use improved fsatz on various elliptic curve things

partial correctness of projective addition

stronger projective addition proof

fixup

1 files changed, 82 insertions, 0 deletions

diff --git a/src/MontgomeryX.v b/src/MontgomeryX.v
new file mode 100644
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+++ b/src/MontgomeryX.v
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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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