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Require Coq.ZArith.BinInt Coq.ZArith.Znumtheory.
Require Crypto.CompleteEdwardsCurve.Pre.
Require Import Crypto.Spec.ModularArithmetic.
Local Open Scope F_scope.
Class TwistedEdwardsParams := {
q : BinInt.Z;
a : F q;
d : F q;
prime_q : Znumtheory.prime q;
two_lt_q : BinInt.Z.lt 2 q;
nonzero_a : a <> 0;
square_a : exists sqrt_a, sqrt_a^2 = a;
nonsquare_d : forall x, x^2 <> d
}.
Module E.
Section TwistedEdwardsCurves.
Context {prm:TwistedEdwardsParams}.
(* 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>
*)
Definition onCurve P := let '(x,y) := P in a*x^2 + y^2 = 1 + d*x^2*y^2.
Definition point := { P | onCurve P}.
Definition zero : point := exist _ (0, 1) (@Pre.zeroOnCurve _ _ _ prime_q).
Definition add' P1' P2' :=
let '(x1, y1) := P1' in
let '(x2, y2) := P2' in
(((x1*y2 + y1*x2)/(1 + d*x1*x2*y1*y2)) , ((y1*y2 - a*x1*x2)/(1 - d*x1*x2*y1*y2))).
Definition add (P1 P2 : point) : point :=
let 'exist P1' pf1 := P1 in
let 'exist P2' pf2 := P2 in
exist _ (add' P1' P2')
(@Pre.unifiedAdd'_onCurve _ _ _ prime_q two_lt_q nonzero_a square_a nonsquare_d _ _ pf1 pf2).
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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