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Require Import Crypto.Algebra.Field.
Require Import Crypto.Util.GlobalSettings Crypto.Util.Notations.
Require Import Crypto.Util.Sum Crypto.Util.Prod Crypto.Util.LetIn.
Require Import Crypto.Util.Decidable.
Require Import Crypto.Spec.MontgomeryCurve Crypto.Curves.Montgomery.Affine.
Module M.
Section MontgomeryCurve.
Import BinNat.
Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
{field:@Algebra.Hierarchy.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
{Feq_dec:Decidable.DecidableRel Feq}
{char_ge_5:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul 5}.
Local Infix "=" := Feq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope.
Local Infix "+" := Fadd. Local Infix "*" := Fmul.
Local Infix "-" := Fsub. Local Infix "/" := Fdiv.
Local Notation "x ^ 2" := (x*x).
Local Notation "0" := Fzero. Local Notation "1" := Fone.
Local Notation "'∞'" := (inr tt) : core_scope.
Local Notation "( x , y )" := (inl (pair x y)).
Context {a b: F} {b_nonzero:b <> 0}.
Local Notation add := (M.add(b_nonzero:=b_nonzero)).
Local Notation opp := (M.opp(b_nonzero:=b_nonzero)).
Local Notation point := (@M.point F Feq Fadd Fmul a b).
Program Definition to_xz (P:point) : F*F :=
match M.coordinates P with
| (x, y) => pair x 1
| ∞ => pair 1 0
end.
Let char_ge_3:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul (BinNat.N.succ_pos (BinNat.N.two)).
Proof. eapply Algebra.Hierarchy.char_ge_weaken; eauto; vm_decide. Qed.
(* From Curve25519 paper by djb, appendix B. Credited to Montgomery *)
Context {a24:F} {a24_correct:(1+1+1+1)*a24 = a-(1+1)}.
Definition xzladderstep (x1:F) (Q Q':F*F) : ((F*F)*(F*F)) :=
match Q, Q' with
pair x z, pair x' z' =>
dlet A := x+z in
dlet B := x-z in
dlet AA := A^2 in
dlet BB := B^2 in
dlet x2 := AA*BB in
dlet E := AA-BB in
dlet z2 := E*(AA + a24*E) in
dlet C := x'+z' in
dlet D := x'-z' in
dlet CB := C*B in
dlet DA := D*A in
dlet x3 := (DA+CB)^2 in
dlet z3 := x1*(DA-CB)^2 in
(pair (pair x2 z2) (pair x3 z3))
end.
End MontgomeryCurve.
End M.
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