Module halo2_proofs::arithmetic
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This module provides common utilities, traits and structures for group, field and polynomial arithmetic.
Structs
- The affine coordinates of a point on an elliptic curve.
Traits
- This trait is the affine counterpart to
Curve
and is used for serialization, storage in memory, and inspection of $x$ and $y$ coordinates. - This trait is a common interface for dealing with elements of an elliptic curve group in a “projective” form, where that arithmetic is usually more efficient.
- This represents an element of a group with basic operations that can be performed. This allows an FFT implementation (for example) to operate generically over either a field or elliptic curve group.
- This trait represents an element of a field.
Functions
- Performs a radix-$2$ Fast-Fourier Transformation (FFT) on a vector of size $n = 2^k$, when provided
log_n
= $k$ and an element of multiplicative order $n$ calledomega
($\omega$). The result is that the vectora
, when interpreted as the coefficients of a polynomial of degree $n - 1$, is transformed into the evaluations of this polynomial at each of the $n$ distinct powers of $\omega$. This transformation is invertible by providing $\omega^{-1}$ in place of $\omega$ and dividing each resulting field element by $n$. - Performs a multi-exponentiation operation.
- This computes the inner product of two vectors
a
andb
. - This evaluates a provided polynomial (in coefficient form) at
point
. - Divides polynomial
a
inX
byX - b
with no remainder. - Returns coefficients of an n - 1 degree polynomial given a set of n points and their evaluations. This function will panic if two values in
points
are the same. - This simple utility function will parallelize an operation that is to be performed over a mutable slice.
- This perform recursive butterfly arithmetic
- Performs a small multi-exponentiation operation. Uses the double-and-add algorithm with doublings shared across points.