The advantage of truncated permutations

Shoni Gilboa, Shay Gueron

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 (formula presented) was suggested by Hall et al. (1998). They conjectured that the distinguishing advantage of an adversary with q quesires, (formula presented), is small if (formula presented), established an upper bound on (formula presented) that confirms the conjecture for (formula presented), and also declared a general lower bound (formula presented). The conjecture was essentially confirmed by Bellare and Impagliazzo in 1999. Nevertheless, the problem of estimating (formula presented) remained open. Combining the trivial bound 1, the birthday bound, and a result by Stam (1978) leads to the following upper bound: (formula presented) 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 (formula presented). In this paper we show that this upper bound is tight for every (formula presented) and (formula presented). This, in turn, verifies that the converse to the conjecture of Hall et al. is also correct, i.e., that (formula presented) is negligible only for (formula presented).

Original languageEnglish
Title of host publicationCyber Security Cryptography and Machine Learning - 3rd International Symposium, CSCML 2019, Proceedings
EditorsShlomi Dolev, Danny Hendler, Sachin Lodha, Moti Yung
PublisherSpringer Verlag
Number of pages10
ISBN (Print)9783030209506
StatePublished - 2019
Event3rd International Symposium on Cyber Security Cryptography and Machine Learning, CSCML 2019 - Beer Sheva, Israel
Duration: 27 Jun 201928 Jun 2019

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11527 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference3rd International Symposium on Cyber Security Cryptography and Machine Learning, CSCML 2019
CityBeer Sheva

Bibliographical note

Publisher Copyright:
© Springer Nature Switzerland AG 2019.


  • Advantage
  • Pseudo random functions
  • Pseudo random permutations

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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