consolidate and rename Edwards curve lemmas

3 files changed, 313 insertions, 206 deletions

diff --git a/src/CompleteEdwardsCurve/CompleteEdwardsCurveTheorems.v b/src/CompleteEdwardsCurve/CompleteEdwardsCurveTheorems.v
index 2fdaff4a5..f70479c3a 100644
--- a/src/CompleteEdwardsCurve/CompleteEdwardsCurveTheorems.v
+++ b/src/CompleteEdwardsCurve/CompleteEdwardsCurveTheorems.v
@@ -7,187 +7,294 @@ Require Import Crypto.ModularArithmetic.PrimeFieldTheorems.
 Require Import Coq.Logic.Eqdep_dec.
 Require Import Crypto.Tactics.VerdiTactics.
 
-Section CompleteEdwardsCurveTheorems.
-  Context {prm:TwistedEdwardsParams}.
-  Local Opaque q a d prime_q two_lt_q nonzero_a square_a nonsquare_d. (* [F_field] calls [compute] *)
-  Existing Instance prime_q.
+Module E.
+  Section CompleteEdwardsCurveTheorems.
+    Context {prm:TwistedEdwardsParams}.
+    Local Opaque q a d prime_q two_lt_q nonzero_a square_a nonsquare_d. (* [F_field] calls [compute] *)
+    Existing Instance prime_q.
+  
+    Add Field Ffield_p' : (@Ffield_theory q _)
+      (morphism (@Fring_morph q),
+       preprocess [Fpreprocess],
+       postprocess [Fpostprocess; try exact Fq_1_neq_0; try assumption],
+       constants [Fconstant],
+       div (@Fmorph_div_theory q),
+       power_tac (@Fpower_theory q) [Fexp_tac]).
+  
+    Add Field Ffield_notConstant : (OpaqueFieldTheory q)
+      (constants [notConstant]).
+  
+    Ltac clear_prm :=
+      generalize dependent a; intro a; intros;
+      generalize dependent d; intro d; intros;
+      generalize dependent prime_q; intro prime_q; intros;
+      generalize dependent q; intro q; intros;
+      clear prm.
+  
+    Lemma point_eq : forall xy1 xy2 pf1 pf2,
+      xy1 = xy2 -> exist E.onCurve xy1 pf1 = exist E.onCurve xy2 pf2.
+    Proof.
+      destruct xy1, xy2; intros; find_injection; intros; subst. apply f_equal.
+      apply UIP_dec, F_eq_dec. (* this is a hack. We actually don't care about the equality of the proofs. However, we *can* prove it, and knowing it lets us use the universal equality instead of a type-specific equivalence, which makes many things nicer. *)
+    Qed. Hint Resolve point_eq.
+  
+    Definition point_eqb (p1 p2:E.point) : bool := andb
+        (F_eqb (fst (proj1_sig p1)) (fst (proj1_sig p2)))
+        (F_eqb (snd (proj1_sig p1)) (snd (proj1_sig p2))).
+  
+    Local Ltac t :=
+      unfold point_eqb;
+        repeat match goal with
+        | _ => progress intros
+        | _ => progress simpl in *
+        | _ => progress subst
+        | [P:E.point |- _ ] => destruct P
+        | [x: (F q * F q)%type |- _ ] => destruct x
+        | [H: _ /\ _ |- _ ] => destruct H
+        | [H: _ |- _ ] => rewrite Bool.andb_true_iff in H
+        | [H: _ |- _ ] => apply F_eqb_eq in H
+        | _ => rewrite F_eqb_refl
+        end; eauto.
+  
+    Lemma point_eqb_sound : forall p1 p2, point_eqb p1 p2 = true -> p1 = p2.
+    Proof.
+      t.
+    Qed.
+  
+    Lemma point_eqb_complete : forall p1 p2, p1 = p2 -> point_eqb p1 p2 = true.
+    Proof.
+      t.
+    Qed.
+  
+    Lemma point_eqb_neq : forall p1 p2, point_eqb p1 p2 = false -> p1 <> p2.
+    Proof.
+      intros. destruct (point_eqb p1 p2) eqn:Hneq; intuition.
+      apply point_eqb_complete in H0; congruence.
+    Qed.
+  
+    Lemma point_eqb_neq_complete : forall p1 p2, p1 <> p2 -> point_eqb p1 p2 = false.
+    Proof.
+      intros. destruct (point_eqb p1 p2) eqn:Hneq; intuition.
+      apply point_eqb_sound in Hneq. congruence.
+    Qed.
+  
+    Lemma point_eqb_refl : forall p, point_eqb p p = true.
+    Proof.
+      t.
+    Qed.
+  
+    Definition point_eq_dec (p1 p2:E.point) : {p1 = p2} + {p1 <> p2}.
+      destruct (point_eqb p1 p2) eqn:H; match goal with
+                                        | [ H: _ |- _ ] => apply point_eqb_sound in H
+                                        | [ H: _ |- _ ] => apply point_eqb_neq in H
+                                        end; eauto.
+    Qed.
+  
+    Lemma point_eqb_correct : forall p1 p2, point_eqb p1 p2 = if point_eq_dec p1 p2 then true else false.
+    Proof.
+      intros. destruct (point_eq_dec p1 p2); eauto using point_eqb_complete, point_eqb_neq_complete.
+    Qed.
+  
+    Ltac Edefn := unfold E.add, E.add', E.zero; intros;
+                    repeat match goal with
+                           | [ p : E.point |- _ ] =>
+                             let x := fresh "x" p in
+                             let y := fresh "y" p in
+                             let pf := fresh "pf" p in
+                             destruct p as [[x y] pf]; unfold E.onCurve in pf
+                    | _ => eapply point_eq, (f_equal2 pair)
+                    | _ => eapply point_eq
+    end.
+    Lemma add_comm : forall A B, (A+B = B+A)%E.
+    Proof.
+      Edefn; apply (f_equal2 div); ring.
+    Qed.
+  
+    Ltac unifiedAdd_nonzero := match goal with
+    | [ |- (?op 1 (d * _ * _ * _ * _ *
+      inv (1 - d * ?xA * ?xB * ?yA * ?yB) * inv (1 + d * ?xA * ?xB * ?yA * ?yB)))%F <> 0%F]
+      => let Hadd := fresh "Hadd" in
+       pose proof (@unifiedAdd'_onCurve _ _ _ _ two_lt_q nonzero_a square_a nonsquare_d (xA, yA) (xB, yB)) as Hadd;
+       simpl in Hadd;
+       match goal with
+       | [H : (1 - d * ?xC * xB * ?yC * yB)%F <> 0%F |- (?op 1 ?other)%F <> 0%F] =>
+         replace other with
+          (d * xC * ((xA * yB + yA * xB) / (1 + d * xA * xB * yA * yB)) 
+          * yC * ((yA * yB - a * xA * xB) / (1 - d * xA * xB * yA * yB)))%F by (subst; unfold div; ring);
+          auto
+      end
+    end.
+  
+    Lemma add_assoc : forall A B C, (A+(B+C) = (A+B)+C)%E.
+    Proof.
+      Edefn; F_field_simplify_eq; try abstract (rewrite ?@F_pow_2_r in *; clear_prm; F_nsatz);
+        pose proof (@edwardsAddCompletePlus _ _ _ _ two_lt_q nonzero_a square_a nonsquare_d);
+        pose proof (@edwardsAddCompleteMinus _ _ _ _ two_lt_q nonzero_a square_a nonsquare_d);
+        cbv beta iota in *;
+        repeat split; field_nonzero idtac; unifiedAdd_nonzero.
+    Qed.
+  
+    Lemma add_0_r : forall P, (P + E.zero = P)%E.
+    Proof.
+      Edefn; repeat rewrite ?F_add_0_r, ?F_add_0_l, ?F_sub_0_l, ?F_sub_0_r,
+             ?F_mul_0_r, ?F_mul_0_l, ?F_mul_1_l, ?F_mul_1_r, ?F_div_1_r; exact eq_refl.
+    Qed.
 
-  Add Field Ffield_p' : (@Ffield_theory q _)
-    (morphism (@Fring_morph q),
-     preprocess [Fpreprocess],
-     postprocess [Fpostprocess; try exact Fq_1_neq_0; try assumption],
-     constants [Fconstant],
-     div (@Fmorph_div_theory q),
-     power_tac (@Fpower_theory q) [Fexp_tac]).
+    Lemma add_0_l : forall P, (E.zero + P)%E = P.
+    Proof.
+      intros; rewrite add_comm. apply add_0_r.
+    Qed.
 
-  Add Field Ffield_notConstant : (OpaqueFieldTheory q)
-    (constants [notConstant]).
+    Lemma mul_0_l : forall P, (0 * P = E.zero)%E.
+    Proof.
+      auto.
+    Qed.
 
-  Ltac clear_prm :=
-    generalize dependent a; intro a; intros;
-    generalize dependent d; intro d; intros;
-    generalize dependent prime_q; intro prime_q; intros;
-    generalize dependent q; intro q; intros;
-    clear prm.
+    Lemma mul_S_l : forall n P, (S n * P)%E = (P + n * P)%E.
+    Proof.
+      auto.
+    Qed.
 
-  Lemma point_eq' : forall xy1 xy2 pf1 pf2,
-    xy1 = xy2 -> exist onCurve xy1 pf1 = exist onCurve xy2 pf2.
-  Proof.
-    destruct xy1, xy2; intros; find_injection; intros; subst. apply f_equal.
-    apply UIP_dec, F_eq_dec. (* this is a hack. We actually don't care about the equality of the proofs. However, we *can* prove it, and knowing it lets us use the universal equality instead of a type-specific equivalence, which makes many things nicer. *)
-  Qed.
+    Lemma mul_add_l : forall a b P, ((a + b)%nat * P)%E = E.add (a * P)%E (b * P)%E.
+    Proof.
+      induction a; intros; rewrite ?plus_Sn_m, ?plus_O_n, ?mul_S_l, ?mul_0_l, ?add_0_l, ?mul_S_, ?IHa, ?add_assoc; auto.
+    Qed.
 
-  Lemma point_eq : forall p1 p2, p1 = p2 -> forall pf1 pf2,
-    mkPoint p1 pf1 = mkPoint p2 pf2.
-  Proof.
-    eauto using point_eq'.
-  Qed.
-  Hint Resolve point_eq' point_eq.
+    Lemma mul_assoc : forall (n m : nat) P, (n * (m * P) = (n * m)%nat * P)%E.
+    Proof.
+      induction n; intros; auto.
+      rewrite ?mul_S_l, ?Mult.mult_succ_l, ?mul_add_l, ?IHn, add_comm. reflexivity.
+    Qed.
 
-  Definition point_eqb (p1 p2:point) : bool := andb
-      (F_eqb (fst (proj1_sig p1)) (fst (proj1_sig p2)))
-      (F_eqb (snd (proj1_sig p1)) (snd (proj1_sig p2))).
-
-  Local Ltac t :=
-    unfold point_eqb;
-      repeat match goal with
-      | _ => progress intros
-      | _ => progress simpl in *
-      | _ => progress subst
-      | [P:point |- _ ] => destruct P
-      | [x: (F q * F q)%type |- _ ] => destruct x
-      | [H: _ /\ _ |- _ ] => destruct H
-      | [H: _ |- _ ] => rewrite Bool.andb_true_iff in H
-      | [H: _ |- _ ] => apply F_eqb_eq in H
-      | _ => rewrite F_eqb_refl
-      end; eauto.
-
-  Lemma point_eqb_sound : forall p1 p2, point_eqb p1 p2 = true -> p1 = p2.
-  Proof.
-    t.
-  Qed.
-
-  Lemma point_eqb_complete : forall p1 p2, p1 = p2 -> point_eqb p1 p2 = true.
-  Proof.
-    t.
-  Qed.
-
-  Lemma point_eqb_neq : forall p1 p2, point_eqb p1 p2 = false -> p1 <> p2.
-  Proof.
-    intros. destruct (point_eqb p1 p2) eqn:Hneq; intuition.
-    apply point_eqb_complete in H0; congruence.
-  Qed.
-
-  Lemma point_eqb_neq_complete : forall p1 p2, p1 <> p2 -> point_eqb p1 p2 = false.
-  Proof.
-    intros. destruct (point_eqb p1 p2) eqn:Hneq; intuition.
-    apply point_eqb_sound in Hneq. congruence.
-  Qed.
-
-  Lemma point_eqb_refl : forall p, point_eqb p p = true.
-  Proof.
-    t.
-  Qed.
-
-  Definition point_eq_dec (p1 p2:point) : {p1 = p2} + {p1 <> p2}.
-    destruct (point_eqb p1 p2) eqn:H; match goal with
-                                      | [ H: _ |- _ ] => apply point_eqb_sound in H
-                                      | [ H: _ |- _ ] => apply point_eqb_neq in H
-                                      end; eauto.
-  Qed.
-
-  Lemma point_eqb_correct : forall p1 p2, point_eqb p1 p2 = if point_eq_dec p1 p2 then true else false.
-  Proof.
-    intros. destruct (point_eq_dec p1 p2); eauto using point_eqb_complete, point_eqb_neq_complete.
-  Qed.
-
-  Ltac Edefn := unfold unifiedAdd, unifiedAdd', zero; intros;
-                  repeat match goal with
-                         | [ p : point |- _ ] =>
-                           let x := fresh "x" p in
-                           let y := fresh "y" p in
-                           let pf := fresh "pf" p in
-                           destruct p as [[x y] pf]; unfold onCurve in pf
-                  | _ => eapply point_eq, (f_equal2 pair)
-                  | _ => eapply point_eq
-  end.
-  Lemma twistedAddComm : forall A B, (A+B = B+A)%E.
-  Proof.
-    Edefn; apply (f_equal2 div); ring.
-  Qed.
-
-  Ltac unifiedAdd_nonzero := match goal with
-  | [ |- (?op 1 (d * _ * _ * _ * _ *
-    inv (1 - d * ?xA * ?xB * ?yA * ?yB) * inv (1 + d * ?xA * ?xB * ?yA * ?yB)))%F <> 0%F]
-    => let Hadd := fresh "Hadd" in
-     pose proof (@unifiedAdd'_onCurve _ _ _ _ two_lt_q nonzero_a square_a nonsquare_d (xA, yA) (xB, yB)) as Hadd;
-     simpl in Hadd;
-     match goal with
-     | [H : (1 - d * ?xC * xB * ?yC * yB)%F <> 0%F |- (?op 1 ?other)%F <> 0%F] =>
-       replace other with
-        (d * xC * ((xA * yB + yA * xB) / (1 + d * xA * xB * yA * yB)) 
-        * yC * ((yA * yB - a * xA * xB) / (1 - d * xA * xB * yA * yB)))%F by (subst; unfold div; ring);
-        auto
-    end
-  end.
-
-  Lemma twistedAddAssoc : forall A B C, (A+(B+C) = (A+B)+C)%E.
-  Proof.
-    (* The Ltac takes ~15s, the Qed no longer takes longer than I have had patience for *)
-    Edefn; F_field_simplify_eq; try abstract (rewrite ?@F_pow_2_r in *; clear_prm; F_nsatz);
-      pose proof (@edwardsAddCompletePlus _ _ _ _ two_lt_q nonzero_a square_a nonsquare_d);
-      pose proof (@edwardsAddCompleteMinus _ _ _ _ two_lt_q nonzero_a square_a nonsquare_d);
-      cbv beta iota in *;
-      repeat split; field_nonzero idtac; unifiedAdd_nonzero.
-  Qed.
-
-  Lemma zeroIsIdentity : forall P, (P + zero = P)%E.
-  Proof.
-    Edefn; repeat rewrite ?F_add_0_r, ?F_add_0_l, ?F_sub_0_l, ?F_sub_0_r,
-           ?F_mul_0_r, ?F_mul_0_l, ?F_mul_1_l, ?F_mul_1_r, ?F_div_1_r; exact eq_refl.
-  Qed.
-
-  (* solve for x ^ 2 *)
-  Definition solve_for_x2 (y : F q) := ((y ^ 2 - 1) / (d * (y ^ 2) - a))%F.
-
-  Lemma d_y2_a_nonzero : (forall y, 0 <> d * y ^ 2 - a)%F.
-    intros ? eq_zero.
-    pose proof prime_q.
-    destruct square_a as [sqrt_a sqrt_a_id].
-    rewrite <- sqrt_a_id in eq_zero.
-    destruct (Fq_square_mul_sub _ _ _ eq_zero) as [ [sqrt_d sqrt_d_id] | a_zero].
-    + pose proof (nonsquare_d sqrt_d); auto.
-    + subst.
-      rewrite Fq_pow_zero in sqrt_a_id by congruence.
-      auto using nonzero_a.
-  Qed.
-
-  Lemma a_d_y2_nonzero : (forall y, a - d * y ^ 2 <> 0)%F.
-  Proof.
-    intros y eq_zero.
-    pose proof prime_q.
-    eapply F_minus_swap in eq_zero.
-    eauto using (d_y2_a_nonzero y).
-  Qed.
-
-  Lemma solve_correct : forall x y, onCurve (x, y) <->
-    (x ^ 2 = solve_for_x2 y)%F.
-  Proof.
-    split.
-    + intro onCurve_x_y.
+    Lemma mul_zero_r : forall m, (m * E.zero = E.zero)%E.
+    Proof.
+      induction m; rewrite ?mul_S_l, ?add_0_l; auto.
+    Qed.
+  
+    (* solve for x ^ 2 *)
+    Definition solve_for_x2 (y : F q) := ((y ^ 2 - 1) / (d * (y ^ 2) - a))%F.
+  
+    Lemma d_y2_a_nonzero : (forall y, 0 <> d * y ^ 2 - a)%F.
+      intros ? eq_zero.
       pose proof prime_q.
-      unfold onCurve in onCurve_x_y.
-      eapply F_div_mul; auto using (d_y2_a_nonzero y).
-      replace (x ^ 2 * (d * y ^ 2 - a))%F with ((d * x ^ 2 * y ^ 2) - (a * x ^ 2))%F by ring.
-      rewrite F_sub_add_swap.
-      replace (y ^ 2 + a * x ^ 2)%F with (a * x ^ 2 + y ^ 2)%F by ring.
-      rewrite onCurve_x_y.
-      ring.
-    + intro x2_eq.
-      unfold onCurve, solve_for_x2 in *.
-      rewrite x2_eq.
-      field.
-      auto using d_y2_a_nonzero.
-  Qed.
-
-End CompleteEdwardsCurveTheorems.
+      destruct square_a as [sqrt_a sqrt_a_id].
+      rewrite <- sqrt_a_id in eq_zero.
+      destruct (Fq_square_mul_sub _ _ _ eq_zero) as [ [sqrt_d sqrt_d_id] | a_zero].
+      + pose proof (nonsquare_d sqrt_d); auto.
+      + subst.
+        rewrite Fq_pow_zero in sqrt_a_id by congruence.
+        auto using nonzero_a.
+    Qed.
+  
+    Lemma a_d_y2_nonzero : (forall y, a - d * y ^ 2 <> 0)%F.
+    Proof.
+      intros y eq_zero.
+      pose proof prime_q.
+      eapply F_minus_swap in eq_zero.
+      eauto using (d_y2_a_nonzero y).
+    Qed.
+  
+    Lemma solve_correct : forall x y, E.onCurve (x, y) <->
+      (x ^ 2 = solve_for_x2 y)%F.
+    Proof.
+      split.
+      + intro onCurve_x_y.
+        pose proof prime_q.
+        unfold E.onCurve in onCurve_x_y.
+        eapply F_div_mul; auto using (d_y2_a_nonzero y).
+        replace (x ^ 2 * (d * y ^ 2 - a))%F with ((d * x ^ 2 * y ^ 2) - (a * x ^ 2))%F by ring.
+        rewrite F_sub_add_swap.
+        replace (y ^ 2 + a * x ^ 2)%F with (a * x ^ 2 + y ^ 2)%F by ring.
+        rewrite onCurve_x_y.
+        ring.
+      + intro x2_eq.
+        unfold E.onCurve, solve_for_x2 in *.
+        rewrite x2_eq.
+        field.
+        auto using d_y2_a_nonzero.
+    Qed.
+  
+  
+    Program Definition opp (P:E.point) : E.point := let '(x, y) := proj1_sig P in (opp x, y).
+    Next Obligation. Proof.
+      pose (proj2_sig P) as H; rewrite <-Heq_anonymous in H; simpl in H.
+      rewrite F_square_opp; trivial.
+    Qed.
+    
+    Definition sub P Q := (P + opp Q)%E.
+    
+    Lemma opp_zero : opp E.zero = E.zero.
+    Proof.
+      pose proof @F_opp_0.
+      unfold opp, E.zero; eapply point_eq; congruence.
+    Qed.
+  
+    Lemma add_opp_r : forall P, (P + opp P = E.zero)%E.
+    Proof.
+      unfold opp; Edefn; rewrite ?@F_pow_2_r in *; (F_field_simplify_eq; [clear_prm; F_nsatz|..]);
+        rewrite <-?@F_pow_2_r in *;
+        pose proof (@edwardsAddCompletePlus _ _ _ _ two_lt_q nonzero_a square_a nonsquare_d _ _ _ _ pfP pfP);
+        pose proof (@edwardsAddCompleteMinus _ _ _ _ two_lt_q nonzero_a square_a nonsquare_d _ _ _ _ pfP pfP);
+        field_nonzero idtac.
+    Qed.
+  
+    Lemma add_opp_l : forall P, (opp P + P = E.zero)%E.
+    Proof.
+      intros. rewrite add_comm. eapply add_opp_r.
+    Qed.
+  
+    Lemma add_cancel_r : forall A B C, (B+A = C+A -> B = C)%E.
+    Proof.
+      intros.
+      assert ((B + A) + opp A = (C + A) + opp A)%E as Hc by congruence.
+      rewrite <-!add_assoc, !add_opp_r, !add_0_r in Hc; exact Hc.
+    Qed.
+  
+    Lemma add_cancel_l : forall A B C, (A+B = A+C -> B = C)%E.
+    Proof.
+      intros.
+      rewrite (add_comm A C) in H.
+      rewrite (add_comm A B) in H.
+      eauto using add_cancel_r.
+    Qed.
+    
+    Lemma shuffle_eq_add_opp : forall P Q R, (P + Q = R <-> Q = opp P + R)%E.
+    Proof.
+      split; intros.
+      { assert (opp P + (P + Q) = opp P + R)%E as Hc by congruence.
+        rewrite add_assoc, add_opp_l, add_comm, add_0_r in Hc; exact Hc. }
+      { subst. rewrite add_assoc, add_opp_r, add_comm, add_0_r; reflexivity. }
+    Qed.
+        
+    Lemma opp_opp : forall P, opp (opp P) = P.
+    Proof.
+      intros.
+      pose proof (add_opp_r P%E) as H.
+      rewrite add_comm in H.
+      rewrite shuffle_eq_add_opp in H.
+      rewrite add_0_r in H.
+      congruence.
+    Qed.
+    
+    Lemma opp_add : forall P Q, opp (P + Q)%E = (opp P + opp Q)%E.
+    Proof.
+      intros.
+      pose proof (add_opp_r (P+Q)%E) as H.
+      rewrite <-!add_assoc in H.
+      rewrite add_comm in H.
+      rewrite <-!add_assoc in H.
+      rewrite shuffle_eq_add_opp in H.
+      rewrite add_comm in H.
+      rewrite shuffle_eq_add_opp in H.
+      rewrite add_0_r in H.
+      assumption.
+    Qed.
+      
+    Lemma opp_mul : forall n P, opp (E.mul n P) = E.mul n (opp P).
+    Proof.
+      pose proof opp_add; pose proof opp_zero.
+      induction n; simpl; intros; congruence.
+    Qed.
+  End CompleteEdwardsCurveTheorems.
+End E.
+Infix "-" := E.sub : E_scope.
\ No newline at end of file
diff --git a/src/CompleteEdwardsCurve/DoubleAndAdd.v b/src/CompleteEdwardsCurve/DoubleAndAdd.v
index d58a06c55..50027349d 100644
--- a/src/CompleteEdwardsCurve/DoubleAndAdd.v
+++ b/src/CompleteEdwardsCurve/DoubleAndAdd.v
@@ -5,26 +5,26 @@ Require Import Coq.Numbers.BinNums Coq.NArith.NArith Coq.NArith.Nnat Coq.ZArith.
 
 Section EdwardsDoubleAndAdd.
   Context {prm:TwistedEdwardsParams}.
-  Definition doubleAndAdd (bound n : nat) (P : point) : point :=
-    iter_op unifiedAdd zero N.testbit_nat (N.of_nat n) P bound.
+  Definition doubleAndAdd (bound n : nat) (P : E.point) : E.point :=
+    iter_op E.add E.zero N.testbit_nat (N.of_nat n) P bound.
 
-  Lemma scalarMult_double : forall n P, scalarMult (n + n) P = scalarMult n (P + P)%E.
+  Lemma scalarMult_double : forall n P, E.mul (n + n) P = E.mul n (P + P)%E.
   Proof.
     intros.
     replace (n + n)%nat with (n * 2)%nat by omega.
     induction n; simpl; auto.
-    rewrite twistedAddAssoc.
+    rewrite E.add_assoc.
     f_equal; auto.
   Qed.
 
   Lemma doubleAndAdd_spec : forall bound n P, N.size_nat (N.of_nat n) <= bound ->
-    scalarMult n P = doubleAndAdd bound n P.
+    E.mul n P = doubleAndAdd bound n P.
   Proof.
     induction n; auto; intros; unfold doubleAndAdd;
       rewrite iter_op_spec with (scToN := fun x => x); (
         unfold Morphisms.Proper, Morphisms.respectful, Equivalence.equiv;
         intros; subst; try rewrite Nat2N.id;
-        reflexivity || assumption || apply twistedAddAssoc
-        || rewrite twistedAddComm; apply zeroIsIdentity).
+        reflexivity || assumption || apply E.add_assoc
+        || rewrite E.add_comm; apply E.add_0_r).
   Qed.
 End EdwardsDoubleAndAdd.
\ No newline at end of file
diff --git a/src/CompleteEdwardsCurve/ExtendedCoordinates.v b/src/CompleteEdwardsCurve/ExtendedCoordinates.v
index a9fbf5e40..5e4fe0b0b 100644
--- a/src/CompleteEdwardsCurve/ExtendedCoordinates.v
+++ b/src/CompleteEdwardsCurve/ExtendedCoordinates.v
@@ -45,7 +45,7 @@ Section ExtendedCoordinates.
   Local Notation "P '~=' rP" := (rep P rP) (at level 70).
 
   Ltac unfoldExtended :=
-    repeat progress (autounfold; unfold onCurve, unifiedAdd, unifiedAdd', rep in *; intros);
+    repeat progress (autounfold; unfold E.onCurve, E.add, E.add', rep in *; intros);
     repeat match goal with
       | [ p : (F q*F q)%type |- _ ] =>
           let x := fresh "x" p in
@@ -83,14 +83,14 @@ Section ExtendedCoordinates.
     solveExtended.
   Qed.
 
-  Definition extendedPoint := { P:extended | rep P (extendedToTwisted P) /\ onCurve (extendedToTwisted P) }.
+  Definition extendedPoint := { P:extended | rep P (extendedToTwisted P) /\ E.onCurve (extendedToTwisted P) }.
 
-  Program Definition mkExtendedPoint : point -> extendedPoint := twistedToExtended.
+  Program Definition mkExtendedPoint : E.point -> extendedPoint := twistedToExtended.
   Next Obligation.
     destruct x; erewrite extendedToTwisted_rep; eauto using twistedToExtended_rep.
   Qed.
 
-  Program Definition unExtendedPoint : extendedPoint -> point := extendedToTwisted.
+  Program Definition unExtendedPoint : extendedPoint -> E.point := extendedToTwisted.
   Next Obligation.
     destruct x; simpl; intuition.
   Qed.
@@ -103,7 +103,7 @@ Section ExtendedCoordinates.
 
   Lemma unExtendedPoint_mkExtendedPoint : forall P, unExtendedPoint (mkExtendedPoint P) = P.
   Proof.
-    destruct P; eapply point_eq; simpl; erewrite extendedToTwisted_rep; eauto using twistedToExtended_rep.
+    destruct P; eapply E.point_eq; simpl; erewrite extendedToTwisted_rep; eauto using twistedToExtended_rep.
   Qed.
 
   Global Instance Proper_mkExtendedPoint : Proper (eq==>equiv) mkExtendedPoint.
@@ -135,12 +135,12 @@ Section ExtendedCoordinates.
       let T3 := E*H in
       let Z3 := F*G in
       (X3, Y3, Z3, T3).
-    Local Hint Unfold unifiedAdd.
+    Local Hint Unfold E.add.
 
     Local Ltac tnz := repeat apply Fq_mul_nonzero_nonzero; auto using (@char_gt_2 q two_lt_q).
 
-    Lemma unifiedAddM1'_rep: forall P Q rP rQ, onCurve rP -> onCurve rQ ->
-      P ~= rP -> Q ~= rQ -> (unifiedAddM1' P Q) ~= (unifiedAdd' rP rQ).
+    Lemma unifiedAddM1'_rep: forall P Q rP rQ, E.onCurve rP -> E.onCurve rQ ->
+      P ~= rP -> Q ~= rQ -> (unifiedAddM1' P Q) ~= (E.add' rP rQ).
     Proof.
       intros P Q rP rQ HoP HoQ HrP HrQ.
       pose proof (@edwardsAddCompletePlus _ _ _ _ two_lt_q nonzero_a square_a nonsquare_d).
@@ -152,9 +152,9 @@ Section ExtendedCoordinates.
         field_nonzero tnz.
     Qed.
 
-    Lemma unifiedAdd'_onCurve : forall P Q, onCurve P -> onCurve Q -> onCurve (unifiedAdd' P Q).
+    Lemma unifiedAdd'_onCurve : forall P Q, E.onCurve P -> E.onCurve Q -> E.onCurve (E.add' P Q).
     Proof.
-      intros; pose proof (proj2_sig (unifiedAdd (mkPoint _ H) (mkPoint _ H0))); eauto.
+      intros; pose proof (proj2_sig (E.add (exist _ _ H) (exist _ _ H0))); eauto.
     Qed.
 
     Program Definition unifiedAddM1 : extendedPoint -> extendedPoint -> extendedPoint := unifiedAddM1'.
@@ -167,9 +167,9 @@ Section ExtendedCoordinates.
         eauto using unifiedAdd'_onCurve.
     Qed.
 
-    Lemma unifiedAddM1_rep : forall P Q, unifiedAdd (unExtendedPoint P) (unExtendedPoint Q) = unExtendedPoint (unifiedAddM1 P Q).
+    Lemma unifiedAddM1_rep : forall P Q, E.add (unExtendedPoint P) (unExtendedPoint Q) = unExtendedPoint (unifiedAddM1 P Q).
     Proof.
-      destruct P, Q; unfold unExtendedPoint, unifiedAdd, unifiedAddM1; eapply point_eq; simpl in *; intuition.
+      destruct P, Q; unfold unExtendedPoint, E.add, unifiedAddM1; eapply E.point_eq; simpl in *; intuition.
       pose proof (unifiedAddM1'_rep x x0 (extendedToTwisted x) (extendedToTwisted x0));
         destruct (unifiedAddM1' x x0);
         unfold rep in *; intuition.
@@ -180,23 +180,23 @@ Section ExtendedCoordinates.
       repeat (econstructor || intro).
       repeat match goal with [H: _ === _ |- _ ] => inversion H; clear H end; unfold equiv, extendedPoint_eq.
       rewrite <-!unifiedAddM1_rep.
-      destruct x, y, x0, y0; simpl in *; eapply point_eq; congruence.
+      destruct x, y, x0, y0; simpl in *; eapply E.point_eq; congruence.
     Qed.
 
-    Lemma unifiedAddM1_0_r : forall P, unifiedAddM1 P (mkExtendedPoint zero) === P.
+    Lemma unifiedAddM1_0_r : forall P, unifiedAddM1 P (mkExtendedPoint E.zero) === P.
       unfold equiv, extendedPoint_eq; intros.
-      rewrite <-!unifiedAddM1_rep, unExtendedPoint_mkExtendedPoint, zeroIsIdentity; auto.
+      rewrite <-!unifiedAddM1_rep, unExtendedPoint_mkExtendedPoint, E.add_0_r; auto.
     Qed.
 
-    Lemma unifiedAddM1_0_l : forall P, unifiedAddM1 (mkExtendedPoint zero) P === P.
+    Lemma unifiedAddM1_0_l : forall P, unifiedAddM1 (mkExtendedPoint E.zero) P === P.
       unfold equiv, extendedPoint_eq; intros.
-      rewrite <-!unifiedAddM1_rep, twistedAddComm, unExtendedPoint_mkExtendedPoint, zeroIsIdentity; auto.
+      rewrite <-!unifiedAddM1_rep, E.add_comm, unExtendedPoint_mkExtendedPoint, E.add_0_r; auto.
     Qed.
 
     Lemma unifiedAddM1_assoc : forall a b c, unifiedAddM1 a (unifiedAddM1 b c) === unifiedAddM1 (unifiedAddM1 a b) c.
     Proof.
       unfold equiv, extendedPoint_eq; intros.
-      rewrite <-!unifiedAddM1_rep, twistedAddAssoc; auto.
+      rewrite <-!unifiedAddM1_rep, E.add_assoc; auto.
     Qed.
 
     Lemma testbit_conversion_identity : forall x i, N.testbit_nat x i = N.testbit_nat ((fun a => a) x) i.
@@ -204,14 +204,14 @@ Section ExtendedCoordinates.
       trivial.
     Qed.
 
-    Definition scalarMultM1 := iter_op unifiedAddM1 (mkExtendedPoint zero) N.testbit_nat.
+    Definition scalarMultM1 := iter_op unifiedAddM1 (mkExtendedPoint E.zero) N.testbit_nat.
     Definition scalarMultM1_spec :=
-      iter_op_spec unifiedAddM1 unifiedAddM1_assoc (mkExtendedPoint zero) unifiedAddM1_0_l
+      iter_op_spec unifiedAddM1 unifiedAddM1_assoc (mkExtendedPoint E.zero) unifiedAddM1_0_l
         N.testbit_nat (fun x => x) testbit_conversion_identity.
-    Lemma scalarMultM1_rep : forall n P, unExtendedPoint (scalarMultM1 (N.of_nat n) P (N.size_nat (N.of_nat n))) = scalarMult n (unExtendedPoint P).
+    Lemma scalarMultM1_rep : forall n P, unExtendedPoint (scalarMultM1 (N.of_nat n) P (N.size_nat (N.of_nat n))) = E.mul n (unExtendedPoint P).
       intros; rewrite scalarMultM1_spec, Nat2N.id; auto.
       induction n; [simpl; rewrite !unExtendedPoint_mkExtendedPoint; reflexivity|].
-      unfold scalarMult; fold scalarMult.
+      unfold E.mul; fold E.mul.
       rewrite <-IHn, unifiedAddM1_rep; auto.
     Qed.