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) not equal y.
We show that an extractor introduced by Chor and Goldreich is non-malleable when the entropy rate is above half. It outputs a linear number of bits when the entropy rate is 1/2 + alpha, for any alpha > 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 delta for any constant delta > 0, our new protocol takes a constant (polynomial in 1/delta) number of rounds. Our protocols run in polynomial time under the above well-known conjecture about primes.
|Title of host publication||2011 IEEE 52ND ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS 2011)|
|Place of Publication||LOS ALAMITOS|
|Publisher||IEEE Computer Society|
|Number of pages||10|
|Publication status||Published - 2011|
|Event||52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS) - Palm Springs, Canada|
Duration: 22 Oct 2011 → 25 Oct 2011
|Name||Annual IEEE Symposium on Foundations of Computer Science|
|Publisher||IEEE COMPUTER SOC|
|Conference||52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS)|
|Period||22/10/11 → 25/10/11|