Enum CurveType 

pub enum CurveType {
    Weierstrass {
        a: FieldElement,
        b: FieldElement,
    },
    Edwards {
        a: FieldElement,
        d: FieldElement,
    },
}

Elliptic curve type enumeration defining supported curve models.

This enum represents the two main families of elliptic curves used in cryptography, each with different mathematical properties and performance characteristics. The choice of curve model affects implementation complexity, performance, and security properties.

Mathematical Foundations

Weierstrass Curves

Equation: y² = x³ + ax + b

Classic curve representation used by most standardized curves including:

• NIST P-curves (P-256, P-384, P-521)
• secp256k1 (Bitcoin curve)
• Brainpool curves

Properties:

• Point addition requires case analysis (doubling vs. addition)
• Point at infinity requires special handling
• More complex implementation but widely supported

Edwards Curves

Equation: ax² + y² = 1 + dx²y² (twisted Edwards form)

Modern curve representation with better properties:

• Ed25519 (digital signatures)
• Curve25519 (key exchange)
• More recent standardized curves

Properties:

• Unified addition formulas (no special cases)
• Complete formulas (no exceptional points)
• Faster arithmetic, simpler implementation
• Better resistance to some implementation attacks

Security Considerations

Implementation Attacks

• Edwards curves: More resistant to timing attacks due to unified formulas
• Weierstrass curves: Require careful handling of point doubling cases

Side-Channel Resistance

• Both models can be implemented with constant-time arithmetic
• Edwards curves have simpler control flow
• Choice affects cache access patterns and power consumption

Performance Characteristics

Addition Speed

• Edwards: Faster due to unified formulas
• Weierstrass: Slightly slower due to case analysis

Implementation Complexity

• Edwards: Simpler code, fewer edge cases
• Weierstrass: More complex, more special cases

Memory Usage

• Both models use similar memory for point operations
• Curve parameters (a,b,d) stored in enum variants

Usage Examples

use clock_curve_math::field::elliptic_curve::CurveType;

// secp256k1 (Bitcoin curve)
let secp256k1 = CurveType::Weierstrass {
    a: FieldElement::from_u64(0),
    b: FieldElement::from_u64(7),
};

// Ed25519 curve
let ed25519 = CurveType::Edwards {
    a: FieldElement::from_u64(0).sub(&FieldElement::from_u64(1)), // a = -1
    d: /* computed d parameter */,
};

Conversion Between Models

Some curves can be represented in both forms:

• Curve25519: Can be converted between Montgomery and Edwards forms
• secp256k1: Weierstrass form only
• NIST P-256: Weierstrass form only

The enum prevents mixing operations between incompatible curve types.

Variants

Weierstrass

Weierstrass curve model: y² = x³ + ax + b

The classical elliptic curve representation used by most standardized cryptographic curves. Point arithmetic uses chord-tangent formulas with special cases for point doubling.

Parameters

• a: Linear coefficient in the curve equation
• b: Constant term in the curve equation

Examples

• secp256k1: a=0, b=7
• NIST P-256: a=-3, b=constant

Fields

a: FieldElement

Coefficient ‘a’ in the curve equation y² = x³ + ax + b

b: FieldElement

Coefficient ‘b’ in the curve equation y² = x³ + ax + b

Edwards

Edwards curve model: ax² + y² = 1 + dx²y²

Modern elliptic curve representation with unified addition formulas that work for all point combinations without special cases. Provides better performance and security properties than Weierstrass curves.

Parameters

• a: Coefficient in the x² term
• d: Coefficient in the x²y² term

Examples

• Ed25519: a=-1, d=specific constant
• Curve25519: a=constant, d=constant

Fields

a: FieldElement

Coefficient ‘a’ in the Edwards equation ax² + y² = 1 + dx²y²

d: FieldElement

Coefficient ‘d’ in the Edwards equation ax² + y² = 1 + dx²y²

Trait Implementations

impl Clone for CurveType

fn clone(&self) -> CurveType

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for CurveType

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

Formats the value using the given formatter.

impl Copy for CurveType