### Abstract

We initiate a study of randomness condensers for sources that are efficiently samplable but may depend on the seed of the condenser. That is, we seek functions Cond : {0,1} ^{n} × {0,1} ^{d} → {0,1} ^{m} such that if we choose a random seed S ← {0,1} ^{d}, and a source is generated by a randomized circuit of size t such that X has min-entropy at least k given S, then Cond(X;S) should have min-entropy at least some k′ given S. The distinction from the standard notion of randomness condensers is that the source X may be correlated with the seed S (but is restricted to be efficiently samplable). Randomness extractors of this type (corresponding to the special case where k′ = m) have been implicitly studied in the past (by Trevisan and Vadhan, FOCS '00). We show that: Unlike extractors, we can have randomness condensers for samplable, seed-dependent sources whose computational complexity is smaller than the size t of the adversarial sampling algorithm . Indeed, we show that sufficiently strong collision-resistant hash functions are seed-dependent condensers that produce outputs with min-entropy , i.e. logarithmic entropy deficiency. Randomness condensers suffice for key derivation in many cryptographic applications: when an adversary has negligible success probability (or negligible "squared advantage" [3]) for a uniformly random key, we can use instead a key generated by a condenser whose output has logarithmic entropy deficiency. Randomness condensers for seed-dependent samplable sources that are robust to side information generated by the sampling algorithm imply soundness of the Fiat-Shamir Heuristic when applied to any constant-round, public-coin interactive proof system.

Original language | English (US) |
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Title of host publication | Theory of Cryptography - 9th Theory of Cryptography Conference, TCC 2012, Proceedings |

Pages | 618-635 |

Number of pages | 18 |

DOIs | |

State | Published - 2012 |

Event | 9th Theory of Cryptography Conference, TCC 2012 - Taormina, Sicily, Italy Duration: Mar 19 2012 → Mar 21 2012 |

### 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 | 7194 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 9th Theory of Cryptography Conference, TCC 2012 |
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Country | Italy |

City | Taormina, Sicily |

Period | 3/19/12 → 3/21/12 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Theory of Cryptography - 9th Theory of Cryptography Conference, TCC 2012, Proceedings*(pp. 618-635). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7194 LNCS). https://doi.org/10.1007/978-3-642-28914-9_35