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<p>&nbsp;<p>
<hr noshade size="5">
<h2><a name="EccEncrypt"></a>
EccEncrypt
</h2>

Encrypt a message (point) using the ElGamal public key cryptosystem
and a subgroup of the group of an elliptic curve.

<p><p>

<h3>
Syntax:
</h3>
<pre>M := EccEncrypt(K,E,B,aB,k,P);</pre>

<p>

<table>
<tr>
<td>finite field </td>
<td><pre>  K  </pre></td>
<td></td>
</tr>
<tr>
<td>list </td>
<td><pre>  E  </pre></td>
<td></td>
</tr>
<tr>
<td>list </td>
<td><pre>  B  </pre></td>
<td></td>
</tr>
<tr>
<td>list </td>
<td><pre>  aB  </pre></td>
<td></td>
</tr>
<tr>
<td>integer </td>
<td><pre>  k  </pre></td>
<td></td>
</tr>
<tr>
<td>list </td>
<td><pre>  P  </pre></td>
<td></td>
</tr>
<tr>
<td>list </td>
<td><pre>  M  </pre></td>
<td></td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#EccPointsAdd">EccPointsAdd</a>, <a href="kashref.html#EccIntPointMult">EccIntPointMult</a>, <a href="kashref.html#EccPointIsOnCurve">EccPointIsOnCurve</a>, <a href="kashref.html#EccDecrypt">EccDecrypt</a>, <a href="kashref.html#FF">FF</a>

<p>

<h3>
Description:
</h3>
Let E be an elliptic curve over a finite field K, let B be a
basepoint of a subgroup
of the group of E. Bob chooses an integer a, as his secret key.
He publishes the public key a_B:=acdot B. Before sending a
message to Bob
Alice picks a random integer k, then she sends him the point pair
[kcdot B,P+kcdot a_B].
The elliptic curve E is
either given by a list of two or five elements of K or integers. If
the equation of E is y^2=x^3+a_4 x+a_6 the curve is represented
by [a_4,a_6], if the equation of E is
y^2+a_1 xy+a_3y=x^3+a_2x^2+a_4 x+a_6 then the representation is
[a_1,a_2,a_3,a_4,a_6]. Points on the curve are given by a pair of
elements of K or integers; the point at infinity is represented by
the empty list [\;].

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>

<pre>

kash> K := FF(11);
Finite field of size 11
kash> E := [0,0,0,1,6];
[ 0, 0, 0, 1, 6 ]
kash> B := [2,7];
[ 2, 7 ]
kash> a := 8;
8
kash> aB := EccIntPointMult(K,E,a,B);
[ 3, 5 ]
kash> k := 7;
7
kash> M := EccEncrypt(K,E,B,aB,k,[3,6]);
[ [ 7, 2 ], [ 10, 9 ] ]
kash> EccDecrypt(K,E,a,M);
[ 3, 6 ]
kash> k := 5;
5
kash> M := EccEncrypt(K,E,B,aB,k,[3,6]);
[ [ 3, 6 ], [ 7, 9 ] ]
kash> EccDecrypt(K,E,a,M);
> [ 3, 6 ]

</pre>

<p>


<hr>

<p>
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