permutations

Toshiya ITOH, Yoshinori TAKEI, Jun TARUI

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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 S_n 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 S_n 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 S_n of ε-approximate k-wise independent permutations such that || = ; (3) for any constant ε> 0 and any n = p^m - 1 with p prime, it is possible to construct a polynomial time samplable family S_n of ε-approximate pairwise independent permutations such that || = O(n(n - 1)/ε); (4) for any constant ε> 0 and any n = p^m with p prime, it is possible to construct a polynomial time samplable family S_n 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.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E87-A No.5 pp.993-1003

Publication Date: 2004/05/01

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Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)

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Toshiya ITOH, Yoshinori TAKEI, Jun TARUI, "Constructing Families of ε-Approximate k-Wise Independent Permutations" in IEICE TRANSACTIONS on Fundamentals, vol. E87-A, no. 5, pp. 993-1003, May 2004, doi: .
Abstract: 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 S_n 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 S_n 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 S_n of ε-approximate k-wise independent permutations such that || = ; (3) for any constant ε> 0 and any n = p^m - 1 with p prime, it is possible to construct a polynomial time samplable family S_n of ε-approximate pairwise independent permutations such that || = O(n(n - 1)/ε); (4) for any constant ε> 0 and any n = p^m with p prime, it is possible to construct a polynomial time samplable family S_n 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.
URL: https://globals.ieice.org/en_transactions/fundamentals/10.1587/e87-a_5_993/_p

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@ARTICLE{e87-a_5_993,
author={Toshiya ITOH, Yoshinori TAKEI, Jun TARUI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Constructing Families of ε-Approximate k-Wise Independent Permutations},
year={2004},
volume={E87-A},
number={5},
pages={993-1003},
abstract={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 S_n 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 S_n 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 S_n of ε-approximate k-wise independent permutations such that || = ; (3) for any constant ε> 0 and any n = p^m - 1 with p prime, it is possible to construct a polynomial time samplable family S_n of ε-approximate pairwise independent permutations such that || = O(n(n - 1)/ε); (4) for any constant ε> 0 and any n = p^m with p prime, it is possible to construct a polynomial time samplable family S_n 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.},
keywords={},
doi={},
ISSN={},
month={May},}

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TY - JOUR
TI - Constructing Families of ε-Approximate k-Wise Independent Permutations
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 993
EP - 1003
AU - Toshiya ITOH
AU - Yoshinori TAKEI
AU - Jun TARUI
PY - 2004
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E87-A
IS - 5
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - May 2004
AB - 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 S_n 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 S_n 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 S_n of ε-approximate k-wise independent permutations such that || = ; (3) for any constant ε> 0 and any n = p^m - 1 with p prime, it is possible to construct a polynomial time samplable family S_n of ε-approximate pairwise independent permutations such that || = O(n(n - 1)/ε); (4) for any constant ε> 0 and any n = p^m with p prime, it is possible to construct a polynomial time samplable family S_n 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.
ER -