Efficient Strongly Universal and Optimally Universal Hashing

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 severail 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.

[1]  Avi Wigderson,et al.  The amazing power of pairwise independence (abstract) , 1994, STOC '94.

[2]  Jörg Keller,et al.  Simulation-based Comparison of Hash Functions for Emulated Shared Memory , 1993, PARLE.

[3]  János Komlós,et al.  Storing a sparse table with O(1) worst case access time , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[4]  Uzi Vishkin,et al.  On Parallel Hashing and Integer Sorting , 1991, J. Algorithms.

[5]  Martin Dietzfelbinger,et al.  Universal Hashing and k-Wise Independent Random Variables via Integer Arithmetic without Primes , 1996, STACS.

[6]  Phillip Rogaway.  Bucket Hashing and its Application to Fast Message Authentication , 1995, CRYPTO.

[7]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[8]  Douglas R. Stinson,et al.  Combinatorial Techniques for Universal Hashing , 1994, J. Comput. Syst. Sci..

[9]  Hugo Krawczyk,et al.  LFSR-based Hashing and Authentication , 1994, CRYPTO.

[10]  Larry Carter,et al.  Universal Classes of Hash Functions , 1979, J. Comput. Syst. Sci..

[11]  Noga Alon,et al.  Addendum to "Simple Construction of Almost k-wise Independent Random Variables" , 1993, Random Struct. Algorithms.

[12]  Friedhelm Meyer auf der Heide,et al.  Dynamic perfect hashing: upper and lower bounds , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[13]  Noga Alon,et al.  Simple Construction of Almost k-wise Independent Random Variables , 1992, Random Struct. Algorithms.

[14]  Martti Penttonen,et al.  A Reliable Randomized Algorithm for the Closest-Pair Problem , 1997, J. Algorithms.

[15]  Dilip V. Sarwate,et al.  A Note on Universal Classes of Hash Functions , 1980, Inf. Process. Lett..

[16]  Larry Carter,et al.  New classes and applications of hash functions , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[17]  Friedhelm Meyer auf der Heide,et al.  Dynamic Perfect Hashing: Upper and Lower Bounds , 1994, SIAM J. Comput..

[18]  Douglas R. Stinson,et al.  On the Connections Between Universal Hashing, Combinatorial Designs and Error-Correcting Codes , 1995, Electron. Colloquium Comput. Complex..

[19]  Rajeev Raman,et al.  Sorting in linear time? , 1995, STOC '95.