Efficient Strongly Universal and Optimally Universal Hashing

Extended Abstract
  • Philipp Woelfel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1672)

Abstract

New hash families are analyzed, mainly consisting of the hash functions ha,b : {0,... , u − 1} → {0,... , r − 1}, x ↦ ((ax + b) mod(kr)) div k. Universal classes of such functions have already been investigated in [5, 6], and used in several applications, e.g. [3,9]. The new constructions which are introduced here, improve in several ways upon the former results. Some of them achieve a smaller universality parameter, i.e., two keys collide under a randomly chosen function with a smaller probability. In fact, an optimally universal hash class is presented, which means that the universality parameter achieves the minimum possible value. Furthermore, the bound of the universality parameter of a known, almost strongly universal hash family is improved, and it is shown how to reduce the size of a known class, retaining its properties. Finally, a new composition technique for constructing hash classes for longer keys is presented. Its application leads to efficient hash families which consist of linear functions over the ring of polynomials over ℤm.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Alon, O. Goldreich, J. Håstad, and R. Peralta. Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms, 3:289–304, 1992.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    N. Alon, O. Goldreich, J. Håstad, and R. Peralta. Addendum to “simple constructions of almost k-wise independent random variables”. Random Structures and Algorithms, 4:119–120, 1993.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    A. Andersson, T. Hagerup, S. Nilsson, and R. Raman. Sorting in linear time? In Proc. of 25th ACM STOC, pages 427–436, 1995.Google Scholar
  4. 4.
    J. L. Carter and M. N. Wegman. Universal classes of hash functions. J. Comp. Syst. Sci., 18:143–154, 1979.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Dietzfelbinger. Universal hashing and-wise independent random variables via integer arithmetic without primes. In Proc. of 13th STACS, pages 569–580, 1996.Google Scholar
  6. 6.
    M. Dietzfelbinger, T. Hagerup, J. Katajainen, and M. Penttonen. A reliable randomized algorithm for the closest-pair problem. J. Alg., 25:19–51, 1997.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    M. Dietzfelbinger and M. Hühne. A dictionary implementation based on dynamic perfect hashing. Proc. of 5th DIMACS Chal. Worksh., 1996. To appear.Google Scholar
  8. 8.
    M. Dietzfelbinger, A. Karlin, K. Mehlhorn, F. M. auf der Heide, H. Rohnert, and R. E. Tarjan. Dynamic perfect hashing: Upper and lower bounds. SIAM J. Comput, 23:738–761, 1994.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Engelmann and Keller. Simulation-based comparison of hash functions for emulated shared memory. In Proc. of 5th PARLE, pages 1–11, 1993.Google Scholar
  10. 10.
    M. L. Fredman, J. Komlós, and E. Szemerédi. Storing a sparse table with O(l) worst case access time. J. Assoc. Comput. Mach., 31:538–544, 1984.MATHMathSciNetGoogle Scholar
  11. 11.
    H. Krawczyk. LFSR-based hashing and authentication. In Advances in Cryptology — CRYPTO’ 94, pages 129–139, 1994.Google Scholar
  12. 12.
    Y. Matias and U. Vishkin. On parallel hashing and integer sorting. J. Algorithms, 12:573–606, 1991.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    P. Rogaway. Bucket hashing and its application to fast message authentication. In Advances in Cryptology — CRYPTO’ 95, pages 29–42, 1995.Google Scholar
  14. 14.
    D. V. Sarwate. A note on universal classes of hash functions. Inf. Proc. Letters, 10:41–45, 1980.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    A. Schönhage and V. Strassen. Schnelle Multiplication großer Zahlen. Computing, 7:281–292, 1971.MATHCrossRefGoogle Scholar
  16. 16.
    D. R. Stinson. Combinatorial techniques for universal hashing. J. Comp. Syst. Sci., 48:337–346, 1994.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    D. R. Stinson. On the connections between universal hashing, combi natorial designs and error-correcting codes. Report TR95-052, ECCC, ftp://ftp.eccc.uni-trier.de/pub/eccc/reports/1995/TR95-052/Paper.ps, 1995.
  18. 18.
    M. N. Wegman and J. L. Carter. New classes and applications of hash functions. In Proc. of 20th IEEE FOCS, pages 175–182, 1979.Google Scholar
  19. 19.
    A. Wigderson. The amazing power of pairwise independence. In Proc. of 26th ACM STOC, pages 574–583, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Philipp Woelfel
    • 1
  1. 1.Lehrstuhl Informatik 2Universität DortmundDortmundGermany

Personalised recommendations