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View all- Beaugrand ACastagnos GLaguillaumie F(2025)Efficient Succinct Zero-Knowledge Arguments in the CL FrameworkJournal of Cryptology10.1007/s00145-024-09534-138:1Online publication date: 3-Jan-2025
On Sigma-Protocols and (Packed) Black-Box Secret Sharing Schemes
Pages 426 - 457
Abstract
-protocols are a widely utilized, relatively simple and well understood type of zero-knowledge proofs. However, the well known Schnorr -protocol for proving knowledge of discrete logarithm in a cyclic group of known prime order, and similar protocols working over this type of groups, are hard to generalize to dealing with other groups, in particular those of hidden order, due to the inability of the knowledge extractor to invert elements modulo the order.
In this paper, we introduce a universal construction of -protocols designed to prove knowledge of preimages of group homomorphisms for any abelian finite group. In order to do this, we first establish a general construction of a -protocol for -module homomorphism given only a linear secret sharing scheme over the ring, where zero knowledge and special soundness can be related to the privacy and reconstruction properties of the secret sharing scheme. Then, we introduce a new construction of 2-out-of-n packed black-box secret sharing scheme capable of sharing k elements of an arbitrary (abelian, finite) group where each share consists of group elements. From these two elements we obtain a generic “batch” -protocol for proving knowledge of k preimages of elements via the same group homomorphism, which communicates elements of the group to achieve knowledge error. For the case of class groups, we show that our -protocol improves in several aspects on existing proofs for knowledge of discrete logarithm and other related statements that have been used in a number of works.
Finally, we extend our constructions from group homomorphisms to the case of ZK-ready functions, introduced by Cramer and Damgård in Crypto 09, which in particular include the case of proofs of knowledge of plaintext (and randomness) for some linearly homomorphic encryption schemes such as Joye-Libert encryption. However, in the case of Joye-Libert, we show a better alternative, using Shamir secret sharing over Galois rings, which achieves knowledge soundness by communicating k ciphertexts to prove k statements.
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Apr 2024
468 pages
© International Association for Cryptologic Research 2024.
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Published: 15 April 2024
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View all- Beaugrand ACastagnos GLaguillaumie F(2025)Efficient Succinct Zero-Knowledge Arguments in the CL FrameworkJournal of Cryptology10.1007/s00145-024-09534-138:1Online publication date: 3-Jan-2025