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IEEE International Conference on Trust, Security and Privacy in Computing and Communications (TrustCom)

Rational Secret Sharing Information -- Theoretically Secure against Adaptive Adversaries

Yang Yu, Zhanfei Zhou

Published Jul 1, 2013• 0 citations
Publisher

Abstract

Most rational secret sharing protocols aim to achieve a more appealing equilibrium or resist adversaries. However, all previous protocols achieve resilience against irrational adversaries at the cost of inducing a weak guarantee of Nash equilibrium surviving iterated deletion, which has been proved weaker than strict Nash equilibrium. Moreover, they achieve equilibrium in a computational way based on the computational limitations of adversaries. In this paper, we design two unconditionally secure rational (t, n) secret sharing protocols. Contrary to prior works, our protocols induce an enhanced notion of strict equilibrium with respect to adversary structure A, and resist the computationally unbounded and adaptive adversary without using cryptographic primitives. Our protocols achieve stronger stability, since rational players have an incentive not to deviate no matter how the adversary behaves, and they are immune to the backward induction. Our protocols are secure against the adversary corrupting less than [n/3] players and the adversary corrupting less than [n/2]-1 players respectively. Both of them achieve utility independence.

DOI
10.1109/trustcom.2013.34

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Most rational secret sharing protocols aim to achieve a more appealing equilibrium or resist adversaries. However, all previous protocols achieve resilience against irrational adversaries at the cost of inducing a weak guarantee of Nash equilibrium surviving iterated deletion, which has been proved weaker than strict Nash equilibrium. Moreover, they achieve equilibrium in a computational way based on the computational limitations of adversaries. In this paper, we design two unconditionally secure rational (t, n) secret sharing protocols. Contrary to prior works, our protocols induce an enhanced notion of strict equilibrium with respect to adversary structure A, and resist the computationally unbounded and adaptive adversary without using cryptographic primitives. Our protocols achieve stronger stability, since rational players have an incentive not to deviate no matter how the adversary behaves, and they are immune to the backward induction. Our protocols are secure against the adversary corrupting less than [n/3] players and the adversary corrupting less than [n/2]-1 players respectively. Both of them achieve utility independence.
The design of rational cryptographic protocols is a recently created research area at the intersection of cryptography and game theory. In this paper, we propose a new m-out-of-n rational secret sharing scheme requiring neither the involvement of the dealer (except during the initial share distribution) nor a trusted mediator. Our protocol leads to a Nash equilibrium surviving the iterated deletion of weakly dominated strategies for m ≥ 4. Our construction is information theoretically secure and it is immune against backward induction attacks. Contrary to Kol and Naor who used a specific cryptographic primitive in their TCC'08 paper (namely, meaningful/meaningless encryption), the immunity of our scheme is based on the use of bivariate polynomials and one-time pads. To the best of our knowledge, it is the first time that such polynomials have been used for rational secret sharing. Our scheme is efficient and does not require any physical assumptions such as envelopes or ballot boxes. As most of existing rational protocols, our construction requires simultaneous broadcast channels. However, our proposed scheme does not require any computational assumption and it provides information theoretical security.
We present a general construction of a rational secret-sharing protocol that converts any rational secret-sharing protocol to a protocol with an expected constant-round reconstruction. Our construction can be applied to protocols for synchronous channels, and preserves a strict Nash equilibrium of the original protocol. Combining with an existing protocol, we obtain the first expected constant-round protocol that achieves a strict Nash equilibrium with the optimal coalition resilience ⌈n2 ⌉ − 1, where n is the number of players. Our construction can be extended to a construction that preserves the immunity to unexpectedly behaving players. Then, for any constantm ≥ 1, we obtain an expected constant-round protocol that achieves a Nash equilibrium with the optimal coalition resilience ⌈n2 ⌉ −m − 1 in the presence of m unexpectedly behaving players. The same protocol also achieves a strict Nash equilibrium with coalition resilience 1. We show that our protocol with immunity achieves the optimal coalition resilience among constant-round protocols with immunity with respect to both Nash and strict Nash equilibria.
Abstract. Rational secret sharing protocols in both the two-party and multi-party settings are proposed. These protocols are built in standard communication networks and with unconditional security. Namely, the protocols run over standard point-to-point networks without requiring physical assumptions or simultaneous channels, and even a computationally unbounded player cannot gain more than ɛ by deviating from the protocol. More precisely, for the 2-out-of-2 protocol the ɛ is a negligible function in the size of the secret, which is caused by the informationtheoretic MACs used for authentication. The t-out-of-n protocol is (t−1)resilient and the ɛ is exponentially small in the number of participants. Although secret recovery cannot be guaranteed in this setting, a participant can at least reduce the Shannon entropy of the secret to less than 1 after the protocol. When the secret-domain is large, every rational player has great incentive to participate in the protocol.
We propose cryptographic rational secret sharing protocols over general networks. In a general network, the dealer may not have direct connections to each player, and players may not have direct connections to each of the other players. We present conditions on the network topology for which our proposed protocols are computational strict Nash equilibria and (k−1)-resilient, along with analysis on their round and communication complexity. We also present new notions of equilibria such as Φ-resilient computational Nash equilibria, whereby a protocol is resilient to coalitions that satisfy conditions in Φ, regardless of the coalition’s size. We also propose (n−1)-key leakage-tolerant equilibria applicable to cryptographic protocols involving secret keys, whereby the equilibrium holds even if some players acquire (n−1) tuples of secret keys.
We present a protocol compiler of rational secret-sharing that converts any rational secret-sharing protocol to a protocol with an expected constant-round reconstruction. Our compiler can be applied to protocols for synchronous channels, and preserves a strict Nash equilibrium of the original protocol. Combining with an existing protocol, we obtain the first expected constant-round protocol that achieves a strict Nash equilibrium with the optimal coalition resilience ⌈n2⌉−1⁠, where n is the number of players. Our compiler can be extended to one that preserves the immunity to unexpectedly behaving players. For any constant m≥1⁠, we obtain an expected constant-round protocol that achieves a Nash equilibrium with the optimal coalition resilience ⌈n2⌉−m−1 in the presence of m unexpectedly behaving players. The protocol also achieves a strict Nash equilibrium. As a negative result, we show that if an expected constant-round protocol has immunity m>0⁠, then it cannot achieve a strict Nash equilibrium with the coalition resilience 2. Thus, our protocol with immunity achieves the optimal coalition resilience with respect to both Nash and strict Nash equilibrium.
Rational secret sharing, first introduced by Halpern and Teague, means that whether players send their shares relies on their utilities, whereas in traditional scheme, players are assumed to contribute their shares willingly. This paper puts forward a rational secret-sharing scheme where players run their reconstruction protocol in point-to-point communication networks other than broadcast channels which are often used in previous papers. Kol and Naor propose an efficient t -out-of- n protocol for rational secret sharing without using cryptographic primitives while it has weak robustness to coalition when short player colludes with any long players. The protocol proposed by Fuchsbauer and Katz runs efficiently in asynchronous or point-to-point networks. Although it achieves strict Nash equilibrium and is stable with respect to trembles, it is susceptible to realise fairness among the whole party. This paper proposes a rational secret-sharing scheme in point-to-point communication networks to achieve fairness and robustness to coalition.