ciphers::elliptic_curve_key_exchange Namespace Reference

namespace elliptic_curve_key_exchange More...

Classes
struct	Point
	Definition of struct Point. More...

Typedefs
typedef struct ciphers::elliptic_curve_key_exchange::Point	Point
	Definition of struct Point. More...

Functions
uint256_t	exp (uint256_t number, uint256_t power, const uint256_t &mod)
	This function calculates number raised to exponent power under modulo mod using Modular Exponentiation. More...
Point	addition (Point a, Point b, const uint256_t &curve_a_coeff, uint256_t mod)
	Addition of points. More...
Point	multiply (const Point &a, const uint256_t &curve_a_coeff, uint256_t p, const uint256_t &mod)
	multiply Point and integer More...

Detailed Description

namespace elliptic_curve_key_exchange

Demonstration of Elliptic Curve Diffie-Hellman key exchange.

Typedef Documentation

Point

typedef struct ciphers::elliptic_curve_key_exchange::Point ciphers::elliptic_curve_key_exchange::Point

Definition of struct Point.

Definition of Point in the curve.

Function Documentation

addition()

Point ciphers::elliptic_curve_key_exchange::addition	(	Point	a,
		Point	b,
		const uint256_t &	curve_a_coeff,
		uint256_t	mod
	)

Addition of points.

Add given point to generate third point. More description can be found here, and here

Parameters

a	First point
b	Second point
curve_a_coeff	Coefficient a of the given curve (y^2 = x^3 + ax + b) % mod
mod	Given field

Returns

the resultant point

Slope

value zero

slope when the line is tangent to curve. This operation is performed while doubling. Taking derivative of y^2 = x^3 + ax + b => 2y dy = (3 * x^2 + a)dx => (dy/dx) = (3x^2 + a)/(2y)

if y co-ordinate is zero, the slope is infinite, return inf. else calculate the slope (here % mod and store in lambda)

                              {
    uint256_t lambda(0);  /// Slope
    uint256_t zero(0);    /// value zero
    lambda = zero = 0;
    uint256_t inf = ~zero;
    if (a.x != b.x || a.y != b.y) {
        // Slope being infinite.
        if (b.x == a.x) {
            return {inf, inf};
        }
        uint256_t num = (b.y - a.y + mod), den = (b.x - a.x + mod);
        lambda = (num * (exp(den, mod - 2, mod))) % mod;
    } else {
        /**
         *  slope when the line is tangent to curve. This operation is performed
         * while doubling. Taking derivative of `y^2 = x^3 + ax + b`
         * => `2y dy = (3 * x^2 + a)dx`
         * => `(dy/dx) = (3x^2 + a)/(2y)`
         */
        /**
         * if y co-ordinate is zero, the slope is infinite, return inf.
         * else calculate the slope (here % mod and store in lambda)
         */
        if (!a.y) {
            return {inf, inf};
        }
        uint256_t axsq = ((a.x * a.x)) % mod;
        // Mulitply by 3 adjustment
        axsq += (axsq << 1);
        axsq %= mod;
        // Mulitply by 2 adjustment
        uint256_t a_2 = (a.y << 1);
        lambda =
            (((axsq + curve_a_coeff) % mod) * exp(a_2, mod - 2, mod)) % mod;
    }
    Point c;
    // new point: x = ((lambda^2) - x1 - x2)
    // y = (lambda * (x1 - x)) - y1
    c.x = ((lambda * lambda) % mod + (mod << 1) - a.x - b.x) % mod;
    c.y = (((lambda * (a.x + mod - c.x)) % mod) + mod - a.y) % mod;
    return c;
}

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exp()

uint256_t ciphers::elliptic_curve_key_exchange::exp	(	uint256_t	number,
		uint256_t	power,
		const uint256_t &	mod
	)

This function calculates number raised to exponent power under modulo mod using Modular Exponentiation.

Parameters

number	integer base
power	unsigned integer exponent
mod	integer modulo

Returns

number raised to power modulo mod

                                                                       {
    if (!power) {
        return uint256_t(1);
    }
    uint256_t ans(1);
    number = number % mod;
    while (power) {
        if ((power & 1)) {
            ans = (ans * number) % mod;
        }
        power >>= 1;
        if (power) {
            number = (number * number) % mod;
        }
    }
    return ans;
}

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multiply()

Point ciphers::elliptic_curve_key_exchange::multiply	(	const Point &	a,
		const uint256_t &	curve_a_coeff,
		uint256_t	p,
		const uint256_t &	mod
	)

multiply Point and integer

Multiply Point by a scalar factor (here it is a private key p). The multiplication is called as double and add method

Parameters

a	Point to multiply
curve_a_coeff	Coefficient of given curve (y^2 = x^3 + ax + b) % mod
p	The scalar value
mod	Given field

Returns

the resultant point

                                     {
    Point N = a;
    N.x %= mod;
    N.y %= mod;
    uint256_t inf{};
    inf = ~uint256_t(0);
    Point Q = {inf, inf};
    while (p) {
        if ((p & 1)) {
            if (Q.x == inf && Q.y == inf) {
                Q.x = N.x;
                Q.y = N.y;
            } else {
                Q = addition(Q, N, curve_a_coeff, mod);
            }
        }
        p >>= 1;
        if (p) {
            N = addition(N, N, curve_a_coeff, mod);
        }
    }
    return Q;
}

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