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A collection of sets displays aproximity gapwith respect to some property if for every set in the collection, either (i) all members areδ-close to the property in relative Hamming distance or (ii) only a tiny fraction of members areδ-close to the property. In particular, no set in the collection has roughly half of its membersδ-close to the property and the othersδ-far from it. We show that the collection of affine spaces displays a proximity gap with respect to Reed–Solomon (RS) codes, even over small fields, of size polynomial in the dimension of the code, and the gap applies to anyδsmaller than the Johnson/Guruswami–Sudan list-decoding bound of the RS code. We also show near-optimal gap results, over fields of (at least)linearsize in the RS code dimension, forδsmaller than the unique decoding radius. Concretely, ifδis smaller than half the minimal distance of an RS code\(V\subset {\mathbb {F}}_q^n \), every affine space is either entirelyδ-close to the code, or alternatively at most an (n/q)-fraction of it isδ-close to the code. Finally, we discuss several applications of our proximity gap results to distributed storage, multi-party cryptographic protocols, and concretely efficient proof systems. We prove the proximity gap results by analyzing the execution of classical algebraic decoding algorithms for Reed–Solomon codes (due to Berlekamp–Welch and Guruswami–Sudan) on aformal elementof an affine space. This involves working with Reed–Solomon codes whose base field is an (infinite) rational function field. Our proofs are obtained by developing an extension (to function fields) of a strategy of Arora and Sudan for analyzing low-degree tests. 

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 ⊆ F^n 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).