1(* Elliptic algebra functions: FEE format.
2
3   y^2 = x^3 + c x^2 + a x + b.
4
5   Montgomery: b = 0, a = 1;
6   Weierstrass: c = 0;
7   Atkin3: c = a = 0;
8   Atkin4: c = b = 0;
9
10   Parameters c, a, b, p must be global.
11 *)
12
13elleven[pt_] := Block[{x1 = pt[[1]], z1 = pt[[2]], e, f },
14  	e = Mod[(x1^2 - a z1^2)^2 - 4 b (2 x1 + c z1) z1^3, p];
15  	f = Mod[4 z1 (x1^3 + c x1^2 z1 + a x1 z1^2 + b z1^3), p];
16  	Return[{e,f}]
17];
18
19ellodd[pt_, pu_, pv_] := Block[
20		{x1 = pt[[1]], z1 = pt[[2]],
21		 x2 = pu[[1]], z2 = pu[[2]],
22		 xx = pv[[1]], zz = pv[[2]], i, j},
23  	     i = Mod[zz ((x1 x2 - a z1 z2)^2 -
24  	          4 b(x1 z2 + x2 z1 + c z1 z2) z1 z2), p];
25  	     j = Mod[xx (x1 z2 - x2 z1)^2, p];
26  		 Return[{i,j}]
27];
28
29bitList[k_] := Block[{li = {}, j = k},
30	While[j > 0,
31	    li = Append[li, Mod[j,2]];
32	    j = Floor[j/2];
33	];
34	Return[Reverse[li]];
35	];
36
37elliptic[pt_, k_] := Block[{porg, ps, pp, q},
38
39	If[k ==1, Return[pt]];
40	If[k ==2, Return[elleven[pt]]];
41	porg = pt;
42	ps = elleven[pt];
43	pp = pt;
44	bitlist = bitList[k];
45	Do[
46	   If[bitlist[[q]] == 1,
47	   	   pp = ellodd[ps, pp, porg];
48	   	   ps = elleven[ps],
49	   	      ps = ellodd[pp, ps, porg];
50		      pp = elleven[pp]
51	   ],
52	   {q,2,Length[bitlist]}
53    ];
54    Return[Mod[pp,p]]
55];
56ellinv[n_] := PowerMod[n,-1,p];
57ex[pt_] := Mod[pt[[1]] * ellinv[pt[[2]]], p];
58