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| 1 | +package com.thealgorithms.ciphers; |
| 2 | + |
| 3 | +import java.math.BigInteger; |
| 4 | +import java.security.SecureRandom; |
| 5 | + |
| 6 | +/** |
| 7 | + * ECC - Elliptic Curve Cryptography |
| 8 | + * Elliptic Curve Cryptography is a public-key cryptography method that uses the algebraic structure of |
| 9 | + * elliptic curves over finite fields. ECC provides a higher level of security with smaller key sizes compared |
| 10 | + * to other public-key methods like RSA, making it particularly suitable for environments where computational |
| 11 | + * resources are limited, such as mobile devices and embedded systems. |
| 12 | + * |
| 13 | + * This class implements elliptic curve cryptography, providing encryption and decryption |
| 14 | + * functionalities based on public and private key pairs. |
| 15 | + * |
| 16 | + * @author xuyang |
| 17 | + */ |
| 18 | +public class ECC { |
| 19 | + |
| 20 | + private BigInteger privateKey; // Private key used for decryption |
| 21 | + private ECPoint publicKey; // Public key used for encryption |
| 22 | + private EllipticCurve curve; // Elliptic curve used in cryptography |
| 23 | + private ECPoint basePoint; // Base point G on the elliptic curve |
| 24 | + |
| 25 | + public EllipticCurve getCurve() { |
| 26 | + return curve; // Returns the elliptic curve |
| 27 | + } |
| 28 | + |
| 29 | + public ECC(int bits) { |
| 30 | + generateKeys(bits); // Generates public-private key pair |
| 31 | + } |
| 32 | + |
| 33 | + // Getter and Setter for private key |
| 34 | + public BigInteger getPrivateKey() { |
| 35 | + return privateKey; |
| 36 | + } |
| 37 | + |
| 38 | + public void setPrivateKey(BigInteger privateKey) { |
| 39 | + this.privateKey = privateKey; |
| 40 | + } |
| 41 | + |
| 42 | + public void setCurve(EllipticCurve curve) { |
| 43 | + this.curve = curve; |
| 44 | + } |
| 45 | + |
| 46 | + /** |
| 47 | + * Encrypts the message using the public key. |
| 48 | + * The message is transformed into an ECPoint and encrypted with elliptic curve operations. |
| 49 | + * |
| 50 | + * @param message The plain message to be encrypted |
| 51 | + * @return The encrypted message as an array of ECPoints (R, S) |
| 52 | + */ |
| 53 | + public synchronized ECPoint[] encrypt(String message) { |
| 54 | + BigInteger m = new BigInteger(message.getBytes()); // Convert message to BigInteger |
| 55 | + SecureRandom r = new SecureRandom(); // Generate random value for k |
| 56 | + BigInteger k = new BigInteger(curve.getFieldSize(), r); // Generate random scalar k |
| 57 | + |
| 58 | + // Calculate point R = k * G, where G is the base point |
| 59 | + ECPoint R = basePoint.multiply(k, curve.getP(), curve.getA()); |
| 60 | + |
| 61 | + // Calculate point S = k * publicKey + encodedMessage |
| 62 | + ECPoint S = publicKey.multiply(k, curve.getP(), curve.getA()).add(curve.encodeMessage(m), curve.getP(), curve.getA()); |
| 63 | + |
| 64 | + return new ECPoint[] { R, S }; // Return encrypted message as two ECPoints |
| 65 | + } |
| 66 | + |
| 67 | + /** |
| 68 | + * Decrypts the encrypted message using the private key. |
| 69 | + * The decryption process is the reverse of encryption, recovering the original message. |
| 70 | + * |
| 71 | + * @param encryptedMessage The encrypted message as an array of ECPoints (R, S) |
| 72 | + * @return The decrypted plain message as a String |
| 73 | + */ |
| 74 | + public synchronized String decrypt(ECPoint[] encryptedMessage) { |
| 75 | + ECPoint R = encryptedMessage[0]; // First part of ciphertext |
| 76 | + ECPoint S = encryptedMessage[1]; // Second part of ciphertext |
| 77 | + |
| 78 | + // Perform decryption: S - R * privateKey |
| 79 | + ECPoint decodedMessage = S.subtract(R.multiply(privateKey, curve.getP(), curve.getA()), curve.getP(), curve.getA()); |
| 80 | + |
| 81 | + BigInteger m = curve.decodeMessage(decodedMessage); // Decode the message from ECPoint |
| 82 | + |
| 83 | + return new String(m.toByteArray()); // Convert BigInteger back to String |
| 84 | + } |
| 85 | + |
| 86 | + /** |
| 87 | + * Generates a new public-private key pair for encryption and decryption. |
| 88 | + * |
| 89 | + * @param bits The size (in bits) of the keys to generate |
| 90 | + */ |
| 91 | + public final synchronized void generateKeys(int bits) { |
| 92 | + SecureRandom r = new SecureRandom(); |
| 93 | + curve = new EllipticCurve(bits); // Initialize a new elliptic curve |
| 94 | + basePoint = curve.getBasePoint(); // Set the base point G |
| 95 | + |
| 96 | + // Generate private key as a random BigInteger |
| 97 | + privateKey = new BigInteger(bits, r); |
| 98 | + |
| 99 | + // Generate public key as the point publicKey = privateKey * G |
| 100 | + publicKey = basePoint.multiply(privateKey, curve.getP(), curve.getA()); |
| 101 | + } |
| 102 | + |
| 103 | + /** |
| 104 | + * Class representing an elliptic curve with the form y^2 = x^3 + ax + b. |
| 105 | + */ |
| 106 | + public static class EllipticCurve { |
| 107 | + private final BigInteger a; // Coefficient a in the curve equation |
| 108 | + private final BigInteger b; // Coefficient b in the curve equation |
| 109 | + private final BigInteger p; // Prime number p, defining the finite field |
| 110 | + private final ECPoint basePoint; // Base point G on the curve |
| 111 | + |
| 112 | + // Constructor with explicit parameters for a, b, p, and base point |
| 113 | + public EllipticCurve(BigInteger a, BigInteger b, BigInteger p, ECPoint basePoint) { |
| 114 | + this.a = a; |
| 115 | + this.b = b; |
| 116 | + this.p = p; |
| 117 | + this.basePoint = basePoint; |
| 118 | + } |
| 119 | + |
| 120 | + // Constructor that randomly generates the curve parameters |
| 121 | + public EllipticCurve(int bits) { |
| 122 | + SecureRandom r = new SecureRandom(); |
| 123 | + this.p = BigInteger.probablePrime(bits, r); // Random prime p |
| 124 | + this.a = new BigInteger(bits, r); // Random coefficient a |
| 125 | + this.b = new BigInteger(bits, r); // Random coefficient b |
| 126 | + this.basePoint = new ECPoint(BigInteger.valueOf(4), BigInteger.valueOf(8)); // Fixed base point G |
| 127 | + } |
| 128 | + |
| 129 | + public ECPoint getBasePoint() { |
| 130 | + return basePoint; |
| 131 | + } |
| 132 | + |
| 133 | + public BigInteger getP() { |
| 134 | + return p; |
| 135 | + } |
| 136 | + |
| 137 | + public BigInteger getA() { |
| 138 | + return a; |
| 139 | + } |
| 140 | + |
| 141 | + public BigInteger getB() { |
| 142 | + return b; |
| 143 | + } |
| 144 | + |
| 145 | + public int getFieldSize() { |
| 146 | + return p.bitLength(); |
| 147 | + } |
| 148 | + |
| 149 | + public ECPoint encodeMessage(BigInteger message) { |
| 150 | + // Simple encoding of a message as an ECPoint (this is a simplified example) |
| 151 | + return new ECPoint(message, message); |
| 152 | + } |
| 153 | + |
| 154 | + public BigInteger decodeMessage(ECPoint point) { |
| 155 | + return point.getX(); // Decode the message from ECPoint (simplified) |
| 156 | + } |
| 157 | + } |
| 158 | + |
| 159 | + /** |
| 160 | + * Class representing a point on the elliptic curve. |
| 161 | + */ |
| 162 | + public static class ECPoint { |
| 163 | + private final BigInteger x; // X-coordinate of the point |
| 164 | + private final BigInteger y; // Y-coordinate of the point |
| 165 | + |
| 166 | + public ECPoint(BigInteger x, BigInteger y) { |
| 167 | + this.x = x; |
| 168 | + this.y = y; |
| 169 | + } |
| 170 | + |
| 171 | + public BigInteger getX() { |
| 172 | + return x; |
| 173 | + } |
| 174 | + |
| 175 | + public BigInteger getY() { |
| 176 | + return y; |
| 177 | + } |
| 178 | + |
| 179 | + @Override |
| 180 | + public String toString() { |
| 181 | + return "ECPoint(x=" + x.toString() + ", y=" + y.toString() + ")"; |
| 182 | + } |
| 183 | + |
| 184 | + /** |
| 185 | + * Add two points on the elliptic curve. |
| 186 | + */ |
| 187 | + public ECPoint add(ECPoint other, BigInteger p, BigInteger a) { |
| 188 | + if (this.x.equals(BigInteger.ZERO) && this.y.equals(BigInteger.ZERO)) { |
| 189 | + return other; // If this point is the identity, return the other point |
| 190 | + } |
| 191 | + if (other.x.equals(BigInteger.ZERO) && other.y.equals(BigInteger.ZERO)) { |
| 192 | + return this; // If the other point is the identity, return this point |
| 193 | + } |
| 194 | + |
| 195 | + BigInteger lambda; |
| 196 | + if (this.equals(other)) { |
| 197 | + // Special case: point doubling |
| 198 | + lambda = (this.x.pow(2).multiply(BigInteger.valueOf(3)).add(a)) |
| 199 | + .multiply(this.y.multiply(BigInteger.valueOf(2)).modInverse(p)).mod(p); |
| 200 | + } else { |
| 201 | + // General case: adding two different points |
| 202 | + lambda = (other.y.subtract(this.y)) |
| 203 | + .multiply(other.x.subtract(this.x).modInverse(p)).mod(p); |
| 204 | + } |
| 205 | + |
| 206 | + BigInteger xr = lambda.pow(2).subtract(this.x).subtract(other.x).mod(p); |
| 207 | + BigInteger yr = lambda.multiply(this.x.subtract(xr)).subtract(this.y).mod(p); |
| 208 | + |
| 209 | + return new ECPoint(xr, yr); |
| 210 | + } |
| 211 | + |
| 212 | + /** |
| 213 | + * Subtract two points on the elliptic curve. |
| 214 | + */ |
| 215 | + public ECPoint subtract(ECPoint other, BigInteger p, BigInteger a) { |
| 216 | + ECPoint negOther = new ECPoint(other.x, p.subtract(other.y)); // Negate the Y coordinate |
| 217 | + return this.add(negOther, p, a); // Add the negated point |
| 218 | + } |
| 219 | + |
| 220 | + /** |
| 221 | + * Multiply a point by a scalar (repeated addition). |
| 222 | + */ |
| 223 | + public ECPoint multiply(BigInteger k, BigInteger p, BigInteger a) { |
| 224 | + ECPoint result = new ECPoint(BigInteger.ZERO, BigInteger.ZERO); // Identity point |
| 225 | + ECPoint addend = this; |
| 226 | + |
| 227 | + while (k.signum() > 0) { |
| 228 | + if (k.testBit(0)) { |
| 229 | + result = result.add(addend, p, a); // Add when k's bit is 1 |
| 230 | + } |
| 231 | + addend = addend.add(addend, p, a); // Double the point |
| 232 | + k = k.shiftRight(1); // Shift k to the right by 1 bit |
| 233 | + } |
| 234 | + |
| 235 | + return result; // Return the resulting point |
| 236 | + } |
| 237 | + } |
| 238 | +} |
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