From a5f12b2990047847cff78424b8b62330bad67454 Mon Sep 17 00:00:00 2001
From: Andres Erbsen <andreser@mit.edu>
Date: Thu, 16 Jun 2016 14:51:21 -0400
Subject: port edwards curve spec

---
 src/Spec/CompleteEdwardsCurve.v | 63 ++++++++++++++++++++---------------------
 1 file changed, 31 insertions(+), 32 deletions(-)

(limited to 'src')

diff --git a/src/Spec/CompleteEdwardsCurve.v b/src/Spec/CompleteEdwardsCurve.v
index 3348be1d9..518d3d551 100644
--- a/src/Spec/CompleteEdwardsCurve.v
+++ b/src/Spec/CompleteEdwardsCurve.v
@@ -1,46 +1,45 @@
-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}.
+
+    Context {F eq Fzero one opp Fadd sub Fmul inv div} `{Algebra.field F eq Fzero one opp Fadd sub Fmul inv div}.
+    Local Infix "=" := eq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope.
+    Local Notation "0" := Fzero.  Local Notation "1" := one.
+    Local Infix "+" := Fadd. Local Infix "*" := Fmul.
+    Local Infix "-" := sub. Local Infix "/" := div.
+    Local Notation "x ^2" := (x*x) (at level 30).
+
+    Context {a d: F}.
+    Class twisted_edwards_params :=
+      {
+        char_gt_2 : 1 + 1 <> 0;
+        nonzero_a : a <> 0;
+        square_a : exists sqrt_a, sqrt_a^2 = a;
+        nonsquare_d : forall x, x^2 <> d
+      }.
+    Context `{twisted_edwards_params}.
   
-    Definition zero : point := exist _ (0, 1) (@Pre.zeroOnCurve _ _ _ prime_q).
+    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.
+
+    Program Definition zero : point := (0, 1).
     
-    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).
-  
+    Program Definition add (P1 P2:point) : point := exist _ (
+      let (x1, y1) := coordinates P1 in
+      let (x2, y2) := coordinates P2 in
+        (((x1*y2  +  y1*x2)/(1 + d*x1*x2*y1*y2)) , ((y1*y2 - a*x1*x2)/(1 - d*x1*x2*y1*y2)))) _.
+
+    (** The described points are indeed on the curve -- see [CompleteEdwardsCurve.Pre] for proof *)
+    Solve All Obligations using intros; exact Pre.zeroOnCurve
+      || exact (Pre.unifiedAdd'_onCurve (char_gt_2:=char_gt_2) (d_nonsquare:=nonsquare_d)
+         (a_nonzero:=nonzero_a) (a_square:=square_a) _ _ (proj2_sig _) (proj2_sig _)).
+
     Fixpoint mul (n:nat) (P : point) : point :=
       match n with
       | O => zero
-- 
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