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(c) Let P = (2 : 5: 1) and Q = (9 : 6: 1) be points on the elliptic curve y? = x +7x +3 defined over the finite field F23. Calculate P +Q.

(c) Let P = (2 : 5: 1) and Q = (9 : 6: 1) be points on the elliptic curve y? = x +7x +3 defined over the finite field F23. Calculate P +Q.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve c part only in 20 min and take a thumb up plz
2. (a) Let C be the conic X2 – 2XY – XZ + Y2 – XZ - YZ = 0 defined over Q. Use the point P = (0 : 0: 1) to find a parametrisation p: pl → C. (b) Let (C, 0) be an elliptic curve over a field k such that O is an inflection point. For an inflection point QE C(k), we define QV = Q, and for any other point Q E C(k), we define Q + Q to be the second point of intersection of To(C) and C. Let PE C(k). Consider the list pv, (P")", ((P")")", ((P")")")", ... Show that if this list contains P, then P is a torsion point. (c) Let P = (2 : 5 : 1) and Q = (9 : 6 : 1) be points on the elliptic curve y? = x +7x +3 defined over the finite field F23. Calculate P + Q. (d) Let p be a prime number. Let O + Q = (xQ YQ 1) E E(Q) be a point on an elliptic curve E given in Weierstrass form y? = x³ +Ax+B with A, B E Z. Suppose that the numerator of to = -xQ/yQ is divisible by p? when written as a reduced fraction. Show that the numerator of tpo is divisible by p³.
Transcribed Image Text:2. (a) Let C be the conic X2 – 2XY – XZ + Y2 – XZ - YZ = 0 defined over Q. Use the point P = (0 : 0: 1) to find a parametrisation p: pl → C. (b) Let (C, 0) be an elliptic curve over a field k such that O is an inflection point. For an inflection point QE C(k), we define QV = Q, and for any other point Q E C(k), we define Q + Q to be the second point of intersection of To(C) and C. Let PE C(k). Consider the list pv, (P")", ((P")")", ((P")")")", ... Show that if this list contains P, then P is a torsion point. (c) Let P = (2 : 5 : 1) and Q = (9 : 6 : 1) be points on the elliptic curve y? = x +7x +3 defined over the finite field F23. Calculate P + Q. (d) Let p be a prime number. Let O + Q = (xQ YQ 1) E E(Q) be a point on an elliptic curve E given in Weierstrass form y? = x³ +Ax+B with A, B E Z. Suppose that the numerator of to = -xQ/yQ is divisible by p? when written as a reduced fraction. Show that the numerator of tpo is divisible by p³.
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