Journal of Cryptology

, Volume 14, Issue 4, pp 231–253 | Cite as

Almost k -Wise Independent Sample Spaces and Their Cryptologic Applications

  • Kaoru  Kurosawa
  • Thomas  Johansson
  • Douglas R.  Stinson


An almost k -wise independent sample space is a small subset of m bit sequences in which any k bits are ``almost independent''. We show that this idea has close relationships with useful cryptologic notions such as multiple authentication codes (multiple A -codes), almost strongly universal hash families, almost k -resilient functions, almost correlation-immune functions, indistinguishable random variables and k -wise decorrelation bias of block ciphers.

We use almost k -wise independent sample spaces to construct new efficient multiple A -codes such that the number of key bits grows linearly as a function of k (where k is the number of messages to be authenticated with a single key). This improves on the construction of Atici and Stinson \cite{AS96}, in which the number of key bits is Ω (k 2 ) .

We introduce the concepts of ɛ -almost k -resilient functions and almost correlation-immune functions, and give a construction for almost k -resilient functions that has parameters superior to k -resilient functions. We also point out the connection between almost k -wise independent sample spaces and pseudorandom functions that can be distinguished from truly random functions, by a distinguisher limited to k oracle queries, with only a small probability. Vaudenay \cite{Vaudenay99} has shown that such functions can be used to construct block ciphers with a small decorrelation bias.

Finally, new bounds (necessary conditions) are derived for almost k -wise independent sample spaces, multiple A -codes and balanced ɛ -almost k -resilient functions.

Key words. Independent sample space, Resilient function, Universal hash family, Authentication code. 


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Copyright information

© International Association for Cryptologic Research 2001

Authors and Affiliations

  • Kaoru  Kurosawa
    • 1
  • Thomas  Johansson
    • 2
  • Douglas R.  Stinson
    • 3
  1. 1.Department of Communication and Integrated Systems, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552, Japan
  2. 2.Department of Information Technology, Lund University, PO Box 118, S-22100 Lund, Sweden thomas@it.lth.seSE
  3. 3.Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 dstinson@uwaterloo.caCA

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