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The Advantage of Truncated Permutations

  • Shoni Gilboa
  • Shay GueronEmail author
Conference paper
  • 545 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11527)

Abstract

Constructing a Pseudo Random Function (PRF) from a pseudorandom permutation is a fundamental problem in cryptology. Such a construction, implemented by truncating the last m bits of permutations of \(\{0, 1\}^{n}\) was suggested by Hall et al. (1998). They conjectured that the distinguishing advantage of an adversary with q quesires, \(\mathbf{Adv}_{n, m} (q)\), is small if \(q = o (2^{(m+n)/2})\), established an upper bound on \(\mathbf{Adv}_{n, m} (q)\) that confirms the conjecture for \(m < n/7\), and also declared a general lower bound \(\mathbf{Adv}_{n,m}(q)=\varOmega (q^2/2^{n+m})\). The conjecture was essentially confirmed by Bellare and Impagliazzo in 1999. Nevertheless, the problem of estimating \(\mathbf{Adv}_{n, m} (q)\) remained open. Combining the trivial bound 1, the birthday bound, and a result by Stam (1978) leads to the following upper bound:
$$\mathbf{Adv}_{n,m}(q) \le O\left( \min \left\{ \frac{q^2}{2^n},\,\frac{q}{2^{\frac{n+m}{2}}},\,1\right\} \right) $$
This upper bound shows that the number of times that a truncated permutation can be used as a PRF can exceed the birthday bound by at least a factor of \(2^{m/2}\). In this paper we show that this upper bound is tight for every \(m<n\) and \(q>1\). This, in turn, verifies that the converse to the conjecture of Hall et al. is also correct, i.e., that \(\mathbf{Adv}_{n, m} (q)\) is negligible only for \(q = o (2^{(m+n)/2})\).

Keywords

Pseudo random permutations Pseudo random functions Advantage 

Notes

Acknowledgments

We thank Ron Peled for fruitful discussion.

This research was partially supported by: The Israel Science Foundation (grant No. 1018/16); The BIU Center for Research in Applied Cryptography and Cyber Security, in conjunction with the Israel National Cyber Bureau in the Prime Minister’s Office; The Center for Cyber Law and Policy at the University of Haifa in conjunction with the Israel National Cyber Directorate in the Prime Minister’s Office.

References

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.The Open University of IsraelRa’ananaIsrael
  2. 2.University of HaifaHaifaIsrael
  3. 3.AmazonSeattleUSA

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