Abstract
Constructions of k-wise almost independent permutations have been receiving a growing amount of attention in recent years. However, unlike the case of k-wise independent functions, the size of previously constructed families of such permutations is far from optimal. This paper gives a new method for reducing the size of families given by previous constructions. Our method relies on pseudorandom generators for space-bounded computations. In fact, all we need is a generator, that produces “pseudorandom walks” on undirected graphs with a consistent labelling. One such generator is implied by Reingold’s log-space algorithm for undirected connectivity (Reingold/Reingold et al. in Proc. of the 37th/38th Annual Symposium on Theory of Computing, pp. 376–385/457–466, 2005/2006). We obtain families of k-wise almost independent permutations, with an optimal description length, up to a constant factor. More precisely, if the distance from uniform for any k tuple should be at most δ, then the size of the description of a permutation in the family is \(O(kn+\log \frac{1}{\delta})\) .
Similar content being viewed by others
References
Aldous, D., Diaconis, P.: Shuffling cards and stopping times. Am. Math. Mon. 93, 333–348 (1986)
Aldous, D., Fill, J.A.: Reversible Markov chains and random walks on graphs. http://www.stat.berkeley.edu/users/aldous/RWG/book.html
Alon, N., Spencer, J.: The Probabilistic Method. Wiley, New York (1992)
Azar, Y., Motwani, R., Naor, J.: Approximating probability distributions using small sample spaces. Combinatorica 18(2), 151–171 (1998)
Bar-Noy, A., Naor, J., Schieber, B.: Pushing dependent data in clients-providers-servers systems. Wirel. Netw. 9(5), 421–430 (2003)
Black, J., Rogaway, P.: Ciphers with arbitrary finite domains. In: Topics in Cryptology—CT-RSA 2002. Lecture Notes in Computer Science, vol. 2271, pp. 114–130. Springer, Berlin (2002)
Broder, A.Z., Charikar, M., Frieze, A.M., Mitzenmacher, M.: Min-wise independent permutations. J. Comput. Syst. Sci. 60(3), 630–659 (2000) (preliminary version STOC 2000)
Broder, A.Z., Glassman, S.C., Manasse, M.S., Zweig, G.: Syntactic clustering of the Web. Comput. Netw. 29, 1157–1166 (1997)
Brodsky, A., Hoory, S.: Simple permutations mix even better. Random Struct. Algorithms 32(3), 274–289 (2007). Arxiv:math.CO/0411098
Cameron, P.J.: Finite permutation groups and finite simple groups. Bull. Lond. Math. Soc. 13, 1–22 (1981)
Dietzfelbinger, M., Woelfel, P.: Almost random graphs with simple hash functions. In: Proc. of the 35th Annual ACM Symposium on Theory of Computing, pp. 629–638 (2003)
Gilbert, A.C., Guha, S., Indyk, P., Kotidis, Y., Muthukrishnan, S., Strauss, M.: Fast, small-space algorithms for approximate histogram maintenance. In: Proc. of the 34th Annual ACM Symposium on Theory of Computing, pp. 389–398 (2002)
Goldreich, O., Goldwasser, S., Nussboim, A.: On the implementation of huge random objects. In: Proc. of the 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 68–79 (2003)
Gowers, W.T.: An almost m-wise independent random permutation of the cube. Comb. Probab. Comput. 5(2), 119–130 (1996)
Hoory, S., Magen, A., Myers, S., Rackoff, C.: Simple permutations mix well. In: The 31st International Colloquium on Automata, Languages and Programming (ICALP). Lecture Notes in Computer Science, vol. 3142, pp. 770–781. Springer, Berlin (2004)
Indyk, P.: Stable distributions, pseudorandom generators, embeddings and data stream computation. In: Proc. of the 41st Annual IEEE Symposium on Foundations of Computer Science, pp. 189–197 (2000)
Itoh, T., Takei, Y., Tarui, J.: On permutations with limited independence. In: Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 137–146 (2000)
Itoh, T., Takei, Y., Tarui, J.: On the sample size of k-restricted min-wise independent permutations and other k-wise distributions. In: Proc. of the 35th Annual ACM Symposium on Theory of Computing, pp. 710–719 (2003)
Kaplan, E., Naor, M., Reingold, O.: Derandomized constructions of k-wise (almost) independent permutations. In: The 9th International Workshop on Randomization and Computation (RANDOM). Lecture Notes in Computer Science, vol. 3624, pp. 354–365. Springer, Berlin (2005)
Kassabov, M.: Symmetric groups and expanders. arXiv:math.GR/0503204
Koller, D., Megiddo, N.: Constructing small sample spaces satisfying given constraints. SIAM J. Discrete Math. 7(2), 260–274 (1994)
Luby, M., Rackoff, C.: How to construct pseudorandom permutations and pseudorandom functions. SIAM J. Comput. 17, 373–386 (1988)
Maurer, U.M., Pietrzak, K.: The security of many-round Luby–Rackoff pseudo-random permutations. In: Advances in Cryptology—EUROCRPYT ’2003. Lecture Notes in Computer Science, vol. 2656, pp. 544–561. Springer, Berlin (2003)
Maurer, U.M., Pietrzak, K.: Composition of random systems: when two weak make one strong. In: First Theory of Cryptography Conference, TCC 2004. Lecture Notes in Computer Science, vol. 2951, pp. 410–427. Springer, Berlin (2004)
Morris, B.: On the mixing time for the Thorp shuffle. In: Proc. of the 37th Annual ACM Symposium on Theory of Computing, pp. 403–412 (2005)
Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, New York (1995)
Myers, S.: Black-box composition does not imply adaptive security. In: Advances in Cryptology—EUROCRYPT ’2004. Lecture Notes in Computer Science, vol. 3027, pp. 189–203. Springer, Berlin (2004)
Naor, M., Reingold, O.: On the construction of pseudorandom permutations: Luby–Rackoff revisited. J. Cryptol. 12(1), 29–66 (1999)
Nisan, N.: Pseudorandom generators for space-bounded computation. Combinatorica 12(4), 449–461 (1992)
Nisan, N., Zuckerman, D.: Randomness is linear in space. J. Comput. Syst. Sci. 52(1), 43–52 (1996)
Ostlin, A., Pagh, R.: Uniform hashing in constant time and linear space. In: Proc. of the 35th Annual ACM Symposium on Theory of Computing, pp. 622–628 (2003)
Patarin, J.: Improved security bounds for pseudorandom permutations. In: Proc. of the 4th ACM Conference on Computer and Communications Security, pp. 142–150 (1997)
Patarin, J.: Luby–Rackoff: 7 rounds are enough for 2n(1−ε) security. In: Advances in Cryptology—CRYPTO 2003. Lecture Notes in Computer Science, vol. 2729, pp. 513–529. Springer, Berlin (2003)
Patarin, J.: Security of random Feistel schemes with 5 or more rounds. In: Advances in Cryptology—CRYPTO ’2004. Lecture Notes in Computer Science, vol. 3152, pp. 106–122. Springer, Berlin (2004)
Pietrzak, K.: Composition does not imply adaptive security. In: Advances in Cryptology—CRYPTO ’2005. Lecture Notes in Computer Science, vol. 3621, pp. 55–65. Springer, Berlin (2005)
Pinkas, B.: Communication preserving cryptographic protocols. Ph.D. dissertation, Weizmann Institute of Science (1999)
Reingold, O.: Undirected ST-connectivity in log-space. In: Proc. of the 37th Annual ACM Symposium on Theory of Computing, pp. 376–385 (2005)
Reingold, O., Trevisan, L., Vadhan, S.: Pseudorandom walks in biregular graphs and the RL vs. L problem. In: Proc. of the 38th Annual ACM Symposium on Theory of Computing, pp. 457–466 (2006)
Robinson, D.J.S.: A Course in the Theory of Groups, 2nd edn. Springer, New York (1996)
Rozenman, E., Vadhan, S.: Derandomized squaring of graphs. In: The 9th International Workshop on Randomization and Computation (RANDOM). Lecture Notes in Computer Science, vol. 3624, pp. 436–447. Springer, Berlin (2005)
Rudich, S.: Limits on the provable consequences of one-way functions. Ph.D. thesis, U.C. Berkeley (1988)
Russell, A., Wang, H.: How to fool an unbounded adversary with a short key. In: Advances in Cryptology—EUROCRYPT ’2002. Lecture Notes in Computer Science, vol. 2332, pp. 133–148. Springer, Berlin (2002)
Saks, M., Srinivasan, A., Zhou, S., Zuckerman, D.: Low discrepancy sets yield approximate min-wise independent permutation families. Inf. Process. Lett. 73, 29–32 (2000)
Siegel, A.: On universal classes of extremely random constant-time hash functions. SIAM J. Comput. 33(3), 505–543 (2004)
Sinclair, A.: Improved bounds for mixing rates of Markov chains and multicommodity flow. Comb. Probab. Comput. 1(4), 351–370 (1992)
Sivakumar, D.: Algorithmic derandomization via complexity theory. In: Proc. of the 34th Annual ACM Symposium on Theory of Computing, pp. 619–626 (2002)
Thorp, E.: Nonrandom shuffling with applications to the game of Faro. J. Am. Stat. Assoc. 68, 842–847 (1973)
Author information
Authors and Affiliations
Corresponding author
Additional information
A preliminary version of this paper appeared in Random 2005 [19].
The research of M. Naor was supported in part by a grant from the Israel Science Foundation. The research of O. Reingold was supported by US–Israel Binational Science Foundation Grants 2002246 and 2006060.
Rights and permissions
About this article
Cite this article
Kaplan, E., Naor, M. & Reingold, O. Derandomized Constructions of k-Wise (Almost) Independent Permutations. Algorithmica 55, 113–133 (2009). https://doi.org/10.1007/s00453-008-9267-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-008-9267-y