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Require Crypto.CompleteEdwardsCurve.Pre.
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} `{Algebra.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}.
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 Infix "/" := Fdiv.
Local Notation "x ^ 2" := (x*x) (at level 30).
Context {a d: F}.
Class twisted_edwards_params :=
{
char_gt_2 : 1 + 1 <> 0;
nonzero_a : a <> 0;
square_a : exists sqrt_a, sqrt_a^2 = a;
nonsquare_d : forall x, x^2 <> d
}.
Context `{twisted_edwards_params}.
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.
(** The following points are indeed on the curve -- see [CompleteEdwardsCurve.Pre] for proof *)
Local Obligation Tactic := intros;
apply (Pre.zeroOnCurve(a_nonzero:=nonzero_a)(char_gt_2:=char_gt_2)) ||
apply (Pre.unifiedAdd'_onCurve (char_gt_2:=char_gt_2) (d_nonsquare:=nonsquare_d)
(a_nonzero:=nonzero_a) (a_square:=square_a) _ _ (proj2_sig _) (proj2_sig _)).
Program Definition zero : point := (0, 1).
Program Definition add (P1 P2:point) : point := exist _ (
let (x1, y1) := coordinates P1 in
let (x2, y2) := coordinates P2 in
(((x1*y2 + y1*x2)/(1 + d*x1*x2*y1*y2)) , ((y1*y2 - a*x1*x2)/(1 - d*x1*x2*y1*y2)))) _.
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.add : E_scope.
Infix "*" := E.mul : E_scope.
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