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We consider the following problem: Given a commitment to a valueσ , prove in zero-knowledge that σ belongs to some discrete set φ . The set φ can perhaps be a list of cities or clubs; often φ canbe a numerical range such as [1,220]. This problemarises in e-cash systems, anonymous credential systems, and variousother practical uses of zero-knowledge protocols. When using commitment schemes relying on RSA-like assumptions,there are solutions to this problem which require only a constantnumber of RSA-group elements to be exchanged between the prover andverifier [5, 15, 16]. However, for many commitment schemes based onbilinear group assumptions, these techniques do not work, and thebest known protocols require O (k ) group elementsto be exchanged where k is a security parameter. In this paper, we present two new approaches to buildingset-membership proofs. The first is based on bilinear groupassumptions. When applied to the case where φ is arange of integers, our protocols require $O(\frac{k}{\log k -\log\log k})$ group elements to be exchanged. Not only is thisresult asymptotically better, but the constants are small enough toprovide significant improvements even for small ranges. Indeed, fora discrete logarithm based setting, our new protocol is an order ofmagnitude more efficient than previously known ones. We also discuss alternative implementations of our membershipproof based on the strong RSA assumption. Depending on theapplication, e.g., when φ is a published set of valuessuch a frequent flyer clubs, cities, or other ad hoc collections,these alternative also outperform prior solutions.

Efficient Protocols for Set Membership and Range Proofs