DEEPFRI: Sampling outside the box improves soundness
Abstract
Motivated by the quest for scalable and succinct zero knowledge arguments, we revisit worstcasetoaveragecase reductions for linear spaces, raised by [Rothblum, Vadhan, Wigderson, STOC 2013]. We first show a sharp quantitative form of a theorem which says that if an affine space $U$ is $\delta$far in relative Hamming distance from a linear code $V$  this is the worstcase assumption  then most elements of $U$ are almost $\delta$far from $V$  this is the average case. This leads to an optimal analysis of the soundness of the FRI protocol of [BenSasson, et.al., eprint 2018] for proving proximity to ReedSolomon codes. To further improve soundness, we sample outside the box. We suggest a new protocol which asks a prover for values of a polynomial at points outside the domain of evaluation of the ReedSolomon code. We call this technique Domain Extending for Eliminating Pretenders (DEEP). We use the DEEP technique to devise two new protocols: (1) An Interactive Oracle Proof of Proximity (IOPP) for RS codes, called DEEPFRI. This soundness of the protocol improves upon that of the FRI protocol while retaining linear arithmetic proving complexity and logarithmic verifier arithmetic complexity. (2) An Interactive Oracle Proof (IOP) for the Algebraic Linking IOP (ALI) protocol used to construct zero knowledge scalable transparent arguments of knowledge (ZKSTARKs) in [BenSasson et al., eprint 2018]. The new protocol, called DEEPALI, improves soundness of this crucial step from a small constant $< 1/8$ to a constant arbitrarily close to $1$.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 arXiv:
 arXiv:1903.12243
 Bibcode:
 2019arXiv190312243B
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Cryptography and Security;
 Computer Science  Information Theory
 EPrint:
 36 pages