International Association
for Cryptologic Research

We consider the following problem: Assuming that Alice and Bob have an integer interval $[a, e]$ and an integer $b$ respectively, for a commitment $c$ to $b$, Alice and Bob jointly check whether $b$ is within $[a, e]$ without revealing their inputs, where either party may behave maliciously. A special case of the problem is the secure integer comparison in the malicious model. This problem mainly arises from location-based access control systems where one party needs to assure to the other party that its location is within some definite area.

Our main result is a constant-round protocol that exhibit the square of $\\log e$ communication and the square of $\\log e$ exponentiations with simulation-based security. At the heart of the construction is perfect $k$-ary index and corresponding zero-knowledge proof techniques.

We consider a more general case of the problem where the interval is substituted by a union of intervals.