Efficient NIZKs for Algebraic Sets

Couteau, Geoffroy; Lipmaa, Helger; Parisella, Roberto; Ødegaard, Arne Tobias

doi:10.1007/978-3-030-92078-4_5

Cited by 4 publications

(1 citation statement)

References 45 publications

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“…Finally, Lipmaa and Parisella [53] (building on [24,26]) construct succinct set (non-)membership NIZKs from falsifaible assumptions. That is, the objective of their work is constructing efficient NIZKs for set (non-)membership that can be proven secure in the standard model and assuming only falsifiable assumptions.…”

Section: Recent Developmentsmentioning

confidence: 99%

Zero-knowledge proofs for set membership: efficient, succinct, modular

Benarroch¹,

Campanelli²,

Fiore³

et al. 2023

Des. Codes Cryptogr.

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We consider the problem of proving in zero knowledge that an element of a public set satisfies a given property without disclosing the element, i.e., for some u, “$$u \in S$$ u ∈ S and P(u) holds”. This problem arises in many applications (anonymous cryptocurrencies, credentials or whitelists) where, for privacy or anonymity reasons, it is crucial to hide certain data while ensuring properties of such data. We design new modular and efficient constructions for this problem through new commit-and-prove zero-knowledge systems for set membership, i.e. schemes proving $$u \in S$$ u ∈ S for a value u that is in a public commitment $$c_u$$ c u . We also extend our results to support non-membership proofs, i.e. proving $$u \notin S$$ u ∉ S . Being commit-and-prove, our solutions can act as plug-and-play modules in statements of the form “$$u \in S$$ u ∈ S and P(u) holds” by combining our set (non-)membership systems with any other commit-and-prove scheme for P(u). Also, they work with Pedersen commitments over prime order groups which makes them compatible with popular systems such as Bulletproofs or Groth16. We implemented our schemes as a software library, and tested experimentally their performance. Compared to previous work that achieves similar properties—the clever techniques combining zkSNARKs and Merkle Trees in Zcash—our solutions offer more flexibility, shorter public parameters and $$3.7 \times $$ 3.7 × –$$30\times $$ 30 × faster proving time for a set of size $$2^{64}$$ 2 64 .

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Section: Recent Developmentsmentioning

confidence: 99%

Zero-knowledge proofs for set membership: efficient, succinct, modular

Benarroch¹,

Campanelli²,

Fiore³

et al. 2023

Des. Codes Cryptogr.

View full text Add to dashboard Cite

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Efficient NIZKs for Algebraic Sets

Couteau¹,

Lipmaa²,

Parisella³

et al. 2021

Lecture Notes in Computer Science

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Significantly extending the framework of (Couteau and Hartmann, Crypto 2020), we propose a general methodology to construct NIZKs for showing that an encrypted vector χ belongs to an algebraic set, i.e., is in the zero locus of an ideal I of a polynomial ring. In the case where I is principal, i.e., generated by a single polynomial F , we first construct a matrix that is a "quasideterminantal representation" of F and then a NIZK argument to show that F (χ) = 0. This leads to compact NIZKs for general computational structures, such as polynomial-size algebraic branching programs. We extend the framework to the case where I is non-principal, obtaining efficient NIZKs for R1CS, arithmetic constraint satisfaction systems, and thus for NP. As an independent result, we explicitly describe the corresponding language of ciphertexts as an algebraic language, with smaller parameters than in previous constructions that were based on the disjunction of algebraic languages. This results in an efficient GL-SPHF for algebraic branching programs.

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Set (Non-)Membership NIZKs from Determinantal Accumulators

Lipmaa,

Parisella

2023

Lecture Notes in Computer Science

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Efficient NIZKs for Algebraic Sets

Cited by 4 publications

References 45 publications

Zero-knowledge proofs for set membership: efficient, succinct, modular

Zero-knowledge proofs for set membership: efficient, succinct, modular

Efficient NIZKs for Algebraic Sets

Set (Non-)Membership NIZKs from Determinantal Accumulators

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