### Abstract

In studying how to communicate over a public channel with an active adversary, Dodis and Wichs introduced the notion of a non-malleable extractor. A non-malleable extractor dramatically strengthens the notion of a strong extractor. A strong extractor takes two inputs, a weakly-random x and a uniformly random seed y, and outputs a string which appears uniform, even given y. For a non-malleable extractor nmExt, the output nmExt(x,y) should appear uniform given y as well as nmExt(x,A(y)), where A is an arbitrary function with A(y) ≠ y. We show that an extractor introduced by Chor and Gold reich is non-malleable when the entropy rate is above half. It outputs a linear number of bits when the entropy rate is 1/2 + α, for any α > 0. Previously, no nontrivial parameters were known for any non-malleable extractor. To achieve a polynomial running time when outputting many bits, we rely on a widely-believed conjecture about the distribution of prime numbers in arithmetic progressions. Our analysis involves a character sum estimate, which may be of independent interest. Using our non-malleable extractor, we obtain protocols for "privacy amplification": key agreement between two parties who share a weakly-random secret. Our protocols work in the presence of an active adversary with unlimited computational power, and have asymptotically optimal entropy loss. When the secret has entropy rate greater than 1/2, the protocol follows from a result of Dodis and Wichs, and takes two rounds. When the secret has entropy rate δ for any constant δ > 0, our new protocol takes a constant (polynomial in 1/δ) number of rounds. Our protocols run in polynomial time under the above well-known conjecture about primes.

Original language | English (US) |
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Title of host publication | Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 |

Pages | 668-677 |

Number of pages | 10 |

DOIs | |

State | Published - 2011 |

Event | 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 - Palm Springs, CA, United States Duration: Oct 22 2011 → Oct 25 2011 |

### Other

Other | 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 |
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Country | United States |

City | Palm Springs, CA |

Period | 10/22/11 → 10/25/11 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011*(pp. 668-677). [6108229] https://doi.org/10.1109/FOCS.2011.67

**Privacy amplification and non-malleable extractors via character sums.** / Dodis, Yevgeniy; Li, Xin; Wooley, Trevor D.; Zuckerman, David.

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

*Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011.*, 6108229, pp. 668-677, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, United States, 10/22/11. https://doi.org/10.1109/FOCS.2011.67

}

TY - GEN

T1 - Privacy amplification and non-malleable extractors via character sums

AU - Dodis, Yevgeniy

AU - Li, Xin

AU - Wooley, Trevor D.

AU - Zuckerman, David

PY - 2011

Y1 - 2011

N2 - In studying how to communicate over a public channel with an active adversary, Dodis and Wichs introduced the notion of a non-malleable extractor. A non-malleable extractor dramatically strengthens the notion of a strong extractor. A strong extractor takes two inputs, a weakly-random x and a uniformly random seed y, and outputs a string which appears uniform, even given y. For a non-malleable extractor nmExt, the output nmExt(x,y) should appear uniform given y as well as nmExt(x,A(y)), where A is an arbitrary function with A(y) ≠ y. We show that an extractor introduced by Chor and Gold reich is non-malleable when the entropy rate is above half. It outputs a linear number of bits when the entropy rate is 1/2 + α, for any α > 0. Previously, no nontrivial parameters were known for any non-malleable extractor. To achieve a polynomial running time when outputting many bits, we rely on a widely-believed conjecture about the distribution of prime numbers in arithmetic progressions. Our analysis involves a character sum estimate, which may be of independent interest. Using our non-malleable extractor, we obtain protocols for "privacy amplification": key agreement between two parties who share a weakly-random secret. Our protocols work in the presence of an active adversary with unlimited computational power, and have asymptotically optimal entropy loss. When the secret has entropy rate greater than 1/2, the protocol follows from a result of Dodis and Wichs, and takes two rounds. When the secret has entropy rate δ for any constant δ > 0, our new protocol takes a constant (polynomial in 1/δ) number of rounds. Our protocols run in polynomial time under the above well-known conjecture about primes.

AB - In studying how to communicate over a public channel with an active adversary, Dodis and Wichs introduced the notion of a non-malleable extractor. A non-malleable extractor dramatically strengthens the notion of a strong extractor. A strong extractor takes two inputs, a weakly-random x and a uniformly random seed y, and outputs a string which appears uniform, even given y. For a non-malleable extractor nmExt, the output nmExt(x,y) should appear uniform given y as well as nmExt(x,A(y)), where A is an arbitrary function with A(y) ≠ y. We show that an extractor introduced by Chor and Gold reich is non-malleable when the entropy rate is above half. It outputs a linear number of bits when the entropy rate is 1/2 + α, for any α > 0. Previously, no nontrivial parameters were known for any non-malleable extractor. To achieve a polynomial running time when outputting many bits, we rely on a widely-believed conjecture about the distribution of prime numbers in arithmetic progressions. Our analysis involves a character sum estimate, which may be of independent interest. Using our non-malleable extractor, we obtain protocols for "privacy amplification": key agreement between two parties who share a weakly-random secret. Our protocols work in the presence of an active adversary with unlimited computational power, and have asymptotically optimal entropy loss. When the secret has entropy rate greater than 1/2, the protocol follows from a result of Dodis and Wichs, and takes two rounds. When the secret has entropy rate δ for any constant δ > 0, our new protocol takes a constant (polynomial in 1/δ) number of rounds. Our protocols run in polynomial time under the above well-known conjecture about primes.

UR - http://www.scopus.com/inward/record.url?scp=84863321325&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84863321325&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2011.67

DO - 10.1109/FOCS.2011.67

M3 - Conference contribution

AN - SCOPUS:84863321325

SN - 9780769545714

SP - 668

EP - 677

BT - Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011

ER -