Secure Classical Bit Commitment over Finite Channels

If mistrustful parties A and B control two or more appropriately located sites, special relativity can be used to guarantee that a pair of messages exchanged by A and B are independent. Using this fact, an unconditionally secure relativistic bit commitment protocol, RBC1, was recently defined. Security is maintained in RBC1 by exchanging a sequence of messages whose transmission rate increases exponentially in time. We define here a variant, RBC2, which requires a constant transmission rate and can in practice be securely maintained for long periods. We also introduce a zero-knowledge proof protocol which allows A to convince B of the value of any function of N relativistic bit commitments without giving any further information. Applying this to the XOR of two bit commitments made by RBC1 or RBC2, we define two further relativistic bit commitment protocols, RBC3 and RBC4. We describe also protocols more efficient version of this zero-knowledge protocol, based on a technique of Rudich, whose applicability to relativistic bit commitment was suggested by Crepeau. We discuss security against classical and quantum attacks.