Points: 20
Q1. Consider the curve y2 ≡ x3 + 5x + 11 mod 17.
(i) Confirm that it is an elliptic curve.
(ii) Determine points on the curve over real numbers with x=0, 1, 2, and 3.
(iii) Determine all the points on the curve over integer numbers (Hint: Take x=0, 1,2,…,15,16; compute y2. Find y. If you find that y2 is not a perfect square, then keep on trying other mod 17 equivalents of y2. For example, if y2 = 2, then try other equivalents of 2 such as 19, 36, 53, … In this case, we can stop at 36 since it is a perfect square. Sometimes you may not get any perfect square. This means there is no integer point. For example, if y2 = 7, the other mod 17 equivalents are 24, 41, 58, 75, 92, 109, 126, 143, 160, 177, 194, 211, 228, 245,262,279.
None of them are perfect squares. So we have no integer point. Also, you will get a + value and a –ve value since it is a square root. For the –ve value, since it is mod 17, add 17 to make it positive. For example, if y2 = 25, y = +5, –5. So we take it as y= +5, +12. The final answers are (x,y) points on the elliptic curve. Present it in the following format:
x y2 = x3 + 5x + 11 mod 17 (list all possible values until 256) y (integer
values)
(x,y)
0
1
2
3
4
5
6
7
8
9
10
11
12
(iv) For a point P= (3,6), find 2P (or double)
(v) For two of the points P = (3,6) and Q =(7,7), find P+Q
(vi) Find the bound for the number of points on this curve using Hesse’s theorem.
What to submit? Submit a pdf file with your answers via the Blackboard. Show your work
