@Sakura please summarize this article, thanks uwu.

## TLDR

The article explores the development of “circle STARKs”, a new technique for building efficient zero-knowledge proofs over small prime fields like Mersenne31.

## Key Points

- Circle STARKs use a clever trick to enable efficient FFT-like operations over small prime fields, overcoming limitations of regular FRI.
- They work by representing polynomials as evaluations over a circle group, rather than a line.
- This allows for more efficient arithmetic and domain reduction compared to regular STARKs.
- Circle STARKs are a step towards maximizing the efficiency of the “base layer” of STARKs, paving the way for further optimizations at higher levels.

## In-depth Summary

The article starts by discussing the trend towards using smaller prime fields in STARK protocols, as this allows for more efficient arithmetic operations. However, this introduces a challenge - the standard FRI technique used in STARKs relies on being able to repeatedly halve the domain size, which is difficult with small prime fields like Mersenne31.

The article then introduces the concept of “circle STARKs”, which use a clever trick to enable efficient domain reduction. Instead of working with polynomials over a line, circle STARKs represent polynomials as evaluations over a circle group. This allows for a two-to-one map that can be repeatedly applied, similar to the doubling operation in regular FRI.

The article goes into the technical details of how this circle-based FRI and FFT work, explaining the use of Riemann-Roch spaces and the need for “quotienting” to handle the lack of a true line function. It also discusses other nuances like the use of “reverse bit order” to optimize the Merkle tree structure.

Overall, the article paints circle STARKs as an important step towards maximizing the efficiency of the “base layer” of STARKs, paving the way for further optimizations at higher levels like more efficient arithmetization of primitives and VMs.

## ELI5

Circle STARKs are a new way to make zero-knowledge proofs really fast, especially for small numbers. Instead of working with normal lines and polynomials, they use a special circle shape to do the math. This lets them shrink the problem really quickly, making the proofs much faster. It’s a clever trick that helps push the limits of how efficient these zero-knowledge proofs can be.

## Writer’s Main Point

The main point of the article is to explore the development of “circle STARKs” - a new technique for building efficient zero-knowledge proofs over small prime fields. Circle STARKs use a clever trick to enable efficient FFT-like operations, overcoming limitations of regular FRI and paving the way for further optimizations in STARK protocols.