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.
• A 64-bit field with modulus 2⁶⁴ - 2³² + 1. This field is about 15% slower than the 62-bit field described above, but it has a number of other attractive properties. 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 and cubic extensions of supported STARK fields. This can be done by implementing 'ExtensibleField' trait for degrees 2 and 3.

Quadratic extension fields are defined using the following irreducible polynomials:

• For f62 field, the polynomial is x² - x - 1.
• For f64 field, the polynomial is x² - x + 2.
• For f128 field, the polynomial is x² - x - 1.

Cubic extension fields are defined using the following irreducible polynomials:

• For f62 field, the polynomial is x³ + 2x + 2.
• For f64 field, the polynomial is x³ - x - 1.
• For f128 field, cubic extensions are not supported.

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.
• concurrent - implies std and also enables multi-threaded execution for some of the crate functions.
• no_std - does not rely on Rust's standard library and enables compilation to WebAssembly.

To compile with no_std, disable default features via --no-default-features flag.

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.

License

This project is MIT licensed.