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Require Import Crypto.Algebra Crypto.Algebra.Field.
Require Import Crypto.Util.GlobalSettings Crypto.Util.Notations.
Require Import Crypto.Util.Tactics Crypto.Util.Sum Crypto.Util.Prod.
Require Import Crypto.Spec.MontgomeryCurve Crypto.MontgomeryCurveTheorems.
Module M.
Section MontgomeryCurve.
Import BinNat.
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}
{char_ge_3:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul (BinNat.N.succ_pos (BinNat.N.two))}
{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.
(* 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' =>
let A := x+z in
let B := x-z in
let AA := A^2 in
let BB := B^2 in
let x2 := AA*BB in
let E := AA-BB in
let z2 := E*(AA + a24*E) in
let C := x'+z' in
let D := x'-z' in
let CB := C*B in
let DA := D*A in
let x3 := (DA+CB)^2 in
let z3 := x1*(DA-CB)^2 in
(pair (pair x2 z2) (pair x3 z3))
end.
Ltac t :=
repeat
match goal with
| _ => solve [ contradiction | trivial ]
| _ => progress intros
| _ => progress subst
| _ => progress Tactics.DestructHead.destruct_head' @M.point
| _ => progress Tactics.DestructHead.destruct_head' @prod
| _ => progress Tactics.DestructHead.destruct_head' @sum
| _ => progress Tactics.DestructHead.destruct_head' @and
| _ => progress Sum.inversion_sum
| _ => progress Prod.inversion_prod
| _ => progress Tactics.BreakMatch.break_match_hyps
| _ => progress Tactics.BreakMatch.break_match
| _ => progress cbv [fst snd M.coordinates M.add M.zero M.eq M.opp proj1_sig xzladderstep to_xz] in *
| |- _ /\ _ => split
end.
Lemma xzladderstep_correct
(Q Q':point) x z x' z' x1 x2 z2 x3 z3
(Hl:Logic.eq (pair(pair x2 z2)(pair x3 z3)) (xzladderstep x1 (pair x z) (pair x' z')))
(H:match M.coordinates Q with∞=>z=0/\x<>0|(xQ,y)=>xQ=x/z/\z<>0 (* TODO *) /\ y <> 0 (* TODO: prove this from non-squareness of a^2 - 4 *) end)
(H':match M.coordinates Q' with∞=>z'=0/\x'<>0|(xQ',_)=>xQ'=x'/z'/\z'<>0 end)
(H1:match M.coordinates (add Q (opp Q')) with∞=>False|(x,y)=>x=x1/\x<>0 end):
match M.coordinates (add Q Q) with∞=>z2=0/\x2<>0|(xQQ,_)=>xQQ=x2/z2/\z2<>0 end /\
match M.coordinates (add Q Q') with∞=>z3=0/\x3<>0|(xQQ',_)=>xQQ'=x3/z3/\z3<>0 end.
Proof. t; abstract fsatz. Qed.
End MontgomeryCurve.
End M.
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