Struct Zp 

pub struct Zp { /* private fields */ }

Element of finite field Z_p (integers modulo prime p)

This type provides efficient modular arithmetic with automatic normalization. All operations maintain the invariant that value < modulus.

Memory Layout

The struct is 16 bytes (two u64), fitting well in registers and cache lines.

Thread Safety

Zp is Copy, Send, and Sync, making it safe for parallel computation.

Examples

use mathhook_core::algebra::Zp;

// Create field elements mod 7
let a = Zp::new(3, 7);
let b = Zp::new(5, 7);

// Arithmetic
let sum = a + b;  // 3 + 5 = 8 ≡ 1 (mod 7)
assert_eq!(sum.value(), 1);

let product = a * b;  // 3 * 5 = 15 ≡ 1 (mod 7)
assert_eq!(product.value(), 1);

Implementations

impl Zp

pub fn new(value: u64, modulus: u64) -> Self

Create a new finite field element

The value is automatically reduced modulo p.

Arguments

• value - The integer value (will be reduced mod p)
• modulus - The prime modulus p

Examples

use mathhook_core::algebra::Zp;

let a = Zp::new(10, 7);
assert_eq!(a.value(), 3);  // 10 mod 7 = 3

pub fn from_signed(value: i64, modulus: u64) -> Self

Create a new finite field element from a signed integer

Handles negative values correctly using symmetric representation.

Arguments

• value - The signed integer value
• modulus - The prime modulus p

Examples

use mathhook_core::algebra::Zp;

let a = Zp::from_signed(-3, 7);
assert_eq!(a.value(), 4);  // -3 ≡ 4 (mod 7)

pub fn value(&self) -> u64

Get the underlying value

pub fn modulus(&self) -> u64

Get the modulus

pub fn is_zero(&self) -> bool

Check if this element is zero

pub fn is_one(&self) -> bool

Check if this element is one

pub fn zero(modulus: u64) -> Self

Get the zero element of this field

pub fn one(modulus: u64) -> Self

Get the multiplicative identity (one) of this field

pub fn negate(&self) -> Self

Compute the additive inverse (-a mod p)

pub fn inverse(&self) -> FiniteFieldResult<Self>

Compute the multiplicative inverse using extended Euclidean algorithm

Uses Fermat’s little theorem: a^(-1) ≡ a^(p-2) (mod p) for prime p. However, extended GCD is faster for single inversions.

Returns

Ok(inverse) if the element is non-zero, Err otherwise.

Examples

use mathhook_core::algebra::Zp;

let a = Zp::new(3, 7);
let inv = a.inverse().unwrap();
assert_eq!((a * inv).value(), 1);  // 3 * 5 = 15 ≡ 1 (mod 7)

pub fn pow(&self, exp: u64) -> Self

Compute a^n mod p using binary exponentiation

This is O(log n) multiplications.

Arguments

• exp - The exponent

Examples

use mathhook_core::algebra::Zp;

let a = Zp::new(2, 7);
let result = a.pow(3);
assert_eq!(result.value(), 1);  // 2^3 = 8 ≡ 1 (mod 7)

pub fn to_symmetric(&self) -> i64

Convert to signed representation in [-p/2, p/2]

This is the symmetric representation, useful for CRT reconstruction.

Examples

use mathhook_core::algebra::Zp;

let a = Zp::new(6, 7);
assert_eq!(a.to_symmetric(), -1);  // 6 ≡ -1 (mod 7)

Trait Implementations

impl Add for Zp

type Output = Zp

The resulting type after applying the + operator.

fn add(self, rhs: Self) -> Self

Performs the + operation.

impl Clone for Zp

fn clone(&self) -> Zp

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for Zp

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Display for Zp

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Div for Zp

type Output = Zp

The resulting type after applying the / operator.

fn div(self, rhs: Self) -> Self

Performs the / operation.

impl Hash for Zp

fn hash<__H: Hasher>(&self, state: &mut __H)

Feeds this value into the given Hasher.

fn hash_slice<H>(data: &[Self], state: &mut H)

where H: Hasher, Self: Sized,

Feeds a slice of this type into the given Hasher.

impl Mul for Zp

type Output = Zp

The resulting type after applying the * operator.

fn mul(self, rhs: Self) -> Self

Performs the * operation.

impl Neg for Zp

type Output = Zp

The resulting type after applying the - operator.

fn neg(self) -> Self

Performs the unary - operation.

impl PartialEq for Zp

fn eq(&self, other: &Zp) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl Sub for Zp

type Output = Zp

The resulting type after applying the - operator.

fn sub(self, rhs: Self) -> Self

Performs the - operation.

impl Copy for Zp

impl Eq for Zp

impl StructuralPartialEq for Zp