# Trait ff::PrimeField

``````pub trait PrimeField: Field + From<u64> {
type Repr: Copy + Default + Send + Sync + 'static + AsRef<[u8]> + AsMut<[u8]>;

const NUM_BITS: u32;
const CAPACITY: u32;
const S: u32;

// Required methods
fn from_repr(repr: Self::Repr) -> CtOption<Self>;
fn to_repr(&self) -> Self::Repr;
fn is_odd(&self) -> Choice;
fn multiplicative_generator() -> Self;
fn root_of_unity() -> Self;

// Provided methods
fn from_str_vartime(s: &str) -> Option<Self> { ... }
fn from_repr_vartime(repr: Self::Repr) -> Option<Self> { ... }
fn is_even(&self) -> Choice { ... }
}``````
Expand description

This represents an element of a prime field.

## Required Associated Types§

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#### type Repr: Copy + Default + Send + Sync + 'static + AsRef<[u8]> + AsMut<[u8]>

The prime field can be converted back and forth into this binary representation.

## Required Associated Constants§

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#### const NUM_BITS: u32

How many bits are needed to represent an element of this field.

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#### const CAPACITY: u32

How many bits of information can be reliably stored in the field element.

This is usually `Self::NUM_BITS - 1`.

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#### const S: u32

An integer `s` satisfying the equation `2^s * t = modulus - 1` with `t` odd.

This is the number of leading zero bits in the little-endian bit representation of `modulus - 1`.

## Required Methods§

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#### fn from_repr(repr: Self::Repr) -> CtOption<Self>

Attempts to convert a byte representation of a field element into an element of this prime field, failing if the input is not canonical (is not smaller than the field’s modulus).

The byte representation is interpreted with the same endianness as elements returned by `PrimeField::to_repr`.

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#### fn to_repr(&self) -> Self::Repr

Converts an element of the prime field into the standard byte representation for this field.

The endianness of the byte representation is implementation-specific. Generic encodings of field elements should be treated as opaque.

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#### fn is_odd(&self) -> Choice

Returns true iff this element is odd.

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#### fn multiplicative_generator() -> Self

Returns a fixed multiplicative generator of `modulus - 1` order. This element must also be a quadratic nonresidue.

It can be calculated using SageMath as `GF(modulus).primitive_element()`.

Implementations of this method MUST ensure that this is the generator used to derive `Self::root_of_unity`.

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#### fn root_of_unity() -> Self

Returns the `2^s` root of unity.

It can be calculated by exponentiating `Self::multiplicative_generator` by `t`, where `t = (modulus - 1) >> Self::S`.

## Provided Methods§

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#### fn from_str_vartime(s: &str) -> Option<Self>

Interpret a string of numbers as a (congruent) prime field element. Does not accept unnecessary leading zeroes or a blank string.

##### §Security

This method provides no constant-time guarantees.

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#### fn from_repr_vartime(repr: Self::Repr) -> Option<Self>

Attempts to convert a byte representation of a field element into an element of this prime field, failing if the input is not canonical (is not smaller than the field’s modulus).

The byte representation is interpreted with the same endianness as elements returned by `PrimeField::to_repr`.

##### §Security

This method provides no constant-time guarantees. Implementors of the `PrimeField` trait may optimise this method using non-constant-time logic.

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#### fn is_even(&self) -> Choice

Returns true iff this element is even.

## Object Safety§

This trait is not object safe.