Detection of Cheaters in Non-interactive Polynomial Evaluation

In this paper, we consider both theoretical and practical aspects of robust NI-PE (noninteractive 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 2 = 0), 2. any polynomial of degree d can be evaluated, 3. the total size of shares of the output for some participating set is o(m)× log |F|, only if T is of type Qd+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 Qd+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 2 = 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.