Need d) & e)

Consider the elliptic curve E given by the equation y2-x3-x defined over the real numbers. Tet F(R.) denote its set of points. (a) Draw a picture of E(R). If P is the point (0,0) i E(R), what is P H (usually denoted 2P)? (b) For the remainder of this question, consider the elliptic curve E : y2-X3-X defined over Zp, where p is an odd prime. If P is the point (0, 0), what is 2P? Find 3P (c) Let p -5. Find all points on the elliptic curve (don't forget its point at infinity). This set of points is denoted E (Zp). Give its addition table. (d) Suppose p (mod 4) is 3. Assuming that x2--1 (mod p) has no solution (this follows since the order of z in Zp divides the size, p-1, of Zp, but you don't need to prove this), show that E(Zp) contains exactly p+1 points [hint: x3-x is an odd function of x - consider what happens when you replace x-a by x- -a. Together, how many points in E(Zp) have x-coordinate a or -a, for a given a?] (e) There is a powerful attack (the "MOV attack") that works best when the *-1 for some k much smaller than p (it actually turns ECDLP in E(Zp) into DLP in the multiplica- tive group of the field with p* elements). Explain why this implies that an elliptic curve cryptosystem that uses E : y-x3-x defined over Zp with p (mod 4) 3 elliptic curve cryptosystem employs a point P whose order divides p is a poor idea. Consider the elliptic curve E given by the equation y2-x3-x defined over the real numbers. Tet F(R.) denote its set of points. (a) Draw a picture of E(R). If P is the point (0,0) i E(R), what is P H (usually denoted 2P)? (b) For the remainder of this question, consider the elliptic curve E : y2-X3-X defined over Zp, where p is an odd prime. If P is the point (0, 0), what is 2P? Find 3P (c) Let p -5. Find all points on the elliptic curve (don't forget its point at infinity). This set of points is denoted E (Zp). Give its addition table. (d) Suppose p (mod 4) is 3. Assuming that x2--1 (mod p) has no solution (this follows since the order of z in Zp divides the size, p-1, of Zp, but you don't need to prove this), show that E(Zp) contains exactly p+1 points [hint: x3-x is an odd function of x - consider what happens when you replace x-a by x- -a. Together, how many points in E(Zp) have x-coordinate a or -a, for a given a?] (e) There is a powerful attack (the "MOV attack") that works best when the *-1 for some k much smaller than p (it actually turns ECDLP in E(Zp) into DLP in the multiplica- tive group of the field with p* elements). Explain why this implies that an elliptic curve cryptosystem that uses E : y-x3-x defined over Zp with p (mod 4) 3 elliptic curve cryptosystem employs a point P whose order divides p is a poor idea