math

Winter math

This crate contains modules with mathematical operations needed in STARK proof generation and verification.

Finite field

Finite field module implements arithmetic operations in STARK-friendly finite fields. The operation include:

• Basic arithmetic operations: addition, multiplication, subtraction, division, inversion.
• Drawing random and pseudo-random elements from the field.
• Computing roots of unity of a given order.

Currently, there are two implementations of finite fields:

• A 128-bit field with modulus 2¹²⁸ - 45 * 2⁴⁰ + 1. This field was not chosen with any significant thought given to performance, and the implementation of most operations is sub-optimal as well. Proofs generated in this field can support security level of ~100 bits. If higher level of security is desired, proofs must be generated in a quadratic extension of the field.
• A 62-bit field with modulus 2⁶² - 111 * 2³⁹ + 1. This field supports very fast modular arithmetic including branchless multiplication and addition. To achieve adequate security (i.e. ~100 bits), proofs must be generated in a quadratic extension of this field. For higher levels of security, a cubic extension field should be used.

Extension fields

Currently, the library provides a generic way to create quadratic extensions of STARK fields. An extension element is defined as α + β * φ, where φ is a root of the polynomial x² - x - 1, and α and β are base field elements.

Support for cubic extension fields is not yet available.

Polynomials

Polynomials module implements basic polynomial operations such as:

• Evaluation of a polynomial at a single point.
• Interpolation of a polynomial from a set of points (using Lagrange interpolation).
• Addition, multiplication, subtraction, and division of polynomials.
• Synthetic polynomial division (using Ruffini's method).

Fast Fourier transform

FFT module contains operations for computing Fast Fourier transform in a prime field (also called Number-theoretic transform). This can be used to interpolate and evaluate polynomials in O(n log n) time as long as the domain of the polynomial is a multiplicative subgroup with size which is a power of 2.

Crate features

This crate can be compiled with the following features:

• std - enabled by default and relies on the Rust standard library.
• std + concurrent - same as std but enables multi-threaded execution for some of the crate functions.
• no_std + alloc - does not rely on Rust's standard library and enables compilation to WebAssembly.

Concurrent execution

When compiled with concurrent feature enabled, the following operations will be executed in multiple threads:

• fft module:
  • evaluate_poly()
  • evaluate_poly_with_offset()
  • interpolate_poly()
  • interpolate_poly_with_offset()
  • get_twiddles()
  • get_inv_twiddles()
• utils module:
  • get_power_series()
  • get_power_series_with_offset()
  • add_in_place()
  • mul_acc()
  • batch_inversion()

The number of threads can be configured via RAYON_NUM_THREADS environment variable, and usually defaults to the number of logical cores on the machine.

WebAssembly support

To compile this crate to WebAssembly, disable default features and enable the alloc feature.

License

This project is MIT licensed.