16 unstable releases (6 breaking)
|new 0.7.1||Nov 17, 2023|
|0.6.5||Aug 10, 2023|
|0.6.4||May 26, 2023|
|0.6.1||Mar 29, 2023|
|0.2.0||Aug 24, 2021|
#343 in Cryptography
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Used in 34 crates (12 directly)
This crate contains modules with mathematical operations needed in STARK proof generation and verification.
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 three 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 supports very fast modular arithmetic (comparable to the 62-bit field described above), provides a fully constant-time implementation, and 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.
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:
f62field, the polynomial is x2 - x - 1.
f64field, the polynomial is x2 - x + 2.
f128field, the polynomial is x2 - x - 1.
Cubic extension fields are defined using the following irreducible polynomials:
f62field, the polynomial is x3 + 2x + 2.
f64field, the polynomial is x3 - x - 1.
f128field, cubic extensions are not supported.
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.
This crate can be compiled with the following features:
std- enabled by default and relies on the Rust standard library.
stdand 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
When compiled with
concurrent feature enabled, the following operations will be executed in multiple threads:
- fft module:
- utils module:
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.
This project is MIT licensed.