rename-everything

4 files changed, 433 insertions, 0 deletions

diff --git a/src/Curves/Weierstrass/Affine.v b/src/Curves/Weierstrass/Affine.v
new file mode 100644
index 000000000..90bb3bdbc
--- /dev/null
+++ b/src/Curves/Weierstrass/Affine.v
@@ -0,0 +1,18 @@
+Require Import Crypto.Spec.WeierstrassCurve.
+Require Import Crypto.Algebra.Field.
+Require Import Crypto.Util.Decidable Crypto.Util.Tactics.DestructHead Crypto.Util.Tactics.BreakMatch.
+
+Module W.
+  Section W.
+    Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv} {a b:F}
+            {field:@Algebra.Hierarchy.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
+            {Feq_dec:DecidableRel Feq}.
+
+    Program Definition opp (P:@W.point F Feq Fadd Fmul a b) : @W.point F Feq Fadd Fmul a b
+      := match W.coordinates P return F*F+_ with
+         | inl (x1, y1) => inl (x1, Fopp y1)
+         | _ => P
+         end.
+    Next Obligation. destruct P as [[[??]|[]]?]; cbv; trivial; fsatz. Qed.
+  End W.
+End W.
\ No newline at end of file
diff --git a/src/Curves/Weierstrass/AffineProofs.v b/src/Curves/Weierstrass/AffineProofs.v
new file mode 100644
index 000000000..81583d88f
--- /dev/null
+++ b/src/Curves/Weierstrass/AffineProofs.v
@@ -0,0 +1,196 @@
+Require Import Coq.Numbers.BinNums.
+Require Import Coq.Classes.Morphisms.
+Require Import Crypto.Spec.WeierstrassCurve Crypto.Curves.Weierstrass.Affine.
+Require Import Crypto.Algebra.Field Crypto.Algebra.Hierarchy.
+Require Import Crypto.Util.Decidable Crypto.Util.Tactics.DestructHead Crypto.Util.Tactics.BreakMatch.
+Require Import Coq.PArith.BinPos.
+
+Module W.
+  Section W.
+    Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv} {a b:F}
+            {field:@Algebra.Hierarchy.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
+            {Feq_dec:DecidableRel Feq}
+            {char_ge_12:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul 12%positive}. (* FIXME: shouldn't need we need 4, not 12? *)
+    Let char_ge_3 : @Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul 3.
+    Proof. eapply Algebra.Hierarchy.char_ge_weaken; eauto; vm_decide. Qed.
+    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 "-" := Fsub. Local Infix "*" := Fmul.
+    Local Notation "4" := (1+1+1+1). Local Notation "27" := (4*4 + 4+4 +1+1+1).
+
+    Global Instance commutative_group {discriminant_nonzero:id(4*a*a*a + 27*b*b <> 0)} : abelian_group(eq:=W.eq(a:=a)(b:=b))(op:=W.add(char_ge_3:=char_ge_3))(id:=W.zero)(inv:=W.opp).
+    Proof using Type.
+      Time
+        repeat match goal with
+               | _ => solve [ contradiction | trivial | exact _ ]
+               | _ => intro
+               | |- Equivalence _ => split
+               | |- abelian_group => split | |- group => split | |- monoid => split
+               | |- is_associative => split | |- is_commutative => split
+               | |- is_left_inverse => split | |- is_right_inverse => split
+               | |- is_left_identity => split | |- is_right_identity => split
+               | _ => progress destruct_head' @W.point
+               | _ => progress destruct_head' sum
+               | _ => progress destruct_head' prod
+               | _ => progress destruct_head' unit
+               | _ => progress destruct_head' and
+               | _ => progress cbv [W.opp W.eq W.zero W.add W.coordinates proj1_sig]in*
+               | _ => progress break_match
+               end.
+      (* Finished transaction in 2.098 secs (2.099u,0.s) (successful) *)
+      all:  try split.
+      (* Finished transaction in 0.052 secs (0.053u,0.s) (successful) *)
+
+      (* The [discriminant_nonzero] hypothesis makes [fsatz] slow but
+      is necessary in some cases. Thus, we wrap it in [id] by detault
+      to hide it from [nsatz] but unfold it when normal [fsatz] fails. *)
+      (* Variable re-ordering is a micro-optimization *)
+      (* TODO: why does par not work here? *)
+      Ltac s := abstract (
+                    match goal with [H:id _ |- _] => move H at bottom end;
+                    move b at bottom;
+                    move a at bottom;
+                    repeat match goal with [H: ?x = Fopp ?y |- _] => is_var x; is_var y; revert H end; intros;
+                    repeat match goal with [H: ?x = ?y |- _] => is_var x; is_var y; revert H end; intros;
+                    repeat split;
+                    solve
+                      [ fsatz
+                      | cbv [id] in *; fsatz]
+                  ).
+      Time s. (* Finished transaction in 0.099 secs (0.096u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 0.094 secs (0.093u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.48 secs (0.48u,0.s) (successful) *)
+      Time s. (* Finished transaction in 2.229 secs (2.226u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 3.164 secs (3.153u,0.01s) (successful) *)
+      Time s. (* Finished transaction in 2.218 secs (2.199u,0.019s) (successful) *)
+      Time s. (* Finished transaction in 3.499 secs (3.486u,0.01s) (successful) *)
+      Time s. (* Finished transaction in 1.164 secs (1.16u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 1.971 secs (1.953u,0.016s) (successful) *)
+      Time s. (* Finished transaction in 2.344 secs (2.343u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 1.287 secs (1.286u,0.s) (successful) *)
+      Time s. (* Finished transaction in 1.781 secs (1.783u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.497 secs (0.496u,0.s) (successful) *)
+      Time s. (* Finished transaction in 1.859 secs (1.856u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 1.499 secs (1.499u,0.s) (successful) *)
+      Time s. (* Finished transaction in 1.6 secs (1.6u,0.s) (successful) *)
+      Time s. (* Finished transaction in 1.446 secs (1.443u,0.s) (successful) *)
+      Time s. (* Finished transaction in 1.56 secs (1.563u,0.s) (successful) *)
+      Time s. (* Finished transaction in 1.62 secs (1.616u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 1.973 secs (1.966u,0.006s) (successful) *)
+      Time s. (* Finished transaction in 7.66 secs (7.663u,0.s) (successful) *)
+      Time s. (* Finished transaction in 7.645 secs (7.643u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 5.956 secs (5.949u,0.006s) (successful) *)
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+      Time s. (* Finished transaction in 1.893 secs (1.893u,0.s) (successful) *)
+      Time s. (* Finished transaction in 10.23 secs (10.229u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 11.059 secs (11.036u,0.02s) (successful) *)
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+      Time s. (* Finished transaction in 9.539 secs (9.539u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 2.019 secs (2.013u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 2.907 secs (2.9u,0.01s) (successful) *)
+      Time s. (* Finished transaction in 1.622 secs (1.613u,0.01s) (successful) *)
+      Time s. (* Finished transaction in 13.205 secs (13.203u,0.003s) (successful) *)
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+      Time s. (* Finished transaction in 1.389 secs (1.386u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 10.331 secs (10.329u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 12.182 secs (12.176u,0.006s) (successful) *)
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+      Time s. (* Finished transaction in 13.709 secs (13.703u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 1.059 secs (1.06u,0.s) (successful) *)
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+      Time s. (* Finished transaction in 6.196 secs (6.196u,0.s) (successful) *)
+      Time s. (* Finished transaction in 6.949 secs (6.949u,0.s) (successful) *)
+      Time s. (* Finished transaction in 4.685 secs (4.68u,0.006s) (successful) *)
+      Time s. (* Finished transaction in 6.483 secs (6.483u,0.s) (successful) *)
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+      Time s. (* Finished transaction in 7.186 secs (7.186u,0.s) (successful) *)
+      Time s. (* Finished transaction in 8.817 secs (8.819u,0.s) (successful) *)
+      Time s. (* Finished transaction in 1.251 secs (1.25u,0.s) (successful) *)
+      Time s. (* Finished transaction in 1.569 secs (1.569u,0.s) (successful) *)
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+      Time s. (* Finished transaction in 11.45 secs (11.446u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 17.968 secs (17.969u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 12.418 secs (12.366u,0.046s) (successful) *)
+      Time s. (* Finished transaction in 15.323 secs (15.316u,0.01s) (successful) *)
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+      Time s. (* Finished transaction in 11.887 secs (11.889u,0.s) (successful) *)
+      Time s. (* Finished transaction in 7.284 secs (7.283u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 8.75 secs (8.753u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.291 secs (0.29u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.348 secs (0.346u,0.s) (successful) *)
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+      Time s. (* Finished transaction in 0.296 secs (0.296u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.737 secs (0.736u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.227 secs (0.226u,0.s) (successful) *)
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+      Time s. (* Finished transaction in 0.057 secs (0.06u,0.s) (successful) *)
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+      Time s. (* Finished transaction in 0.053 secs (0.056u,0.s) (successful) *)
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+      Time s. (* Finished transaction in 1.256 secs (1.256u,0.s) (successful) *)
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+      Time s. (* Finished transaction in 0.12 secs (0.119u,0.s) (successful) *)
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+      Time s. (* Finished transaction in 0.775 secs (0.773u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.527 secs (0.526u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.625 secs (0.623u,0.003s) (successful) *)
+      Time s. (* Finished transaction in 0.106 secs (0.106u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.586 secs (0.586u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.687 secs (0.686u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.189 secs (0.189u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.21 secs (0.209u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.066 secs (0.066u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.078 secs (0.08u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.083 secs (0.083u,0.s) (successful) *)
+      Time s. (* Finished transaction in 0.068 secs (0.066u,0.s) (successful) *)
+      (* Total: 414.396 seconds, roughly 7 minutes*)
+
+    Time Qed. (* Finished transaction in 390.998 secs (390.783u,0.276s) (successful) *)
+  End W.
+End W.
diff --git a/src/Curves/Weierstrass/Pre.v b/src/Curves/Weierstrass/Pre.v
new file mode 100644
index 000000000..6647d8e76
--- /dev/null
+++ b/src/Curves/Weierstrass/Pre.v
@@ -0,0 +1,62 @@
+Require Import Coq.Classes.Morphisms. Require Coq.Setoids.Setoid.
+Require Import Crypto.Algebra.Field.
+Require Import Crypto.Util.Tactics.DestructHead.
+Require Import Crypto.Util.Tactics.BreakMatch.
+Require Import Crypto.Util.Notations.
+Require Import Crypto.Util.Decidable.
+Import BinNums.
+
+Local Open Scope core_scope.
+
+Section Pre.
+  Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
+          {field:@Algebra.Hierarchy.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
+          {char_ge_3:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul (BinNat.N.succ_pos (BinNat.N.two))}
+          {eq_dec: DecidableRel Feq}.
+  Local Infix "=" := Feq. Local Notation "a <> b" := (not (a = b)).
+  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 "/" := Fdiv.
+  Local Notation "- x" := (Fopp x).
+  Local Notation "x ^ 2" := (x*x). Local Notation "x ^ 3" := (x*x^2).
+  Local Notation "'∞'" := unit : type_scope.
+  Local Notation "'∞'" := (inr tt) : core_scope.
+  Local Notation "2" := (1+1). Local Notation "3" := (1+2).
+  Local Notation "( x , y )" := (inl (pair x y)).
+
+  Context {a:F}.
+  Context {b:F}.
+
+  (* the canonical definitions are in Spec *)
+  Let onCurve (P:F*F + ∞) := match P with
+                             | (x, y) => y^2 = x^3 + a*x + b
+                             | ∞ => True
+                             end.
+  Let add (P1' P2':F*F + ∞) : F*F + ∞ :=
+    match P1', P2' return _ with
+    | (x1, y1), (x2, y2) =>
+      if dec (x1 = x2)
+      then
+        if dec (y2 = -y1)
+        then ∞
+        else let k := (3*x1^2+a)/(2*y1) in
+             let x3 := k^2-x1-x1 in
+             let y3 := k*(x1-x3)-y1 in
+             (x3, y3)
+      else let k := (y2-y1)/(x2-x1) in
+           let x3 := k^2-x1-x2 in
+           let y3 := k*(x1-x3)-y1 in
+           (x3, y3)
+    | ∞, ∞ => ∞
+    | ∞, _ => P2'
+    | _, ∞ => P1'
+    end.
+
+  Lemma add_onCurve P1 P2 (_:onCurve P1) (_:onCurve P2) :
+    onCurve (add P1 P2).
+  Proof using a b char_ge_3 eq_dec field.
+    destruct_head' sum; destruct_head' prod;
+      cbv [onCurve add] in *; break_match; trivial; [|]; fsatz.
+  Qed.
+End Pre.
diff --git a/src/Curves/Weierstrass/Projective.v b/src/Curves/Weierstrass/Projective.v
new file mode 100644
index 000000000..20866ca5d
--- /dev/null
+++ b/src/Curves/Weierstrass/Projective.v
@@ -0,0 +1,157 @@
+Require Import Crypto.Spec.WeierstrassCurve.
+Require Import Crypto.Util.Decidable Crypto.Algebra.Field.
+Require Import Crypto.Util.Tactics.BreakMatch.
+Require Import Crypto.Util.Tactics.DestructHead.
+Require Import Crypto.Util.Tactics.SpecializeBy.
+Require Import Crypto.Util.Tactics.SetoidSubst.
+Require Import Crypto.Util.Notations Crypto.Util.FixCoqMistakes.
+Require Import Crypto.Util.Sum Crypto.Util.Prod Crypto.Util.Sigma.
+
+Module Projective.
+  Section Projective.
+    Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv} {a b:F}
+            {field:@Algebra.Hierarchy.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
+            {char_ge_3:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul (BinNat.N.succ_pos (BinNat.N.two))}
+            {Feq_dec: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 "-" := Fsub.
+    Local Infix "*" := Fmul. Local Infix "/" := Fdiv.
+    Local Notation "x ^ 2" := (x*x). Local Notation "x ^ 3" := (x*x^2).
+    Local Notation Wpoint := (@W.point F Feq Fadd Fmul a b).
+
+    (* Originally from
+    <http://www.mat.uniroma3.it/users/pappa/CORSI/CR510_13_14/BosmaLenstra.pdf>
+    "Commplete Systems of Addition Laws" by Bosma and Lenstra;
+    optimized in <https://eprint.iacr.org/2015/1060.pdf> "Complete
+    addition formulas for prime order elliptic curves" Algorithm 1
+    "Complete, projective point addition for arbitrary prime order
+    short Weierstrass curves" by Joost Renes, Craig Costello, and
+    Lejla Batina. *)
+
+    Ltac t :=
+      repeat match goal with
+             | _ => solve [ contradiction | trivial ]
+             | _ => progress cbv zeta
+             | _ => progress intros
+             | _ => progress destruct_head' @W.point
+             | _ => progress destruct_head' sum
+             | _ => progress destruct_head' prod
+             | _ => progress destruct_head' unit
+             | _ => progress destruct_head' and
+             | _ => progress specialize_by assumption
+             | _ => progress cbv [W.eq W.add W.coordinates proj1_sig] in *
+             | _ => progress break_match_hyps
+             | _ => progress break_match
+             | |- _ /\ _ => split
+             end.
+
+    Definition point : Type := { P : F*F*F | let '(X,Y,Z) := P in Y^2*Z = X^3 + a*X*Z^2 + b*Z^3 /\ (Z = 0 -> Y <> 0) }.
+
+    Program Definition to_affine (P:point) : Wpoint :=
+      match proj1_sig P return F*F+_ with
+      | (X, Y, Z) =>
+        if dec (Z = 0) then inr tt
+        else inl (X/Z, Y/Z)
+      end.
+    Next Obligation. Proof. t. fsatz. Qed.
+
+    Program Definition of_affine (P:Wpoint) : point :=
+      match W.coordinates P return F*F*F with
+      | inl (x, y) => (x, y, 1)
+      | inr _ => (0, 1, 0)
+      end.
+    Next Obligation. Proof. t; fsatz. Qed.
+
+    Program Definition opp (P:point) : point :=
+      match proj1_sig P return F*F*F with
+      | (X, Y, Z) => (X, Fopp Y, Z)
+      end.
+    Next Obligation. Proof. t; fsatz. Qed.
+
+    Context (three_b:F) (three_b_correct: three_b = b+b+b).
+    Local Notation "4" := (1+1+1+1). Local Notation "27" := (4*4 + 4+4 +1+1+1).
+    Context {discriminant_nonzero: id(4*a*a*a + 27*b*b <> 0)}.
+
+    Program Definition add (P Q:point)
+            (y_PmQ_nz: match W.coordinates (W.add (to_affine P) (to_affine (opp Q))) return Prop with
+                       | inr _ => True
+                       | inl (_, y) => y <> 0
+                       end) : point :=
+      match proj1_sig P, proj1_sig Q return F*F*F with (X1, Y1, Z1), (X2, Y2, Z2) =>
+        let t0 := X1*X2 in
+        let t1 := Y1*Y2 in
+        let t2 := Z1*Z2 in
+        let t3 := X1+Y1 in
+        let t4 := X2+Y2 in
+        let t3 := t3*t4 in
+        let t4 := t0+t1 in
+        let t3 := t3-t4 in
+        let t4 := X1+Z1 in
+        let t5 := X2+Z2 in
+        let t4 := t4*t5 in
+        let t5 := t0+t2 in
+        let t4 := t4-t5 in
+        let t5 := Y1+Z1 in
+        let X3 := Y2+Z2 in
+        let t5 := t5*X3 in
+        let X3 := t1+t2 in
+        let t5 := t5-X3 in
+        let Z3 := a*t4 in
+        let X3 := three_b*t2 in
+        let Z3 := X3+Z3 in
+        let X3 := t1-Z3 in
+        let Z3 := t1+Z3 in
+        let Y3 := X3*Z3 in
+        let t1 := t0+t0 in
+        let t1 := t1+t0 in
+        let t2 := a*t2 in
+        let t4 := three_b*t4 in
+        let t1 := t1+t2 in
+        let t2 := t0-t2 in
+        let t2 := a*t2 in
+        let t4 := t4+t2 in
+        let t0 := t1*t4 in
+        let Y3 := Y3+t0 in
+        let t0 := t5*t4 in
+        let X3 := t3*X3 in
+        let X3 := X3-t0 in
+        let t0 := t3*t1 in
+        let Z3 := t5*Z3 in
+        let Z3 := Z3+t0 in
+        (X3, Y3, Z3)
+      end.
+    Next Obligation.
+    Proof.
+      destruct P as [p ?]; destruct p as [p Z1]; destruct p as [X1 Y1].
+      destruct Q as [q ?]; destruct q as [q Z2]; destruct q as [X2 Y2].
+      t.
+      all: try abstract fsatz.
+      (* FIXME: the final fsatz starts requiring 56 <> 0 if
+           - the next assert block is removed
+           - the assertion is changed to [Y2 = Fopp Y1] *)
+      assert (Y2 / Z2 = Fopp (Y1 / Z1)) by (
+        assert (forall pfP pfQ, match W.coordinates (W.add (to_affine (exist _ (X1,Y1,Z1) pfP)) (to_affine (exist _ (X2,Y2,Z2) pfQ))) with inl _ => False | _ => True end) by (cbv [to_affine]; t; fsatz); cbv [to_affine] in *; t; specialize_by (t;fsatz); t; fsatz).
+      unfold id in discriminant_nonzero; fsatz.
+    Qed.
+
+    Lemma to_affine_add P Q H :
+      W.eq
+        (to_affine (add P Q H))
+        (WeierstrassCurve.W.add (to_affine P) (to_affine Q)).
+    Proof using Type.
+      destruct P as [p ?]; destruct p as [p Z1]; destruct p as [X1 Y1].
+      destruct Q as [q ?]; destruct q as [q Z2]; destruct q as [X2 Y2].
+      cbv [add opp to_affine] in *; t.
+      all: try abstract fsatz.
+
+      (* zero + P = P   -- cases for x and y *)
+      assert (X1 = 0) by (setoid_subst_rel Feq; Nsatz.nsatz_power 3%nat); t; fsatz.
+      assert (X1 = 0) by (setoid_subst_rel Feq; Nsatz.nsatz_power 3%nat); t; fsatz.
+
+      (* P  + zero = P   -- cases for x and y *)
+      assert (X2 = 0) by (setoid_subst_rel Feq; Nsatz.nsatz_power 3%nat); t; fsatz.
+      assert (X2 = 0) by (setoid_subst_rel Feq; Nsatz.nsatz_power 3%nat); t; fsatz.
+    Qed.
+  End Projective.
+End Projective.