answer:To find the number of points on an elliptic curve over a finite field, we need to know the curve equation and the field size. In this case, we are given two prime numbers p and q, but we don't have the curve equation. However, if we assume that the merchant wants to use these prime numbers to define the field size, we can proceed as follows: 1. If the field size is defined by the product of p and q, then the field size is n = p * q = 107 * 137 = 14659. 2. To find the number of points on the elliptic curve over the field of size n, we need to use Hasse's theorem, which states that the number of points N on the curve is bounded by the following inequality: n + 1 - 2√n ≤ N ≤ n + 1 + 2√n 3. Calculate the bounds using Hasse's theorem: 14659 + 1 - 2√14659 ≤ N ≤ 14659 + 1 + 2√14659 14660 - 2(121) ≤ N ≤ 14660 + 2(121) 14660 - 242 ≤ N ≤ 14660 + 242 14418 ≤ N ≤ 14902 So, the number of points on the elliptic curve is between 14418 and 14902. However, without the curve equation, we cannot determine the exact number of points on the elliptic curve.

answer:An elliptic curve over a finite field 𝔽p (in this case, 𝔽7) is defined by the equation: y^2 ≡ x^3 + ax + b (mod p) Given the point (1, 3) is on the curve, and the coefficients a = -1 and b = 1, we can substitute these values into the equation: (3)^2 ≡ (1)^3 - 1(1) + 1 (mod 7) Now, we can simplify and solve the equation: 9 ≡ 1 - 1 + 1 (mod 7) 9 ≡ 1 (mod 7) Since 9 ≡ 1 (mod 7), the equation for the elliptic curve defined over the finite field of integers modulo 7 (𝔽7) with the point (1, 3) on the curve and the coefficients a = -1 and b = 1 is: y^2 ≡ x^3 - x + 1 (mod 7)

answer:To perform scalar multiplication on an elliptic curve, we will use the point addition and point doubling formulas. The point addition formula for two points P=(x_1, y_1) and Q=(x_2, y_2) on an elliptic curve is: x_3 = lambda^2 - x_1 - x_2 pmod{p} y_3 = lambda(x_1 - x_3) - y_1 pmod{p} where lambda = (y_2 - y_1)(x_2 - x_1)^{-1} pmod{p} The point doubling formula for a point P=(x_1, y_1) on an elliptic curve is: x_3 = lambda^2 - 2x_1 pmod{p} y_3 = lambda(x_1 - x_3) - y_1 pmod{p} where lambda = (3x_1^2 + a)(2y_1)^{-1} pmod{p} For our given elliptic curve y^2 = x^3 + 5x + 7 over mathbb{F}_{11}, we have a = 5 and p = 11. We are given the starting point P=(2,3) and scalar n=7. We will perform scalar multiplication using the double-and-add algorithm. 1. Initialize Q = P and R = O (the point at infinity). 2. For each bit in the binary representation of n (from left to right): a. If the bit is 1, add Q to R. b. Double Q. The binary representation of n=7 is 111. We will perform the double-and-add algorithm step by step: 1. Q = (2,3), R = O 2. First bit is 1: a. R = R + Q = O + (2,3) = (2,3) b. Q = 2Q = (2,3) + (2,3) (point doubling) For point doubling, we compute lambda: lambda = (3x_1^2 + a)(2y_1)^{-1} pmod{11} = (3cdot2^2 + 5)(2cdot3)^{-1} pmod{11} = (12 + 5)(6)^{-1} pmod{11} = 17cdot6^{-1} pmod{11} = 6cdot2 pmod{11} = 12 pmod{11} = 1 Now we compute the new Q: x_3 = lambda^2 - 2x_1 pmod{11} = 1^2 - 2cdot2 pmod{11} = 1 - 4 pmod{11} = -3 pmod{11} = 8 y_3 = lambda(x_1 - x_3) - y_1 pmod{11} = 1(2 - 8) - 3 pmod{11} = -6 - 3 pmod{11} = -9 pmod{11} = 2 So, Q = (8,2). 3. Second bit is 1: a. R = R + Q = (2,3) + (8,2) (point addition) For point addition, we compute lambda: lambda = (y_2 - y_1)(x_2 - x_1)^{-1} pmod{11} = (2 - 3)(8 - 2)^{-1} pmod{11} = (-1)(6)^{-1} pmod{11} = -1cdot2 pmod{11} = -2 pmod{11} = 9 Now we compute the new R: x_3 = lambda^2 - x_1 - x_2 pmod{11} = 9^2 - 2 - 8 pmod{11} = 81 - 10 pmod{11} = 71 pmod{11} = 6 y_3 = lambda(x_1 - x_3) - y_1 pmod{11} = 9(2 - 6) - 3 pmod{11} = -36 - 3 pmod{11} = -39 pmod{11} = 6 So, R = (6,6). b. Q = 2Q = (8,2) + (8,2) (point doubling) For point doubling, we compute lambda: lambda = (3x_1^2 + a)(2y_1)^{-1} pmod{11} = (3cdot8^2 + 5)(2cdot2)^{-1} pmod{11} = (3cdot64 + 5)(4)^{-1} pmod{11} = (192 + 5)(4)^{-1} pmod{11} = 197cdot4^{-1} pmod{11} = 9cdot4^{-1} pmod{11} = 9cdot3 pmod{11} = 27 pmod{11} = 5 Now we compute the new Q: x_3 = lambda^2 - 2x_1 pmod{11} = 5^2 - 2cdot8 pmod{11} = 25 - 16 pmod{11} = 9 pmod{11} = 9 y_3 = lambda(x_1 - x_3) - y_1 pmod{11} = 5(8 - 9) - 2 pmod{11} = -5 - 2 pmod{11} = -7 pmod{11} = 4 So, Q = (9,4). 4. Third bit is 1: a. R = R + Q = (6,6) + (9,4) (point addition) For point addition, we compute lambda: lambda = (y_2 - y_1)(x_2 - x_1)^{-1} pmod{11} = (4 - 6)(9 - 6)^{-1} pmod{11} = (-2)(3)^{-1} pmod{11} = -2cdot4 pmod{11} = -8 pmod{11} = 3 Now we compute the new R: x_3 = lambda^2 - x_1 - x_2 pmod{11} = 3^2 - 6 - 9 pmod{11} = 9 - 15 pmod{11} = -6 pmod{11} = 5 y_3 = lambda(x_1 - x_3) - y_1 pmod{11} = 3(6 - 5) - 6 pmod{11} = 3 - 6 pmod{11} = -3 pmod{11} = 8 So, R = (5,8). After performing the scalar multiplication, we find that the resulting point P = (5,8).

answer:To find the order of an elliptic curve over a finite field, we need to count the number of points on the curve, including the point at infinity. The given elliptic curve equation is: y^2 = x^3 + 4x + 4 We will check all possible x and y values in the finite field 𝔽13 and see if they satisfy the equation. If they do, we count them as a point on the curve. Here are the points on the curve: (0, 2), (0, 11), (1, 5), (1, 8), (2, 3), (2, 10), (3, 1), (3, 12), (4, 4), (4, 9), (5, 0), (6, 6), (6, 7), (8, 1), (8, 12), (9, 5), (9, 8), (10, 2), (10, 11), (11, 3), (11, 10), (12, 6), (12, 7) There are 23 points on the curve, and we also need to include the point at infinity. So, the order of the elliptic curve over the finite field 𝔽13 is 24.