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**2014/889**

**Title:**

Efficient Zero-Knowledge Proofs for Commitments from Learning With Errors over Rings

**Authors:**
Fabrice Benhamouda, Stephan Krenn, Vadim Lyubashevsky, Krzysztof Pietrzak

**Abstract:**

We design an efficient commitment scheme, and companion zero-knowledge proofs of knowledge, based on the learning with errors over rings (RLWE) problem. In particular, for rings in which almost all elements have inverses, we construct a perfectly binding commitment scheme whose hiding property relies on the RLWE assumption. Our scheme maps elements from the ring (or equivalently, n elements from F_q) to a small constant number of ring elements. We then construct Sigma-protocols for proving, in a zero-knowledge manner, knowledge of the message contained in a commitment. We are able to further extend our basic protocol to allow us to prove additive and multiplicative relations among committed values. Our protocols have a communication complexity of O(Mn\log q) and achieve a negligible knowledge error in one run. Here M is the constant from a rejection sampling technique that we employ, and can be set close to 1 by adjusting other parameters. Previously known Sigma-protocols for LWE-related languages either relied on ``smudgingâ€™â€™ out the error (which necessitates working over large fields, resulting in poor efficiency) or only achieved a noticeable or even constant knowledge error (thus requiring many repetitions of the protocol).

**ePrint:**
https://eprint.iacr.org/2014/889

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