采用对数随机性的邻近性测试

Benjamin E. Diamond, Jim Posen
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引用次数: 5

摘要

一个可追溯到利杰罗(Des. Codes Cryptogr. '23)的基本结果证明,相对于仿射子空间集合而言,每个固定线性块编码都表现出接近性差距,即每个给定子空间要么完全靠近编码,要么只包含靠近编码的一小部分。特别是,任何给定子空间如果没有完全靠近代码,那么该子空间中均匀随机抽样的元素必然会以很高的概率见证这一点。我们研究了这种现象的一种变体,即见证者不是从子空间中均匀抽样,而是从子空间中一个小得多的子集中抽样。我们的研究表明,只需对数数量的随机场元素(在子空间的维度上)就足以实现类似的接近性测试,而且在可能出现的假证人方面也只有对数(乘法)的损失。我们将讨论最近基于线性代码的非交互式证明的应用,包括 Brakedown(CRYPTO '23)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Proximity Testing with Logarithmic Randomness
A fundamental result dating to Ligero (Des. Codes Cryptogr. '23) establishes that each fixed linear block code exhibits proximity gaps with respect to the collection of affine subspaces, in the sense that each given subspace either resides entirely close to the code, or else contains only a small portion which resides close to the code. In particular, any given subspace's failure to reside entirely close to the code is necessarily witnessed, with high probability, by a uniformly randomly sampled element of that subspace. We investigate a variant of this phenomenon in which the witness is not sampled uniformly from the subspace, but rather from a much smaller subset of it. We show that a logarithmic number of random field elements (in the dimension of the subspace) suffice to effect an analogous proximity test, with moreover only a logarithmic (multiplicative) loss in the possible prevalence of false witnesses. We discuss applications to recent noninteractive proofs based on linear codes, including Brakedown (CRYPTO '23).
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