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 $\nm$, the output $\nm(x,y)$ should appear uniform given $y$ as well as $\nm(x,\adv(y))$, where $\adv$ is an arbitrary function with $\adv(y) \neq y$. 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 $1/2 + \alpha$, 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 & quot;: 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.