Struct FieldElement

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pub struct FieldElement(/* private fields */);

Expand description

P-224 field element representing values in F_p

Internally stored as 7 little-endian 32-bit limbs for efficient arithmetic. All operations maintain the invariant that values are reduced modulo p.

Implementations§

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impl FieldElement

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pub fn zero() -> Self

The additive identity element: 0

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pub fn one() -> Self

The multiplicative identity element: 1

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pub fn from_bytes(bytes: &[u8; 28]) -> Result<Self>

Create a field element from big-endian byte representation

Validates that the input represents a value less than the field modulus p. Returns an error if the value is >= p.

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pub fn to_bytes(&self) -> [u8; 28]

Convert field element to big-endian byte representation

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pub fn is_valid(&self) -> bool

Constant-time validation that the field element is in canonical form (< p)

Uses constant-time subtraction to check if self < p without branching. Returns true if the element is valid (< p), false otherwise.

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pub fn add(&self, other: &Self) -> Self

Constant-time field addition: (self + other) mod p

Algorithm:

Perform full 224-bit addition with carry detection
Conditionally subtract p if result >= p
Ensure result is in canonical form

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pub fn sub(&self, other: &Self) -> Self

Constant-time field subtraction: (self - other) mod p

Algorithm:

Perform limb-wise subtraction
If subtraction borrows, add p to get the correct positive result

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pub fn mul(&self, other: &Self) -> Self

Field multiplication: (self * other) mod p

Algorithm:

Compute the full 448-bit product using schoolbook multiplication
Perform carry propagation to get proper limb representation
Apply NIST P-224 specific fast reduction

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pub fn square(&self) -> Self

Field squaring: self² mod p

Optimized version of multiplication for the case where both operands are the same. Currently implemented as self.mul(self) but could be optimized further with dedicated squaring algorithms.

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pub fn invert(&self) -> Result<Self>

Compute the modular multiplicative inverse using Fermat’s Little Theorem

For prime fields, a^(p-1) ≡ 1 (mod p), so a^(p-2) ≡ a^(-1) (mod p). Uses binary exponentiation (square-and-multiply) for efficiency.

Returns an error if attempting to invert zero (which has no inverse).

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pub fn is_zero(&self) -> bool

Check if the field element represents zero

Constant-time check across all limbs to determine if the field element is the additive identity.

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pub fn is_odd(&self) -> bool

Return true if the field element is odd (least-significant bit set)

Used for point compression to determine the sign of the y-coordinate. The parity is determined by the least significant bit of the canonical representation.

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pub fn sqrt(&self) -> Option<Self>

Compute the modular square root using Tonelli‑Shanks.

Because the P‑224 prime satisfies p ≡ 1 (mod 4), we cannot use the simple (p+1)/4 exponent trick. Instead we implement the general Tonelli‑Shanks algorithm which works for any odd prime.

‑ Returns Some(root) with any square‑root of self when the element is a quadratic residue. ‑ Returns None when no square‑root exists.