**Additive Combinatorics and Discrete Logarithm Based Range Protocols**

*Rafik Chaabouni and Helger Lipmaa and abhi shelat*

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

**Category / Keywords: **Additive combinatorics, cryptographic range proof, sumset, zero knowledge

**Publication Info: **ACISP 2010

**Date: **received 23 Sep 2009, last revised 23 Apr 2010

**Contact author: **lipmaa at research cyber ee

**Available format(s): **PDF | BibTeX Citation

**Note: **This corresponds to the published version

**Version: **20100423:151358 (All versions of this report)

**Short URL: **ia.cr/2009/469

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