## Abstract

Most cryptographic primitives require randomness (for example, to generate their secret keys). Usually, one assumes that perfect randomness is available, but, conceivably, such primitives might be built under weaker, more realistic assumptions. This is known to be true for many authentication applications, when entropy alone is typically sufficient. In contrast, all known techniques for achieving privacy seem to fundamentally require (nearly) perfect randomness. We ask the question whether this is just a coincidence, or, perhaps, privacy inherently requires true randomness? We completely resolve this question for the case of (information-theoretic) private-key encryption, where parties wish to encrypt a b-bit value using a shared secret key sampled from some imperfect source of randomness script J sign. Our main result shows that if such n-bit source script J sign allows for a secure encryption of b bits, where b > log n, then one can deterministically extract nearly b almost perfect random bits from script J sign. Further, the restriction that b > log n is nearly tight: there exist sources script J sign allowing one to perfectly encrypt (log n -loglog n) bits, but not to deterministically extract even a single slightly unbiased bit. Hence, to a large extent, true randomness is inherent for encryption: either the key length must be exponential in the message length b, or one can deterministically extract nearly b almost unbiased random bits from the key. In particular, the one-time pad scheme is essentially "universal". Our technique also extends to related computational primitives which are perfectly-binding, such as perfectly-binding commitment and computationally secure private- or public-key encryption, showing the necessity to efficiently extract almost b pseudorandom bits.

Original language | English (US) |
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Title of host publication | Theory of Cryptography - 4th Theory of Cryptography Conference, TCC 2007, Proceedings |

Publisher | Springer Verlag |

Pages | 1-20 |

Number of pages | 20 |

ISBN (Print) | 9783540709350 |

DOIs | |

State | Published - 2007 |

Event | 4th Theory of Cryptography Conference, TCC 2OO7 - Amsterdam, Netherlands Duration: Feb 21 2007 → Feb 24 2007 |

### 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 | 4392 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 4th Theory of Cryptography Conference, TCC 2OO7 |
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Country/Territory | Netherlands |

City | Amsterdam |

Period | 2/21/07 → 2/24/07 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science