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Title:
Detection of Cheaters in Non-interactive Polynomial Evaluation
Authors: Maki Yoshida, Satoshi Obana
Abstract:In this paper, we consider both theoretical and practical aspects of robust NI-PE (non-interactive polynomial evaluation with detection of cheaters). First, we give a necessary condition of adversary structures for which perfectly robust NI-PE with small communication complexity exists. More precisely, we show that for any positive integers n, m and d>1, an n-player access structure U, and an n-player adversary structure T, there exists a U-participating NI-PE scheme for m-variate polynomials over a finite field F with T-private inputs such that (1) perfectly robust (i.e., successful cheating probability \epsilon=0), (2) any polynomial of degree d can be evaluated, and (3) the total size of shares of the output for some participating set is o(m)\times \log |F|, {\em only if} T is of type Q_{d+1} for U, meaning that no d+1 sets in T cover any set in U. Second, we give constructions of perfectly robust NI-PE schemes against threshold adversary and general adversary, respectively. All the proposed schemes ensure perfect robustness against Q_{d+1} adversary, and computability of arbitrary polynomial of degree d. Third, we show that efficient robust NI-PE schemes against general adversary can be constructed by allowing cheaters very small chance of successful cheating. Namely, we construct two robust NI-PE schemes with \epsilon=1/|F| and the total size for shares of the output is only three times larger compared to the perfectly robust NI-PE scheme against threshold adversary.
ePrint: https://eprint.iacr.org/2013/032
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