Homomorphic Trapdoor Commitments to Group Elements

We present homomorphic trapdoor commitments to group elements. In contrast, previous homomorphic trapdoor commitment schemes only allow the messages to be exponents. Our commitment schemes are length-reducing, we can make a short commitment to many group elements at once, and they are perfectly hiding and computationally binding. The commitment schemes are based on groups with a bilinear map. We can commit to elements from a base group, whereas the commitments belong to the target group. We present two constructions based on simple computational intractability assumptions, which we call respectively the double pairing assumption and the simultaneous triple pairing assumption. While the assumptions are new, we demonstrate that they are implied by well-known assumptions; respectively the decision Diffie-Hellman assumption and the decision linear assumption.

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