### Abstract

If we have a problem that is mildly hard, can we create a problem that is significantly harder? A natural approach to hardness amplification is the "direct product"; instead of asking an attacker to solve a single instance of a problem, we ask the attacker to solve several independently generated ones. Interestingly, proving that the direct product amplifies hardness is often highly non-trivial, and in some cases may be false. For example, it is known that the direct product (i.e. "parallel repetition") of general interactive games may not amplify hardness at all. On the other hand, positive results show that the direct product does amplify hardness for many basic primitives such as one-way functions, weakly-verifiable puzzles, and signatures. Even when positive direct product theorems are shown to hold for some primitive, the parameters are surprisingly weaker than what we may have expected. For example, if we start with a weak one-way function that no poly-time attacker can break with probability >1/2, then the direct product provably amplifies hardness to some negligible probability. Naturally, we would expect that we can amplify hardness exponentially, all the way to 2
^{-∈n} probability, or at least to some fixed/known negligible such as n
^{-∈logn} in the security parameter n, just by taking sufficiently many instances of the weak primitive. Although it is known that such parameters cannot be proven via black-box reductions, they may seem like reasonable conjectures, and, to the best of our knowledge, are widely believed to hold. In fact, a conjecture along these lines was introduced in a survey of Goldreich, Nisan and Wigderson (ECCC '95). In this work, we show that such conjectures are false by providing simple but surprising counterexamples. In particular, we construct weakly secure signatures and one-way functions, for which standard hardness amplification results are known to hold, but for which hardness does not amplify beyond just negligible. That is, for any negligible function , we instantiate these primitives so that the direct product can always be broken with probability , no matter how many copies we take.

Original language | English (US) |
---|---|

Title of host publication | Theory of Cryptography - 9th Theory of Cryptography Conference, TCC 2012, Proceedings |

Pages | 476-493 |

Number of pages | 18 |

Volume | 7194 LNCS |

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) |
---|---|

Volume | 7194 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 9th Theory of Cryptography Conference, TCC 2012 |
---|---|

Country | Italy |

City | Taormina, Sicily |

Period | 3/19/12 → 3/21/12 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Theory of Cryptography - 9th Theory of Cryptography Conference, TCC 2012, Proceedings*(Vol. 7194 LNCS, pp. 476-493). (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_27

**Counterexamples to hardness amplification beyond negligible.** / Dodis, Yevgeniy; Jain, Abhishek; Moran, Tal; Wichs, Daniel.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Theory of Cryptography - 9th Theory of Cryptography Conference, TCC 2012, Proceedings.*vol. 7194 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7194 LNCS, pp. 476-493, 9th Theory of Cryptography Conference, TCC 2012, Taormina, Sicily, Italy, 3/19/12. https://doi.org/10.1007/978-3-642-28914-9_27

}

TY - GEN

T1 - Counterexamples to hardness amplification beyond negligible

AU - Dodis, Yevgeniy

AU - Jain, Abhishek

AU - Moran, Tal

AU - Wichs, Daniel

PY - 2012

Y1 - 2012

N2 - If we have a problem that is mildly hard, can we create a problem that is significantly harder? A natural approach to hardness amplification is the "direct product"; instead of asking an attacker to solve a single instance of a problem, we ask the attacker to solve several independently generated ones. Interestingly, proving that the direct product amplifies hardness is often highly non-trivial, and in some cases may be false. For example, it is known that the direct product (i.e. "parallel repetition") of general interactive games may not amplify hardness at all. On the other hand, positive results show that the direct product does amplify hardness for many basic primitives such as one-way functions, weakly-verifiable puzzles, and signatures. Even when positive direct product theorems are shown to hold for some primitive, the parameters are surprisingly weaker than what we may have expected. For example, if we start with a weak one-way function that no poly-time attacker can break with probability >1/2, then the direct product provably amplifies hardness to some negligible probability. Naturally, we would expect that we can amplify hardness exponentially, all the way to 2 -∈n probability, or at least to some fixed/known negligible such as n -∈logn in the security parameter n, just by taking sufficiently many instances of the weak primitive. Although it is known that such parameters cannot be proven via black-box reductions, they may seem like reasonable conjectures, and, to the best of our knowledge, are widely believed to hold. In fact, a conjecture along these lines was introduced in a survey of Goldreich, Nisan and Wigderson (ECCC '95). In this work, we show that such conjectures are false by providing simple but surprising counterexamples. In particular, we construct weakly secure signatures and one-way functions, for which standard hardness amplification results are known to hold, but for which hardness does not amplify beyond just negligible. That is, for any negligible function , we instantiate these primitives so that the direct product can always be broken with probability , no matter how many copies we take.

AB - If we have a problem that is mildly hard, can we create a problem that is significantly harder? A natural approach to hardness amplification is the "direct product"; instead of asking an attacker to solve a single instance of a problem, we ask the attacker to solve several independently generated ones. Interestingly, proving that the direct product amplifies hardness is often highly non-trivial, and in some cases may be false. For example, it is known that the direct product (i.e. "parallel repetition") of general interactive games may not amplify hardness at all. On the other hand, positive results show that the direct product does amplify hardness for many basic primitives such as one-way functions, weakly-verifiable puzzles, and signatures. Even when positive direct product theorems are shown to hold for some primitive, the parameters are surprisingly weaker than what we may have expected. For example, if we start with a weak one-way function that no poly-time attacker can break with probability >1/2, then the direct product provably amplifies hardness to some negligible probability. Naturally, we would expect that we can amplify hardness exponentially, all the way to 2 -∈n probability, or at least to some fixed/known negligible such as n -∈logn in the security parameter n, just by taking sufficiently many instances of the weak primitive. Although it is known that such parameters cannot be proven via black-box reductions, they may seem like reasonable conjectures, and, to the best of our knowledge, are widely believed to hold. In fact, a conjecture along these lines was introduced in a survey of Goldreich, Nisan and Wigderson (ECCC '95). In this work, we show that such conjectures are false by providing simple but surprising counterexamples. In particular, we construct weakly secure signatures and one-way functions, for which standard hardness amplification results are known to hold, but for which hardness does not amplify beyond just negligible. That is, for any negligible function , we instantiate these primitives so that the direct product can always be broken with probability , no matter how many copies we take.

UR - http://www.scopus.com/inward/record.url?scp=84858324616&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858324616&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-28914-9_27

DO - 10.1007/978-3-642-28914-9_27

M3 - Conference contribution

AN - SCOPUS:84858324616

SN - 9783642289132

VL - 7194 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 476

EP - 493

BT - Theory of Cryptography - 9th Theory of Cryptography Conference, TCC 2012, Proceedings

ER -