blob: 770a2d02db170cb12f28e4cdfb6eab72891e8401 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
|
Require Crypto.CompleteEdwardsCurve.Pre.
Require Crypto.Util.Decidable.
Module E.
Section TwistedEdwardsCurves.
(* Twisted Edwards curves with complete addition laws. References:
* <https://eprint.iacr.org/2008/013.pdf>
* <http://ed25519.cr.yp.to/ed25519-20110926.pdf>
* <https://eprint.iacr.org/2015/677.pdf>
*)
Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv} `{field:@Algebra.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv} {Feq_dec:Decidable.DecidableRel Feq}.
Local Infix "=" := Feq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope.
Local Notation "0" := Fzero. Local Notation "1" := Fone.
Local Infix "+" := Fadd. Local Infix "*" := Fmul.
Local Infix "-" := Fsub.
Local Notation "x ^ 2" := (x*x) (at level 30).
Local Notation "n / d" := (fst (Fdiv n d, (_:(d <> 0)))). (* force nonzero d *)
Context {a d: F}.
Class twisted_edwards_params :=
{
char_gt_2 : @Ring.char_gt F Feq Fzero Fone Fopp Fadd Fsub Fmul (BinNat.N.succ_pos BinNat.N.one);
nonzero_a : a <> 0;
square_a : exists sqrt_a, sqrt_a^2 = a;
nonsquare_d : forall x, x^2 <> d
}.
Context `{twisted_edwards_params}. (* TODO: name me *)
Definition point := { P | let '(x,y) := P in a*x^2 + y^2 = 1 + d*x^2*y^2 }.
Definition coordinates (P:point) : (F*F) := proj1_sig P.
Definition eq (P Q:point) :=
fst (coordinates P) = fst (coordinates Q) /\
snd (coordinates P) = snd (coordinates Q).
Program Definition zero : point := (0, 1).
Next Obligation. destruct H; eauto using Pre.onCurve_zero. Qed.
Program Definition add (P1 P2:point) : point :=
let x1y1 := coordinates P1 in let x1 := fst x1y1 in let y1 := snd x1y1 in
let x2y2 := coordinates P2 in let x2 := fst x2y2 in let y2 := snd x2y2 in
(((x1*y2 + y1*x2)/(1 + d*x1*x2*y1*y2)) , ((y1*y2 - a*x1*x2)/(1 - d*x1*x2*y1*y2))).
Next Obligation. destruct P1 as [[??]?], P2 as [[??]?], H. eauto using Pre.denominator_nonzero_x. Qed.
Next Obligation. destruct P1 as [[??]?], P2 as [[??]?], H; eauto using Pre.denominator_nonzero_y. Qed.
Next Obligation. destruct P1 as [[??]?], P2 as [[??]?], H; eauto using Pre.onCurve_add. Qed.
Fixpoint mul (n:nat) (P : point) : point :=
match n with
| O => zero
| S n' => add P (mul n' P)
end.
End TwistedEdwardsCurves.
End E.
Delimit Scope E_scope with E.
Infix "=" := E.eq : E_scope.
Infix "+" := E.add : E_scope.
Infix "*" := E.mul : E_scope.
|