> ## Documentation Index
> Fetch the complete documentation index at: https://starkware-9575960b-odednaor-patch-1.mintlify.site/llms.txt
> Use this file to discover all available pages before exploring further.

# S-two: Under the Hood

> This section provides an overview of the core components and protocols that make up S-two. It is intended for readers who want to understand the technical details and inner workings of the system implementation. Throughout this section, we describe the implementation as available in this [commit](https://github.com/starkware-libs/stwo/tree/0790eba46b8af5697083d84fb75bd34b08a0b31f).

The following topics are covered:

* [**Mersenne Primes**](./mersenne-prime): Introduction to the Mersenne31 prime field used in S-two for efficient arithmetic.
* [**Circle Group**](./circle-group): Explains the algebraic structure underlying FFT and polynomial operations.
* [**Circle Polynomials**](./circle-polynomials/index): Details the representation and evaluation of polynomials in the circle group.
* [**Circle FFT**](./circle-fft/index): Describes the fast Fourier transform algorithm adapted for the circle group.
* [**Vector Commitment Scheme (VCS)**](./vcs/index): Covers the use of Merkle trees for committing to vectors and enabling efficient proofs of inclusion.
* [**AIR to Composition Polynomial**](./air/index): Shows how algebraic constraints are encoded as polynomials for proof generation.
* [**Circle FRI**](./circle-fri/index): Explains the FRI protocol for low-degree testing of polynomials over the circle group.
* [**Polynomial Commitment Scheme (PCS)**](./pcs/index): Describes the protocol for committing to and opening polynomials with soundness guarantees.
* [**Proof Generation and Verification**](./stark_proof/index): Walks through the process of generating and verifying a STARK proof in S-two.
* [**Lookups**](./lookups): Discusses lookup arguments and their implementation in S-two for efficient constraint checking.