Welcome to the resource topic for
**2023/1705**

**Title:**

BaseFold: Efficient Field-Agnostic Polynomial Commitment Schemes from Foldable Codes

**Authors:**
Hadas Zeilberger, Binyi Chen, Ben Fisch

**Abstract:**

Interactive Oracle Proof of Proximity (IOPPs) are a powerful tool for constructing succinct non-interactive arguments of knowledge (SNARKs) in the random oracle model, which are fast and plausibly post-quantum secure. The Fast Reed Solomon IOPP (FRI) is the most widely used in practice, while tensor-code IOPPs (such as Brakedown) achieve significantly faster prover times at the cost of much larger proofs. IOPPs are used to construct polynomial commitment schemes (PCS), which are not only an important building block for SNARKs but also have a wide range of independent applications.

This work introduces Basefold, a generalization of the FRI IOPP to a broad class of linear codes beyond Reed-Solomon, which we call \textit{foldable linear codes}. We construct a new family of foldable linear codes, which are a special type of randomly punctured Reed-Muller code, and prove tight bounds on their minimum distance. Finally, we introduce a new construction of a multilinear PCS from any foldable linear code, which is based on interleaving Basefold with the classical sumcheck protocol for multilinear polynomial evaluation. As a special case, this gives a new multilinear PCS from FRI.

In addition to these theoretical contributions, the Basefold PCS instantiated with our new foldable linear codes offers a more reasonable tradeoff between prover time, proof size, and verifier time than prior constructions. For instance, for polynomials over a 64-bit field with 12 variables, the Basefold prover is faster than both Brakedown and FRI-PCS (2 times faster than Brakedown and 3 times faster than FRI-PCS), and its proof is 4 times smaller than Brakedown’s. On the other hand, for polynomials with 25 variables, Basefold’s prover is 6.5 times faster than FRI-PCS, it’s proof is 2.5 times smaller than Brakedown’s and its verifier is 7.5 times faster. Using Basefold to compile the Hyperplonk PIOP [CBBZ23] results in an extremely fast implementation of Hyperplonk, which in addition to having competitive performance on general circuits, is particularly fast for circuits with high-degree custom gates (e.g., signature verification and table lookups). Hyperplonk with Basefold is approximately equivalent to the speed of Hyperplonk with Brakedown, but with a proof size that is more than 5 times smaller. Finally, Basefold maintains performance across a wider variety of field choices than FRI, which requires FFT-friendly fields. Thus, Basefold can have an extremely fast prover compared to SNARKs from FRI for special applications. Benchmarking a circom ECDSA verification circuit with curve secp256k1, Hyperplonk with Basefold has a prover time that is more than 200\times faster than with FRI and its proof size is 5.8 times smaller than Hyperplonk with Brakedown.

**ePrint:**
https://eprint.iacr.org/2023/1705

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