We consider the related notions of two-prover and of relativistic commitment schemes. In recent work, Lunghi et al. proposed a new relativistic commitment scheme with a multi-round sustain phase that keeps the binding property alive as long as the sustain phase is running. They prove security of their scheme against classical attacks; however, the proven bound on the error parameter is very weak: it blows up double exponentially in the number of rounds. In this work, we give a new analysis of the multi-round scheme of Lunghi et al., and we show a linear growth of the error parameter instead (also considering classical attacks only). Our analysis is based on a new composition theorem for two-prover commitment schemes. The proof of our composition theorem is based on a better understanding of the binding property of two-prover commitments that we provide in the form of new definitions and relations among them. As an additional consequence of these new insights, our analysis is actually with respect to a strictly stronger notion of security than considered by Lunghi et al.

Advances in Cryptology - EUROCRYPT

Fehr, S, & Fillinger, M.J. (2016). On the composition of two-prover commitments, and applications to multi-round relativistic commitments. In 35th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Vienna, Austria, May 8-12, 2016, Proceedings, Part II (pp. 477–496). doi:10.1007/978-3-662-49896-5_17