Arithmetic Sketching

Dan Boneh, Elette Boyle, Henry Corrigan-Gibbs, Niv Gilboa, Yuval Ishai

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

This paper introduces arithmetic sketching, an abstraction of a primitive that several previous works use to achieve lightweight, low-communication zero-knowledge verification of secret-shared vectors. An arithmetic sketching scheme for a language L⊆ Fn consists of (1) a randomized linear function compressing a long input x to a short “sketch,” and (2) a small arithmetic circuit that accepts the sketch if and only if x∈ L, up to some small error. If the language L has an arithmetic sketching scheme with short sketches, then it is possible to test membership in L using an arithmetic circuit with few multiplication gates. Since multiplications are the dominant cost in protocols for computation on secret-shared, encrypted, and committed data, arithmetic sketching schemes give rise to lightweight protocols in each of these settings. Beyond the formalization of arithmetic sketching, our contributions are: A general framework for constructing arithmetic sketching schemes from algebraic varieties. This framework unifies schemes from prior work and gives rise to schemes for useful new languages and with improved soundness error.The first arithmetic sketching schemes for languages of sparse vectors: vectors with bounded Hamming weight, bounded L1 norm, and vectors whose few non-zero values satisfy a given predicate.A method for “compiling” any arithmetic sketching scheme for a language L into a low-communication malicious-secure multi-server protocol for securely testing that a client-provided secret-shared vector is in L. We also prove the first nontrivial lower bounds showing limits on the sketch size for certain languages (e.g., vectors of Hamming-weight one) and proving the non-existence of arithmetic sketching schemes for others (e.g., the language of all vectors that contain a specific value).

Original languageEnglish
Title of host publicationAdvances in Cryptology – CRYPTO 2023 - 43rd Annual International Cryptology Conference, CRYPTO 2023, Proceedings
EditorsHelena Handschuh, Anna Lysyanskaya
PublisherSpringer Science and Business Media Deutschland GmbH
Pages171-202
Number of pages32
ISBN (Print)9783031385568
DOIs
StatePublished - 1 Jan 2023
EventAdvances in Cryptology – CRYPTO 2023 - 43rd Annual International Cryptology Conference, CRYPTO 2023, Proceedings - Santa Barbara, United States
Duration: 20 Aug 202324 Aug 2023

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume14081 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

ConferenceAdvances in Cryptology – CRYPTO 2023 - 43rd Annual International Cryptology Conference, CRYPTO 2023, Proceedings
Country/TerritoryUnited States
CitySanta Barbara
Period20/08/2324/08/23

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science

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