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Consider the elliptic curve E:$y^2 = x^3 + 3x + 11\,\, mod\,\, 19$. Two questions:

  1. Let the cardinality of the set of points on the elliptic curve( including $O$ ) be $|E| = 25$. How many points are generators and why?
  2. 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?

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    $\begingroup$ One subtlety is that an elliptic curve group need not be a cyclic group; in this case, it might be isomorphic to $Z/5 \times Z/5$ (which also has 25 elements). Have you shown that that's not the case? $\endgroup$ – poncho Apr 23 '17 at 21:50
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    $\begingroup$ Please ask separate questions. If your question title becomes too generic the question becomes hard to handle on this site. $\endgroup$ – Maarten - reinstate Monica Apr 23 '17 at 21:50
  • $\begingroup$ groupprops.subwiki.org/wiki/… $\endgroup$ – pg1989 Apr 23 '17 at 23:19
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No, he really means the number of points on $E$. So the number of solutions to the given equation plus 1 (for the point at infinity). He also gives the answer as $25.$

You need to apply your idea about number of generators to the correct group size.

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