## Abstract

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, Adv_{n,m}(q), is small if q=o(2^{(n+m)∕2}), established an upper bound on Adv_{n,m}(q) that confirms the conjecture for m<n∕7, and also declared a general lower bound Adv_{n,m}(q)=Ω(q^{2}∕2^{n+m}). The conjecture was essentially confirmed by Bellare and Impagliazzo (1999). Nevertheless, the problem of estimating Adv_{n,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 Adv_{n,m}(q) is negligible only for q=o(2^{(n+m)∕2}).

Original language | English |
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Pages (from-to) | 214-223 |

Number of pages | 10 |

Journal | Discrete Applied Mathematics |

Volume | 294 |

DOIs | |

State | Published - 15 May 2021 |

### Bibliographical note

Publisher Copyright:© 2021 Elsevier B.V.

## Keywords

- Pseudo random function advantage

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics