Feat: Implementation of Elliptic Curve Diffie Hellman Key Exchange. (T…

…heAlgorithms#1479) * updating DIRECTORY.md * Feat: Elliptic Curve Diffie Hellman Key Exchange, Ciphers * updating DIRECTORY.md * Feat: Elliptic Curve Diffie Hellman Key Exchange, Ciphers: Error handling * Feat: Elliptic Curve Diffie Hellman Key Exchange, Ciphers: Error handling * Feat: Elliptic Curve Diffie Hellman Key Exchange, Ciphers: Error handling-2 * Feat: Elliptic Curve Diffie Hellman Key Exchange, Ciphers: Error handling-bit handling * Feat: Elliptic Curve Diffie Hellman Key Exchange, Ciphers: Error handling-bit handling * Feat: Elliptic Curve Diffie Hellman Key Exchange, Ciphers: Error handling-bit handling * Type checks and destructor added * Type checks and integer shift checked * clang-format and clang-tidy fixes for 276fde9 * Comment modification * clang-format and clang-tidy fixes for ae6a048 * Comment modification * Wrong return * clang-format and clang-tidy fixes for bb40ea4 * Type checks * windows error * clang-format and clang-tidy fixes for 2c41f11 * Error handling * Error handling-2 * Comments * Comment modifications * Update ciphers/uint128_t.hpp Co-authored-by: David Leal <[email protected]> * Comment modifications-2 * Comment modifications-3 * Empty commit * Comments * Additional comments * clang-format and clang-tidy fixes for f7daaa1 * Empty commit for build * Additional test correction and comment modification Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com> Co-authored-by: David Leal <[email protected]>
shiv-tyagi · Apr 14, 2021 · d9a97f1 · d9a97f1
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diff --git a/DIRECTORY.md b/DIRECTORY.md
@@ -16,8 +16,11 @@
 ## Ciphers
   * [Base64 Encoding](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/ciphers/base64_encoding.cpp)
   * [Caesar Cipher](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/ciphers/caesar_cipher.cpp)
+  * [Elliptic Curve Key Exchange](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/ciphers/elliptic_curve_key_exchange.cpp)
   * [Hill Cipher](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/ciphers/hill_cipher.cpp)
   * [Morse Code](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/ciphers/morse_code.cpp)
+  * [Uint128 T](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/ciphers/uint128_t.hpp)
+  * [Uint256 T](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/ciphers/uint256_t.hpp)
   * [Vigenere Cipher](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/ciphers/vigenere_cipher.cpp)
   * [Xor Cipher](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/ciphers/xor_cipher.cpp)
 

diff --git a/ciphers/elliptic_curve_key_exchange.cpp b/ciphers/elliptic_curve_key_exchange.cpp
@@ -0,0 +1,325 @@
+/**
+ * @file
+ * @brief Implementation of [Elliptic Curve Diffie Hellman Key
+ * Exchange](https://cryptobook.nakov.com/asymmetric-key-ciphers/ecdh-key-exchange).
+ *
+ * @details
+ * The ECDH (Elliptic Curve Diffie–Hellman Key Exchange) is anonymous key
+ * agreement scheme, which allows two parties, each having an elliptic-curve
+ * public–private key pair, to establish a shared secret over an insecure
+ * channel.
+ * ECDH is very similar to the classical DHKE (Diffie–Hellman Key Exchange)
+ * algorithm, but it uses ECC point multiplication instead of modular
+ * exponentiations. ECDH is based on the following property of EC points:
+ * (a * G) * b = (b * G) * a
+ * If we have two secret numbers a and b (two private keys, belonging to Alice
+ * and Bob) and an ECC elliptic curve with generator point G, we can exchange
+ * over an insecure channel the values (a * G) and (b * G) (the public keys of
+ * Alice and Bob) and then we can derive a shared secret:
+ * secret = (a * G) * b = (b * G) * a.
+ * Pretty simple. The above equation takes the following form:
+ * alicePubKey * bobPrivKey = bobPubKey * alicePrivKey = secret
+ * @author [Ashish Daulatabad](https://github.com/AshishYUO)
+ */
+#include <cassert>   /// for assert
+#include <iostream>  /// for IO operations
+
+#include "uint256_t.hpp"  /// for 256-bit integer
+
+/**
+ * @namespace ciphers
+ * @brief Cipher algorithms
+ */
+namespace ciphers {
+/**
+ * @brief namespace elliptic_curve_key_exchange
+ * @details Demonstration of [Elliptic Curve
+ * Diffie-Hellman](https://cryptobook.nakov.com/asymmetric-key-ciphers/ecdh-key-exchange)
+ * key exchange.
+ */
+namespace elliptic_curve_key_exchange {
+
+/**
+ * @brief Definition of struct Point
+ * @details Definition of Point in the curve.
+ */
+typedef struct Point {
+    uint256_t x, y;  /// x and y co-ordinates
+
+    /**
+     * @brief operator == for Point
+     * @details check whether co-ordinates are equal to the given point
+     * @param p given point to be checked with this
+     * @returns true if x and y are both equal with Point p, else false
+     */
+    inline bool operator==(const Point &p) { return x == p.x && y == p.y; }
+
+    /**
+     * @brief ostream operator for printing Point
+     * @param op ostream operator
+     * @param p Point to be printed on console
+     * @returns op, the ostream object
+     */
+    friend std::ostream &operator<<(std::ostream &op, const Point &p) {
+        op << p.x << " " << p.y;
+        return op;
+    }
+} Point;
+
+/**
+ * @brief This function calculates number raised to exponent power under modulo
+ * mod using [Modular
+ * Exponentiation](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/modular_exponentiation.cpp).
+ * @param number integer base
+ * @param power unsigned integer exponent
+ * @param mod integer modulo
+ * @return number raised to power modulo mod
+ */
+uint256_t exp(uint256_t number, uint256_t power, const uint256_t &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;
+}
+
+/**
+ * @brief Addition of points
+ * @details Add given point to generate third point. More description can be
+ * found
+ * [here](https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Point_addition),
+ * and
+ * [here](https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Point_doubling)
+ * @param a First point
+ * @param b Second point
+ * @param curve_a_coeff Coefficient `a` of the given curve (y^2 = x^3 + ax + b)
+ * % mod
+ * @param mod Given field
+ * @return the resultant point
+ */
+Point addition(Point a, Point b, const uint256_t &curve_a_coeff,
+               uint256_t mod) {
+    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;
+}
+
+/**
+ * @brief multiply Point and integer
+ * @details Multiply Point by a scalar factor (here it is a private key p). The
+ * multiplication is called as [double and add
+ * method](https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Double-and-add)
+ * @param a Point to multiply
+ * @param curve_a_coeff Coefficient of given curve (y^2 = x^3 + ax + b) % mod
+ * @param p The scalar value
+ * @param mod Given field
+ * @returns the resultant point
+ */
+Point multiply(const Point &a, const uint256_t &curve_a_coeff, uint256_t p,
+               const uint256_t &mod) {
+    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;
+}
+}  // namespace elliptic_curve_key_exchange
+}  // namespace ciphers
+
+/**
+ * @brief Function to test the
+ * uint128_t header
+ * @returns void
+ */
+static void uint128_t_tests() {
+    // 1st test: Operations test
+    uint128_t a("122"), b("2312");
+    assert(a + b == 2434);
+    assert(b - a == 2190);
+    assert(a * b == 282064);
+    assert(b / a == 18);
+    assert(b % a == 116);
+    assert((a & b) == 8);
+    assert((a | b) == 2426);
+    assert((a ^ b) == 2418);
+    assert((a << 64) == uint128_t("2250502776992565297152"));
+    assert((b >> 7) == 18);
+
+    // 2nd test: Operations test
+    a = uint128_t("12321421424232142122");
+    b = uint128_t("23123212");
+    assert(a + b == uint128_t("12321421424255265334"));
+    assert(a - b == uint128_t("12321421424209018910"));
+    assert(a * b == uint128_t("284910839733861759501135864"));
+    assert(a / b == 532859423865LL);
+    assert(a % b == 3887742);
+    assert((a & b) == 18912520);
+    assert((a | b) == uint128_t("12321421424236352814"));
+    assert((a ^ b) == uint128_t("12321421424217440294"));
+    assert((a << 64) == uint128_t("227290107637132170748078080907806769152"));
+}
+
+/**
+ * @brief Function to test the
+ * uint256_t header
+ * @returns void
+ */
+static void uint256_t_tests() {
+    // 1st test: Operations test
+    uint256_t a("122"), b("2312");
+    assert(a + b == 2434);
+    assert(b - a == 2190);
+    assert(a * b == 282064);
+    assert(b / a == 18);
+    assert(b % a == 116);
+    assert((a & b) == 8);
+    assert((a | b) == 2426);
+    assert((a ^ b) == 2418);
+    assert((a << 64) == uint256_t("2250502776992565297152"));
+    assert((b >> 7) == 18);
+
+    // 2nd test: Operations test
+    a = uint256_t("12321423124513251424232142122");
+    b = uint256_t("23124312431243243215354315132413213212");
+    assert(a + b == uint256_t("23124312443564666339867566556645355334"));
+    // Since a < b, the value is greater
+    assert(a - b == uint256_t("115792089237316195423570985008687907853246860353"
+                              "221642219366742944204948568846"));
+    assert(a * b == uint256_t("284924437928789743312147393953938013677909398222"
+                              "169728183872115864"));
+    assert(b / a == uint256_t("1876756621"));
+    assert(b % a == uint256_t("2170491202688962563936723450"));
+    assert((a & b) == uint256_t("3553901085693256462344"));
+    assert((a | b) == uint256_t("23124312443564662785966480863388892990"));
+    assert((a ^ b) == uint256_t("23124312443564659232065395170132430646"));
+    assert((a << 128) == uint256_t("4192763024643754272961909047609369343091683"
+                                   "376561852756163540549632"));
+}
+
+/**
+ * @brief Function to test the
+ * provided algorithm above
+ * @returns void
+ */
+static void test() {
+    // demonstration of key exchange using curve secp112r1
+
+    // Equation of the form y^2 = (x^3 + ax + b) % P (here p is mod)
+    uint256_t a("4451685225093714772084598273548424"),
+        b("2061118396808653202902996166388514"),
+        mod("4451685225093714772084598273548427");
+
+    // Generator value: is pre-defined for the given curve
+    ciphers::elliptic_curve_key_exchange::Point ptr = {
+        uint256_t("188281465057972534892223778713752"),
+        uint256_t("3419875491033170827167861896082688")};
+
+    // Shared key generation.
+    // For alice
+    std::cout << "For alice:\n";
+    // Alice's private key (can be generated randomly)
+    uint256_t alice_private_key("164330438812053169644452143505618");
+    ciphers::elliptic_curve_key_exchange::Point alice_public_key =
+        multiply(ptr, a, alice_private_key, mod);
+    std::cout << "\tPrivate key: " << alice_private_key << "\n";
+    std::cout << "\tPublic Key: " << alice_public_key << std::endl;
+
+    // For Bob
+    std::cout << "For Bob:\n";
+    // Bob's private key (can be generated randomly)
+    uint256_t bob_private_key("1959473333748537081510525763478373");
+    ciphers::elliptic_curve_key_exchange::Point bob_public_key =
+        multiply(ptr, a, bob_private_key, mod);
+    std::cout << "\tPrivate key: " << bob_private_key << "\n";
+    std::cout << "\tPublic Key: " << bob_public_key << std::endl;
+
+    // After public key exchange, create a shared key for communication.
+    // create shared key:
+    ciphers::elliptic_curve_key_exchange::Point alice_shared_key = multiply(
+                                                    bob_public_key, a,
+                                                    alice_private_key, mod),
+                                                bob_shared_key = multiply(
+                                                    alice_public_key, a,
+                                                    bob_private_key, mod);
+
+    std::cout << "Shared keys:\n";
+    std::cout << alice_shared_key << std::endl;
+    std::cout << bob_shared_key << std::endl;
+
+    // Check whether shared keys are equal
+    assert(alice_shared_key == bob_shared_key);
+}
+
+/**
+ * @brief Main function
+ * @returns 0 on exit
+ */
+int main() {
+    uint128_t_tests();  // running predefined 128-bit unsigned integer tests
+    uint256_t_tests();  // running predefined 256-bit unsigned integer tests
+    test();             // running self-test implementations
+    return 0;
+}