New binding-concealing trade-offs for quantum string commitment

Abstract

String commitment schemes are similar to the well studied bit commitment schemes in cryptography with the difference that the committing party, say Alice, is supposed to commit a long string instead of a single bit, to another party say Bob. Similar to bit commitment schemes, such schemes are supposed to be binding, i.e Alice cannot change her choice after committing, and concealing i.e. Bob cannot find Alice's committed string before Alice reveals it. Ideal commitment schemes are known to be impossible. Even if some degrees of cheating is allowed, Buhrman, Christandl, Hayden, Lo and Wehner (quant-ph/0504078) have recently shown that there are some binding-concealing trade-offs that any quantum string commitment scheme (QSC) must follow. They showed trade-offs both in the scenario of single execution of the protocol and in the asymptotic regime of sufficiently large number of parallel executions of the protocol. We present here new trade-offs in the scenario of single execution of a QSC protocol. Our trade-offs also immediately imply the trade-off shown by Buhrman et al. in the asymptotic regime. We show our results by making a central use of an important information theoretic tool called the substate theorem due to Jain, Radhakrishnan and Sen. Our techniques are quite different from that of Buhrman et al. and may be of independent interest.