# Elliptic curves

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Given an elliptic curve equation y2 = x3 + 25x + 17 (mod 29), answer the following questions.
For the point P = (4, 6) and Q = (5, 8), work out P+Q and 2P by hand and verify that P+Q and 2P are still on the curve.
4 marks
Use maple to find all the points on this curve. How many points are there in the EC-based group and then plot all the points of this curve (you need to show your maple code of how you get the points).
4 marks
If the curve is defined over real numbers, i.e., y2 = x3 + 25x + 17, plot the curve with -5
4 marks
An important usage of the elliptic curves is to factorize big integers. Comparing to the difference of squares method, the advantage of EC-based factorization is that it can be parallelized easily. This question asks you to practice integer factorization with EC-based method.
The smallest 3-digit prime is p = 101. And you need to find another prime q as follows. Take the last three digits of your student ID, and then run the maple command “nextprime()” and set the result as q. For example, if my ID is “7654321”, then the last three digits are “321”, then q = nextprime(321)= 331. Now, set n = p*q (note that the value q must be derived from your own student ID but not copy this constant 331).
Set up two elliptic curves randomly (so they are up to your own choice) and factorize the number n=p*q you obtained above. Observe your maple result, which curve gives you the factors p, q faster?

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