[Math] Elliptic Curve: point of prime order $n$

cryptographyelliptic-curvesfinite-fields

I am attempting to understand ElGamal Encryption on Elliptic Curves (beginnning of this paper)

I have an elliptic curve $E\left( \mathbb{F}_{11} \right)$ defined by $y^2 = x^3 + 2x + 3$

I need to choose a point $P$ on $E\left(\mathbb{F}_{11}\right)$ of prime order $n$

I haven't been able to find anything to help me understand what is meant by 'prime order $n$'. Can someone point me in the direction of anything useful or explain to me how to find such a point $P$ please?

I am a computer scientist, so assume limited knowledge of any complex maths/theorems – all my knowledge on elliptic curves is self-taught through Googling…

Best Answer

You were already given a nice answer in comments, but lets add a little detail.

We find all of the points on the curve (Note that $O$ is the point at infinity)

$$(x, y) = O, (0, 5), (0, 6), (2, 2), (2, 9), (3, 5), (3, 6), (4, 3), (4, 8), (6, 0), (8, 5), (8, 6), (10, 0)$$

You can verify each of these by checking that

$$y^2 = x^3 + 2x + 3 \pmod{11}$$

If we take a point $P$ and write out $P, P+P, \ldots$, we get

  • $1P = (2, 2)$
  • $2P = (0, 5)$
  • $3P = (3, 5)$
  • $4P = (4, 3)$
  • $5P = (8, 6)$
  • $6P = (10, 0)$
  • $7P = (8, 5)$
  • $8P = (4, 8)$
  • $9P = (3, 6)$
  • $10P = (0, 6)$
  • $11P = (2, 9)$
  • $12P = O$

Compare this list of points with the list above. What do you notice? Every one of the points must be a point on the curve.

Next, how many points did we cycle through, which is the order?

This elliptic curve has order $\#E = |E| = 12$ since it contains $12$ points in its cyclic group.

There is a theorem called Hasse‘s Theorem: Given an elliptic curve module $p$, the number of points on the curve is denoted by $\#E$ and is bounded by $$p+1-2\sqrt{p} ≤ \#E ≤ p+1+2 \sqrt{p}$$

Interpretation: The number of points is close to the prime $p$.

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