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

Decrypt a message that was encrypted using the ElGamal public key
cryptosystem and a subgroup of the group of an elliptic curve.

<p><p>

<h3>
Syntax:
</h3>
<pre>P := EccDecrypt(K,E,a,M);</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>integer </td>
<td><pre>  a  </pre></td>
<td></td>
</tr>
<tr>
<td>list </td>
<td><pre>  M  </pre></td>
<td></td>
</tr>
<tr>
<td>list </td>
<td><pre>  P  </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#EccEncrypt">EccEncrypt</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.
The message M=[P_1,P_2] consisting of two points on E
is decrypted to a point P by setting P=P_1-acdot P_2.
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> field := FF(751);
Finite field of size 751
kash> ec := [0,0,1,-1,0];
[ 0, 0, 1, -1, 0 ]
kash> base_point := [0,0];
[ 0, 0 ]
kash> secret_key := 58;
58
kash> public_key := EccIntPointMult(field,ec,secret_key,base_point);
[ 201, 380 ]
kash> plaintext := [[562,576],[581,395],[484,214],[501,220],[1,0]];
[ [ 562, 576 ], [ 581, 395 ], [ 484, 214 ], [ 501, 220 ], [ 1, 0 ] ]
kash> k := [254,180,99,472,275];
[ 254, 180, 99, 472, 275 ]
kash> ciphertext := List([1..5], \
> x-> EccEncrypt(field,ec,base_point,public_key,k[x],plaintext[x]));
[ [ [ 268, 146 ], [ 378, 547 ] ], [ [ 680, 469 ], [ 409, 94 ] ], 
  [ [ 710, 395 ], [ 195, 432 ] ], [ [ 747, 222 ], [ 101, 371 ] ], 
  [ [ 13, 246 ], [ 386, 303 ] ] ]
kash> List(ciphertext, x-> EccDecrypt(field,ec,secret_key,x));
> [ [ 562, 576 ], [ 581, 395 ], [ 484, 214 ], [ 501, 220 ], [ 1, 0 ] ]

</pre>

<p>


<hr>

<p>
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