Consider the elliptic curve group based on the equation y=x+ax + b mod p ere a = 10, 6 = 1, and p = 13. is curve contains the point P = (0, 1). The order of this elliptic curve group is the prime number 19, and therefore we can be sure that Pis a primitive ment. Another element in this group is Q = (2. 9). The index of Q with respect to P is the least positive integer d such that Q = dP. What is d, the tex of Q?

(3) Let the elliptic curve & be y² = x³ + x + 6 (mod 11). (a) Given the point a = (2,7) on the elliptic curve E, calculate 2a and 3a. (b) Consider the elliptic curve ElGamal cryptosystem. Let the private key be s = 7. (i) What is the public key? (Hint: You don't need to include the known information E, p, a in the public key, but have to calculate = sa. ) (ii) Encoding: Suppose the text has been converted to a point x = (10, 9) on E. Choose the random number k 3. What are the points (or numbers) to be transmitted? = (iii) Decoding: When receiving information from (ii), how to recover the original text?

Only need the multiple-choice answers, not the steps.

1 And the radios of Co for the pover Serie for the Power Serie vergence 00 xh (2"+1) (n?+1)