Consider the elliptic curve E:$y^2 = x^3 + 3x + 11\,\, mod\,\, 19$. Two questions:
- Let the cardinality of the set of points on the elliptic curve( including $O$ ) be $|E| = 25$. How many points are generators and why?
- Find an element of this group that has order $5$. The group operation is addition of points.
Answer of (1): First, when the author says points, I assume he is being sloppy and really means group. The number of generators for a cyclic group with $n$ elements in it, are the number of integers from $[1,n-1]$ that are relatively prime to $n$. Therefore, in this case, the answer is $19$. Is my answer right?
Regarding question (2): I am thinking about guessing at the answer and doing a brute force approach (maybe doing the brute force approach in software) but it seems to me that there should be a shorter way to do it. Could you comment on my idea?
I think I just found a trick for solving the second question. Let $a$ be a generator of the group. Then I claim that $a^5$ is an element of order $5$. Is my claim correct?