answer:To perform elliptic curve addition, we first need to know the equation of the elliptic curve. The general form of an elliptic curve is: y^2 = x^3 + ax + b Please provide the values of a and b for the elliptic curve.

answer:To find the public key Q = kP for the private key k = 7, we need to perform elliptic curve point multiplication. This means we need to add the point P to itself k times. In this case, we will add P to itself 7 times. Elliptic curve point addition involves finding the slope of the line between two points and using that slope to find the third point of intersection with the curve. The slope formula for point addition is: λ = (y2 - y1) / (x2 - x1) However, when adding a point to itself (point doubling), we use the following formula for the slope: λ = (3x^2 + a) / 2y Given the elliptic curve E: y^2 = x^3 + 2x + 2 over the field F_p, with p = 17, and the base point P = (5,1), we can calculate the public key Q = 7P as follows: 1. Calculate the slope λ for point doubling P: λ = (3 * (5^2) + 2) / (2 * 1) = (75 + 2) / 2 = 77 / 2 ≡ 8 / 2 (mod 17) = 4 2. Calculate the x-coordinate of 2P: x_2P = λ^2 - x1 - x2 = 4^2 - 5 - 5 = 16 - 10 = 6 (mod 17) 3. Calculate the y-coordinate of 2P: y_2P = λ(x1 - x_2P) - y1 = 4(5 - 6) - 1 = -4 - 1 = -5 ≡ 12 (mod 17) Now we have 2P = (6, 12). 4. Calculate the slope λ for adding P and 2P: λ = (12 - 1) / (6 - 5) = 11 / 1 = 11 (mod 17) 5. Calculate the x-coordinate of 3P: x_3P = λ^2 - x1 - x_2P = 11^2 - 5 - 6 = 121 - 11 = 110 ≡ 15 (mod 17) 6. Calculate the y-coordinate of 3P: y_3P = λ(x1 - x_3P) - y1 = 11(5 - 15) - 1 = -110 - 1 = -111 ≡ 6 (mod 17) Now we have 3P = (15, 6). We need to calculate 7P, so we need to add 3P and 4P. Since 4P = 2(2P), we can double 2P to get 4P. 7. Calculate the slope λ for point doubling 2P: λ = (3 * (6^2) + 2) / (2 * 12) = (108 + 2) / 24 = 110 / 24 ≡ 7 / 7 (mod 17) = 1 8. Calculate the x-coordinate of 4P: x_4P = λ^2 - x_2P - x_2P = 1^2 - 6 - 6 = 1 - 12 = -11 ≡ 6 (mod 17) 9. Calculate the y-coordinate of 4P: y_4P = λ(x_2P - x_4P) - y_2P = 1(6 - 6) - 12 = 0 - 12 = -12 ≡ 5 (mod 17) Now we have 4P = (6, 5). Finally, we need to add 3P and 4P to get 7P. 10. Calculate the slope λ for adding 3P and 4P: λ = (5 - 6) / (6 - 15) = -1 / -9 = 1 / 9 ≡ 1 * 2 (mod 17) = 2 11. Calculate the x-coordinate of 7P: x_7P = λ^2 - x_3P - x_4P = 2^2 - 15 - 6 = 4 - 21 = -17 ≡ 0 (mod 17) 12. Calculate the y-coordinate of 7P: y_7P = λ(x_3P - x_7P) - y_3P = 2(15 - 0) - 6 = 30 - 6 = 24 ≡ 7 (mod 17) Now we have 7P = (0, 7). So, the public key Q = kP for the private key k = 7 is Q = (0, 7).

answer:To encrypt a message using elliptic curve cryptography (ECC), we need to follow the ElGamal encryption scheme. Let's denote the elliptic curve as E, the base point as G, Alice's private key as a, and Bob's public key as B. We are given: - E: elliptic curve defined over the field of prime numbers modulo 283 - G: (5, 71) - a: 173 - B: (145, 64) Now, let's assume that Alice wants to encrypt a message M, which is also a point on the elliptic curve E. The ElGamal encryption scheme consists of the following steps: 1. Choose a random integer k from the range [1, n-1], where n is the order of the elliptic curve. 2. Calculate the ephemeral key EK = k * G. 3. Calculate the shared secret S = k * B. 4. Calculate the ciphertext C = M + S. Since we don't have the message M, we cannot compute the ciphertext C. However, we can describe the process Alice would follow to encrypt the message using the given information.

answer:To decrypt the message using Elliptic Curve Cryptography (ECC), we need to find the original point (x, y) on the curve that represents the message. We will use the given information to perform the decryption. Given: - Elliptic curve equation: y^2 = x^3 + 2x + 2 (mod 23) - Private key: d = 7 - Public key: P = (6, 11) - Encrypted message: C = (19, 12) First, we need to find the point R such that R = d * C. In ECC, scalar multiplication is performed using point addition and point doubling operations. 1. Calculate R = 7 * (19, 12) on the elliptic curve. To do this, we will perform point addition and point doubling operations. 1.1. Double (19, 12) to get 2 * (19, 12) = (8, 7). 1.2. Add (8, 7) to (19, 12) to get 3 * (19, 12) = (20, 20). 1.3. Add (20, 20) to (19, 12) to get 4 * (19, 12) = (14, 19). 1.4. Double (14, 19) to get 8 * (19, 12) = (13, 10). 1.5. Subtract (19, 12) from (13, 10) to get 7 * (19, 12) = (13, 13). Now we have R = (13, 13). 2. Calculate the original message point M. To find the original message point M, we need to subtract the public key point P from the point R. M = R - P = (13, 13) - (6, 11) 2.1. Calculate the inverse of P: P_inv = (6, -11) = (6, 12) (mod 23). 2.2. Add R and P_inv: M = (13, 13) + (6, 12) = (10, 6). So, the original message point is M = (10, 6).