Additive Combinatorics and Discrete Logarithm Based Range Protocols

Abstract

We show how to express an arbitrary integer interval $\mathcal{I} = [0, H]$ as a sumset $\mathcal{I} = \sum_{i=1}^\ell G_i * [0, u - 1] + [0, H']$ of smaller integer intervals for some small values ℓ, u, and H′ < u − 1, where b * A = {b a : a ∈ A} and $A + B = \{a + b : a \in A \land b \in B\}$. We show how to derive such expression of $\mathcal{I}$ as a sumset for any value of 1 < u < H, and in particular, how the coefficients G _i can be found by using a nontrivial but efficient algorithm. This result may be interesting by itself in the context of additive combinatorics. Given the sumset-representation of $\mathcal{I}$, we show how to decrease both the communication complexity and the computational complexity of the recent pairing-based range proof of Camenisch, Chaabouni and shelat from ASIACRYPT 2008 by a factor of 2. Our results are important in applications like e-voting where a voting server has to verify thousands of proofs of e-vote correctness per hour. Therefore, our new result in additive combinatorics has direct relevance in practice.