Let be the elliptic curve
Find the sum
Find the sum
Using the result of part (b), find the difference
Find an integer such that
Show that has exactly 20 distinct multiples, including
Using (e) and Exercise 19(d), show that the number of points on is a multiple of 20. Use Hasse’s theorem to show that has exactly 20 points.
You want to represent the message as a point on the curve Write and find a value of the missing last digit of such that there is a point on the curve with this .
Factor 3900353 using elliptic curves.
Try to factor 3900353 using the method of Section 9.4. Using the knowledge of the prime factors obtained from part (a), explain why the method does not work well for this problem.
Let be a point on the elliptic curve
Show that but and
Use Exercise 20 to show that has order 189.
Use Exercise 19(d) and Hasse’s theorem to show that the elliptic curve has 567 points.
Compute the difference on the elliptic curve Note that the answer involves large integers, even though the original points have small coordinates.