# Mathematica.FEE   [plain text]

```(* Elliptic algebra functions: FEE format.

y^2 = x^3 + c x^2 + a x + b.

Montgomery: b = 0, a = 1;
Weierstrass: c = 0;
Atkin3: c = a = 0;
Atkin4: c = b = 0;

Parameters c, a, b, p must be global.
*)

elleven[pt_] := Block[{x1 = pt[[1]], z1 = pt[[2]], e, f },
e = Mod[(x1^2 - a z1^2)^2 - 4 b (2 x1 + c z1) z1^3, p];
f = Mod[4 z1 (x1^3 + c x1^2 z1 + a x1 z1^2 + b z1^3), p];
Return[{e,f}]
];

ellodd[pt_, pu_, pv_] := Block[
{x1 = pt[[1]], z1 = pt[[2]],
x2 = pu[[1]], z2 = pu[[2]],
xx = pv[[1]], zz = pv[[2]], i, j},
i = Mod[zz ((x1 x2 - a z1 z2)^2 -
4 b(x1 z2 + x2 z1 + c z1 z2) z1 z2), p];
j = Mod[xx (x1 z2 - x2 z1)^2, p];
Return[{i,j}]
];

bitList[k_] := Block[{li = {}, j = k},
While[j > 0,
li = Append[li, Mod[j,2]];
j = Floor[j/2];
];
Return[Reverse[li]];
];

elliptic[pt_, k_] := Block[{porg, ps, pp, q},

If[k ==1, Return[pt]];
If[k ==2, Return[elleven[pt]]];
porg = pt;
ps = elleven[pt];
pp = pt;
bitlist = bitList[k];
Do[
If[bitlist[[q]] == 1,
pp = ellodd[ps, pp, porg];
ps = elleven[ps],
ps = ellodd[pp, ps, porg];
pp = elleven[pp]
],
{q,2,Length[bitlist]}
];
Return[Mod[pp,p]]
];
ellinv[n_] := PowerMod[n,-1,p];
ex[pt_] := Mod[pt[[1]] * ellinv[pt[[2]]], p];
```