Privacy Amplification and Non-Malleable Extractors Via Character Sums

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 to 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.