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Zero-knowledge proof protocol is a basic cryptographic technique. And zero-knowledge proof of double discrete logarithm has some particular properties, so it has been widely applied in many security systems. But the efficient problem of zero-knowledge proof of double discrete logarithm has not been solved to this day, since there are some special difficulties in computing this kind of knowledge proof. Hence, the time complexity and the space complexity of existing schemes are all O(k), where k is a security parameter. After redesigning the basic construction of knowledge proof, we provide a new zero-knowledge proof of double discrete logarithm, which is the first scheme with O(1) time complexity and O(1) space complexity. If introducing an off-line TTP (trusted third party), we can provide two additional zeroknowledge proof schemes of double discrete logarithm, one is even more efficient than the first one, the other one solves another open problem, which is how to efficiently prove the equality of double discrete logarithms in zero-knowledge way, and the existing techniques cannot solve this problem. We also provide the detailed security proofs of our designs and efficiency analysis, comparing with the existing schemes. The significant improvement in efficiency of this basic cryptographic technique is also helpful for many security systems.

Efficient Zero-Knowledge Proofs of Knowledge of Double Discrete Logarithm