Additive Combinatorics and Discrete Logarithm Based Range Protocols

We show how to express an arbitrary integer interval I = [0,H] as a sumset I =Σi=1l Gi * [0, u - 1] + [0, H′] of smaller integer intervals for some small values l, u, and H′ < u - 1, where b*A = {ba: a ∈ A} and A+B = {a+b: a ∈ A ∧ b ∈ B}. We show how to derive such expression of I as a sumset for any value of 1 < u < H, and in particular, how the coefficients Gi 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 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.