## 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 (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 language | English |
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Title of host publication | Cyber Security Cryptography and Machine Learning - 3rd International Symposium, CSCML 2019, Proceedings |

Editors | Shlomi Dolev, Danny Hendler, Sachin Lodha, Moti Yung |

Publisher | Springer Verlag |

Pages | 111-120 |

Number of pages | 10 |

ISBN (Print) | 9783030209506 |

DOIs | |

State | Published - 2019 |

Event | 3rd International Symposium on Cyber Security Cryptography and Machine Learning, CSCML 2019 - Beer Sheva, Israel Duration: 27 Jun 2019 → 28 Jun 2019 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11527 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 3rd International Symposium on Cyber Security Cryptography and Machine Learning, CSCML 2019 |
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Country/Territory | Israel |

City | Beer Sheva |

Period | 27/06/19 → 28/06/19 |

### Bibliographical note

Funding Information: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.

Publisher Copyright:

© Springer Nature Switzerland AG 2019.

## Keywords

- Advantage
- Pseudo random functions
- Pseudo random permutations