Additive Combinatorics and Discrete Logarithm Based Range Protocols

  • Rafik Chaabouni
  • Helger Lipmaa
  • Abhi Shelat
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6168)

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 = {ba : 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 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 \(\mathcal{I}\), 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.

Keywords

Additive combinatorics cryptographic range proof sumset zero knowledge 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Rafik Chaabouni
    • 1
  • Helger Lipmaa
    • 2
    • 3
  • Abhi Shelat
    • 4
  1. 1.EPFLSwitzerland
  2. 2.Cybernetica ASEstonia
  3. 3.Tallinn UniversityEstonia
  4. 4.University of VirginiaUSA

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