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Require Export Crypto.Spec.CompleteEdwardsCurve.
Require Import Crypto.ModularArithmetic.FField.
Require Import Crypto.ModularArithmetic.FNsatz.
Require Import Crypto.CompleteEdwardsCurve.Pre.
Require Import Crypto.ModularArithmetic.PrimeFieldTheorems.
Require Import Coq.Logic.Eqdep_dec.
Require Import Crypto.Tactics.VerdiTactics.
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.
Lemma add_0_l : forall P, (E.zero + P)%E = P.
Proof.
intros; rewrite add_comm. apply add_0_r.
Qed.
Lemma mul_0_l : forall P, (0 * P = E.zero)%E.
Proof.
auto.
Qed.
Lemma mul_S_l : forall n P, (S n * P)%E = (P + n * P)%E.
Proof.
auto.
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 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.
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.
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.
|
