WIP

1 files changed, 22 insertions, 25 deletions

diff --git a/src/Spec/CompleteEdwardsCurve.v b/src/Spec/CompleteEdwardsCurve.v
index 770a2d02d..801b31ffc 100644
--- a/src/Spec/CompleteEdwardsCurve.v
+++ b/src/Spec/CompleteEdwardsCurve.v
@@ -9,40 +9,37 @@ Module E.
     * <https://eprint.iacr.org/2015/677.pdf>
     *)
 
-    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}.
+    Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
+            {field:@Algebra.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
+            {char_gt_2 : @Ring.char_gt F Feq Fzero Fone Fopp Fadd Fsub Fmul (BinNat.N.succ_pos BinNat.N.one)}
+            {Feq_dec:Decidable.DecidableRel Feq}.
     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 "-" := Fsub. Local Infix "/" := Fdiv.
     Local Notation "x ^ 2" := (x*x) (at level 30).
-    Local Notation "n / d" := (fst (Fdiv n d, (_:(d <> 0)))). (* force nonzero d *)
-
-    Context {a d: F}.
-    Class twisted_edwards_params :=
-      {
-        char_gt_2 : @Ring.char_gt F Feq Fzero Fone Fopp Fadd Fsub Fmul (BinNat.N.succ_pos BinNat.N.one);
-        nonzero_a : a <> 0;
-        square_a : exists sqrt_a, sqrt_a^2 = a;
-        nonsquare_d : forall x, x^2 <> d
-      }.
-    Context `{twisted_edwards_params}. (* TODO: name me *)
-
-    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.
+
+    Context {a d: F}
+            {nonzero_a : a <> 0}
+            {square_a : exists sqrt_a, sqrt_a^2 = a}
+            {nonsquare_d : forall x, x^2 <> d}.
+
+    Definition point := { xy | let '(x, y) := xy in a*x^2 + y^2 = 1 + d*x^2*y^2 }.
+    Definition coordinates (P:point) : (F*F) := let (xy, xy_onCurve_proof) := P in xy.
     Definition eq (P Q:point) :=
-      fst (coordinates P) = fst (coordinates Q) /\
-      snd (coordinates P) = snd (coordinates Q).
+      match coordinates P, coordinates Q with
+        (x1,y1), (x2,y2) => x1 = x2 /\ y1 = y2
+      end.
 
     Program Definition zero : point := (0, 1).
-    Next Obligation. destruct H; eauto using Pre.onCurve_zero. Qed.
+    Next Obligation. eauto using Pre.onCurve_zero. Qed.
 
     Program Definition add (P1 P2:point) : point :=
-      let x1y1 := coordinates P1 in let x1 := fst x1y1 in let y1 := snd x1y1 in
-      let x2y2 := coordinates P2 in let x2 := fst x2y2 in let y2 := snd x2y2 in
-        (((x1*y2  +  y1*x2)/(1 + d*x1*x2*y1*y2)) , ((y1*y2 - a*x1*x2)/(1 - d*x1*x2*y1*y2))).
-    Next Obligation. destruct P1 as [[??]?], P2 as [[??]?], H. eauto using Pre.denominator_nonzero_x. Qed.
-    Next Obligation. destruct P1 as [[??]?], P2 as [[??]?], H; eauto using Pre.denominator_nonzero_y. Qed.
-    Next Obligation. destruct P1 as [[??]?], P2 as [[??]?], H; eauto using Pre.onCurve_add. Qed.
+      match coordinates P1, coordinates P2 return (F*F) with
+        (x1, y1), (x2, y2) =>
+        (((x1*y2  +  y1*x2)/(1 + d*x1*x2*y1*y2)) , ((y1*y2 - a*x1*x2)/(1 - d*x1*x2*y1*y2)))
+      end.
+    Next Obligation. destruct P1 as [[??]?], P2 as [[??]?]; eapply Pre.onCurve_add; eauto. Qed.
 
     Fixpoint mul (n:nat) (P : point) : point :=
       match n with