       Elliptic curve over reals

• To: mathgroup at smc.vnet.net
• Subject: [mg51596] Elliptic curve over reals
• From: "flip" <flip_alpha at safebunch.com>
• Date: Wed, 27 Oct 2004 01:53:56 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Hello,

I am working on some elliptic curve (EC) cryptography stuff for a
presentation.

I have a routine that selects the the quadratic residues of an EC and than I
pass that to a routine that finds the modular square roots of those QR
points.

Here are those two routines.
***

selectQRList[a_, b_, p_] :=
Select[Range[p] - 1, JacobiSymbol[#^3 + a  # + b, p] != -1 &];

ECPointList[a_, b_, p_, S_List] := Flatten[MapThread[Outer[List, #1, #2] &,
{Partition[S, 1], SqrtModList[#^3 +a # + b, p] & /@ S}], 2];

***

Next, I want to draw the EC over the "reals" that corresponds to the
ECPointList above.

For example:

The EC: y^2 = x^3 + 2x + 1 (mod 5) has the following solutions:

sols = {{0, 1}, {0, 4}, {1, 2}, {1, 3}, {3, 1}, {3, 3}} plus the point at
infinity.

These points over the reals (say without moding out by mod 5) is:

integerpoints = {{0, 1}, {0, -1}, {1, 2}, {1, -2}, {8, 23}, {8, -23}};

Is there any way to be able to show both these lists of points using the
above routines?

Thank you, Flip

email: --------------flip at nethere.com*******************

Remove the "-" and the "*"

```

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