### Abstract

We provide new bounds on how close to regular the map x ↦ x^{e} is on arithmetic progressions in ℤ_{N}, assuming e|Φ(N) and N is composite. We use these bounds to analyze the security of natural cryptographic problems related to RSA, based on the well-studied Φ-Hiding assumption. For example, under this assumption, we show that RSA PKCS #1 v1.5 is secure against chosen-plaintext attacks for messages of length roughly bits, whereas the previous analysis, due to [19], applies only to messages of length less than. In addition to providing new bounds, we also show that a key lemma of [19] is incorrect. We prove a weaker version of the claim which is nonetheless sufficient for most, though not all, of their applications. Our technical results can be viewed as showing that exponentiation in ℤ_{N} is a deterministic extractor for every source that is uniform on an arithmetic progression. Previous work showed this type of statement only on average over a large class of sources, or for much longer progressions (that is, sources with much more entropy).

Original language | English (US) |
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Title of host publication | Theory of Cryptography - 12th Theory of Cryptography Conference, TCC 2015, Proceedings |

Editors | Yevgeniy Dodis, Jesper Buus Nielsen |

Publisher | Springer Verlag |

Pages | 609-628 |

Number of pages | 20 |

ISBN (Electronic) | 9783662464939 |

State | Published - Jan 1 2015 |

Event | 12th Theory of Cryptography Conference, TCC 2015 - Warsaw, Poland Duration: Mar 23 2015 → Mar 25 2015 |

### 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 | 9014 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 12th Theory of Cryptography Conference, TCC 2015 |
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Country | Poland |

City | Warsaw |

Period | 3/23/15 → 3/25/15 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theory of Cryptography - 12th Theory of Cryptography Conference, TCC 2015, Proceedings*(pp. 609-628). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9014). Springer Verlag.