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I have taken elliptic curve $E\colon y^2=x^3-4x+20$, defined over $\mathbb{F}_{29}$. The number of points on the curve, $\left|E(\mathbb{F}_{29})\right|=37$.

I took base point $P=(1,5)$, and got following results:

Array of scalar multiples of point (1,5)

I want the results graphically to know the behaviour of the curve.

Is there any tool for this to show results graphically by giving input as curve points?

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    $\begingroup$ Well, you could simply plot them as Cartesian pairs, but why do you want a graphical representation? Any way you try plotting them will loose information, since they are pairs in $\mathbb{F}_{29}\times\mathbb{F}_{29}$. $\endgroup$ – Cryptographeur Dec 3 '13 at 11:29
  • $\begingroup$ side question: How can you find a base point of a curve?Or because the underlying field is of prime order so all the points of the curve they do form a basis? $\endgroup$ – curious Dec 3 '13 at 17:20
  • $\begingroup$ @curious: In this case it is simple because the curve has a prime number of points. $\endgroup$ – Cryptographeur Dec 3 '13 at 17:30
  • $\begingroup$ What does this mean? $\endgroup$ – curious Dec 3 '13 at 17:32
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    $\begingroup$ @curious: Because the curve has a prime number of points, then (since the points on elliptic curves form groups) the curve must be isomorphic to $C_{37}$, which means any point (other than the point at infinity) is a generator. $\endgroup$ – Cryptographeur Dec 10 '13 at 16:00
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The point is an elliptic curve doesn't actually look anything like a curve when evaluated over a finite field, and indeed there's no real way of visualising points in $\mathbb{F}_{29}\times\mathbb{F}_{29}$ that doesn't loose some of the structure anyway.

The best I can think of would be to plot them simple as points in a $29\times29$ grid and join the dots, but even this would be painfully unclear, since for example when would you 'join' them by overflowing the grid (eg: would the line from $(1,5)\to(4,19)$ go straight from one to the other, or would it go out one side and reappear on the opposite side?).

If you were to evaluate $E(\mathbb{R})$ then the resulting points would form a curve in $\mathbb{R}^2$, which you could then plot.

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As already told the elliptic curve over a finite field is not "really" a curve, but you could find this link (http://arstechnica.com/security/2013/10/a-relatively-easy-to-understand-primer-on-elliptic-curve-cryptography) interesting. In particular, in page 2, an elliptic curve is drawn twice: the first time over the real numbers and the second one over the finite field. There is also an animation of the doubling operation over the finite field. (Hope it helps, even if is not a tool you can use to draw your curve)

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