The advantage of truncated permutations

Shoni Gilboa, Shay Gueron

Research output: Contribution to journalArticlepeer-review


Constructing a Pseudo Random Function (PRF) 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 queries, Advn,m(q), is small if q=o(2(n+m)∕2), established an upper bound on Advn,m(q) that confirms the conjecture for m<n∕7, and also declared a general lower bound Advn,m(q)=Ω(q2∕2n+m). The conjecture was essentially confirmed by Bellare and Impagliazzo (1999). Nevertheless, the problem of estimating Advn,m(q) remained open. Combining the trivial bound 1, the birthday bound, and a result of Stam (1978) leads to the upper bound [Formula presented] In this paper we show that this upper bound is tight for every 0≤m<n and any q. This, in turn, verifies that the converse to the conjecture of Hall et al. is also correct, i.e., that Advn,m(q) is negligible only for q=o(2(n+m)∕2).

Original languageEnglish
Pages (from-to)214-223
Number of pages10
JournalDiscrete Applied Mathematics
StatePublished - 15 May 2021

Bibliographical note

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© 2021 Elsevier B.V.


  • Pseudo random function advantage

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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