Efficient Threshold Secret Sharing Schemes Secure Against Rushing Cheaters

Abstract

In this paper, we consider two very important issues namely detection and identification of k-out-of-n secret sharing schemes against rushing cheaters who are allowed to submit (possibly forged) shares after observing shares of the honest users in the reconstruction phase. Towards this, we present four different schemes. Among these, first we present two k-out-of-n secret sharing schemes, the first one being capable of detecting $(k-1)/3$ cheaters such that $|V_i|=|S|/\epsilon ^3$ and the second one being capable of detecting $n-1$ cheaters such that $|V_i|=|S|/\epsilon ^{k+1}$, where S denotes the set of all possible secrets, $\epsilon$ denotes the successful cheating probability of cheaters and $V_i$ denotes set all possible shares. Next we present two k-out-of-n secret sharing schemes, the first one being capable of identifying $(k-1)/3$ rushing cheaters with share size $|V_i|$ that satisfies $|V_i|=|S|/\epsilon ^k$. This is the first scheme, whose size of shares does not grow linearly with n but only with k, where n is the number of participants. For the second one, in the setting of public cheater identification, we present an efficient optimal cheater resilient k-out-of-n secret sharing scheme against rushing cheaters having the share size $|V_i|= (n-t)^{n+2t}|S|/\epsilon ^{n+2t}$. The proposed scheme achieves flexibility in the sense that the security level (i.e., the cheater(s) success probability) is independent of the secret size. Each of the four proposed schemes has the smallest share size among the existing schemes having the mentioned properties in the respective models.

A. Adhikari—Research is partially supported by National Board for Higher Mathematics, Department of Atomic Energy, Government of India, Grant No. 2/48(10)/2013/NBHM(R.P.)/R&D II/695. The author is also thankful to DST, Govt. of India and JSPS, Govt. of Japan for providing partial support for this collaborative research work under India Japan Cooperative Science Programme (vide Memo no. DST/INT/JSPS/P-191/2014 dated May 27, 2014.

K. Morozov—Research is partially supported by a Kakenhi Grant-in-Aid for Scientific Research (C) 15K00186 from Japan Society for the Promotion of Science. Research is partially supported by JST, CREST.

S. Obana—Research is partially supported by a Kakenhi Grant-in-Aid for Scientific Research (C) 15K00193 from Japan Society for the Promotion of Science.