Additive Combinatorics and Discrete Logarithm Based Range Protocols

We show how to express an arbitrary integer interval $I = [0, H]$ as a sumset $I = \sum_{i=1}^\ell G_i * [0, u - 1] + [0, H']$ of smaller integer intervals for some small values $\ell$, $u$, and $H' < u - 1$, where $b * A = \{b a : a \in A\}$ and $A + B = \{a + b : a \in A \wedge b \in B\}$. 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 $G_i$ 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 $I$, 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.

Published in:
Information Security and Privacy Information Security and Privacy, Proceedings of the 15th Australasian Conference, ACISP 2010, 6168, 336-351
Presented at:
15th Australasian Conference, ACISP 2010, Sydney, Australia, July 5-7, 2010
Berlin Heidelberg, Springer

 Record created 2010-08-23, last modified 2018-03-17

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