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Require Export Crypto.Spec.CompleteEdwardsCurve.
Require Import Crypto.ModularArithmetic.FField.
Require Import Crypto.CompleteEdwardsCurve.Pre.
Require Import Crypto.ModularArithmetic.PrimeFieldTheorems.
Require Import 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.
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]).
Lemma point_eq : forall p1 p2, p1 = p2 -> forall pf1 pf2,
mkPoint p1 pf1 = mkPoint p2 pf2.
Proof.
destruct p1, p2; 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.
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.
Lemma twistedAddAssoc : forall A B C, (A+(B+C) = (A+B)+C)%E.
Proof.
intros; Edefn; F_field_simplify_eq.
- (* nsatz. *) (* requires IntegralDomain (F q), then gives stack overflow *)
admit. (*sage -c "
R.<d,a,xA,yA,xB,yB,xC,yC> = PolynomialRing(QQ,8,order='invlex')
I = R.ideal([
(a * (xA * xA) + yA * yA) - (1 + d * (xA * xA) * (yA * yA)),
(a * (xB * xB) + yB * yB) - (1 + d * (xB * xB) * (yB * yB)),
(a * (xC * xC) + yC * yC) - (1 + d * (xC * xC) * (yC * yC))
])
print ((- xA * xA * xA * yB * yB * yB * yB * yC * yC * xB * xB * xB * xC * d * d * d * yA * yA -
xA * xA * xA * yB * yB * yB * yC * yC * yC * a * xB * xB * xC * xC * d * d +
xA * xA * xA * yB * yB * yB * yC * a * xB * xB * xB * xB * xC * xC * d * d * d * yA * yA -
xA * xA * xA * yB * yB * yB * yC * xB * xB * d * d * yA * yA +
xA * xA * xA * yB * yB * yC * yC * a * a * xB * xB * xB * xC * xC * xC * d * d -
xA * xA * xA * yB * yB * yC * yC * a * xB * xC * d +
xA * xA * xA * yB * yB * a * xB * xB * xB * xC * d * d * yA * yA +
xA * xA * xA * yB * yC * a * a * xB * xB * xC * xC * d +
xA * xA * yB * yB * yB * yB * yC * yC * yC * xB * xC * xC * d * d * yA +
xA * xA * yB * yB * yB * yB * yC * xB * xB * xB * xC * xC * d * d * d * yA * yA * yA +
xA * xA * yB * yB * yB * yC * yC * xB * xB * xB * xB * xC * d * d * d * yA * yA * yA +
xA * xA * yB * yB * yB * yC * yC * xC * d * yA -
xA * xA * yB * yB * yB * xB * xB * xC * d * d * yA * yA * yA -
(1 + 1) * xA * xA * yB * yB * yC * a * xB * xC * xC * d * yA -
xA * xA * yB * yB * yC * xB * xB * xB * d * d * yA * yA * yA +
xA * xA * yB * yC * yC * a * a * xB * xB * xB * xB * xC * xC * xC * d * d * yA -
(1 + 1) * xA * xA * yB * yC * yC * a * xB * xB * xC * d * yA +
xA * xA * yC * a * a * xB * xB * xB * xC * xC * d * yA -
xA * yB * yB * yB * yB * yC * yC * xB * xC * xC * xC * d * d * yA * yA +
xA * yB * yB * yB * yC * xC * xC * d * yA * yA +
xA * yB * yB * yC * yC * xB * xC * d + (1 + 1) * xA * yB * yB * yC * yC * xB * xC * d * yA * yA +
xA * yB * yC * yC * yC * a * xB * xB * xB * xB * xC * xC * d * d * yA * yA -
xA * yB * yC * a * xB * xB * xC * xC * d - (1 + 1) * xA * yB * yC * a * xB * xB * xC * xC * d * yA * yA +
xA * yB * yC - xA * yC * yC * a * xB * xB * xB * xC * d * yA * yA -
xA * a * xB * xC - yB * yB * yB * yC * yC * xB * xB * xC * xC * xC * d * d * yA * yA * yA -
yB * yB * yC * yC * yC * xB * xB * xB * xC * xC * d * d * yA * yA * yA -
yB * yB * yC * xB * xC * xC * d * yA + yB * yB * yC * xB * xC * xC * d * yA * yA * yA -
yB * yC * yC * xB * xB * xC * d * yA + yB * yC * yC * xB * xB * xC * d * yA * yA * yA +
yB * xC * yA + yC * xB * yA - (
- xA * xA * xA * yB * yB * yB * yB * yC * yC * xB * xC * d * d * yA * yA -
xA * xA * xA * yB * yB * yB * yC * yC * yC * xB * xB * d * d * yA * yA +
xA * xA * xA * yB * yB * a * a * xB * xB * xB * xC * xC * xC * d * d * yA * yA +
xA * xA * xA * yB * yC * a * a * xB * xB * xB * xB * xC * xC * d * d * yA * yA +
xA * xA * yB * yB * yB * yB * yC * yC * yC * xB * xB * xB * xC * xC * d * d * d * yA +
xA * xA * yB * yB * yB * yB * yC * xB * xC * xC * d * d * yA * yA * yA +
xA * xA * yB * yB * yB * yC * yC * a * xB * xB * xB * xB * xC * xC * xC * d * d * d * yA +
xA * xA * yB * yB * yB * yC * yC * xC * d * yA -
xA * xA * yB * yB * yB * a * xB * xB * xC * xC * xC * d * d * yA * yA * yA -
xA * xA * yB * yB * yC * yC * yC * xB * xB * xB * d * d * yA * yA * yA +
xA * xA * yB * yB * yC * yC * yC * xB * d * yA -
(1 + 1) * xA * xA * yB * yB * yC * a * xB * xC * xC * d * yA -
xA * xA * yB * yB * yC * xB * d * yA +
xA * xA * yB * yC * yC * a * xB * xB * xB * xB * xC * d * d * yA * yA * yA -
(1 + 1) * xA * xA * yB * yC * yC * a * xB * xB * xC * d * yA +
xA * xA * yB * a * a * xB * xB * xC * xC * xC * d * yA -
xA * xA * yB * a * xB * xB * xC * d * yA + xA * xA * yC * a * a * xB * xB * xB * xC * xC * d * yA -
xA * yB * yB * yB * yB * yC * yC * xB * xB * xB * xC * xC * xC * d * d * d * yA * yA +
xA * yB * yB * yB * yC * yC * yC * xB * xB * xB * xB * xC * xC * d * d * d * yA * yA -
xA * yB * yB * yB * yC * yC * yC * xB * xB * xC * xC * d * d +
xA * yB * yB * yB * yC * xC * xC * d * yA * yA +
xA * yB * yB * yC * yC * a * xB * xB * xB * xC * xC * xC * d * d +
(1 + 1) * xA * yB * yB * yC * yC * xB * xC * d * yA * yA -
xA * yB * yB * a * xB * xC * xC * xC * d * yA * yA +
xA * yB * yB * xB * xC * d * yA * yA + xA * yB * yC * yC * yC * xB * xB * d * yA * yA -
(1 + 1) * xA * yB * yC * a * xB * xB * xC * xC * d * yA * yA -
xA * yB * yC * xB * xB * d * yA * yA + xA * yB * yC -
xA * yC * yC * a * xB * xB * xB * xC * d * yA * yA -
xA * a * xB * xC - yB * yB * yB * yC * yC * xB * xB * xC * xC * xC * d * d * yA -
yB * yB * yC * yC * yC * xB * xB * xB * xC * xC * d * d * yA +
yB * xC * yA + yC * xB * yA))
in I)"*)
- admit. (* nonzero demonimator conditions, to be resolved as in ExtendedCoordinates.v *)
- (* nsatz. *) (* requires IntegralDomain (F q), then gives stack overflow *)
admit. (* sage -c "
R.<d,a,xA,yA,xB,yB,xC,yC> = PolynomialRing(QQ,8,order='invlex')
I = R.ideal([
(a * (xA * xA) + yA * yA) - (1 + d * (xA * xA) * (yA * yA)),
(a * (xB * xB) + yB * yB) - (1 + d * (xB * xB) * (yB * yB)),
(a * (xC * xC) + yC * yC) - (1 + d * (xC * xC) * (yC * yC))
])
print (((- yA * yA * yA * yB * yB * yB * yB * yC * yC * xB * xB * xB * xC * 1 * d * d * d * xA * xA -
yA * yA * yA * yB * yB * yB * yC * yC * yC * xB * xB * xC * xC * d * d +
yA * yA * yA * yB * yB * yB * yC * a * xB * xB * xB * xB * xC * xC * 1 * d * d * d * xA * xA -
yA * yA * yA * yB * yB * yB * yC * xB * xB * 1 * 1 * d * d * xA * xA +
yA * yA * yA * yB * yB * yC * yC * a * xB * xB * xB * xC * xC * xC * d * d -
yA * yA * yA * yB * yB * yC * yC * xB * xC * 1 * d +
yA * yA * yA * yB * yB * a * xB * xB * xB * xC * 1 * 1 * d * d * xA * xA +
yA * yA * yA * yB * yC * a * xB * xB * xC * xC * 1 * d -
yA * yA * yB * yB * yB * yB * yC * yC * yC * xB * xC * xC * d * d * xA -
yA * yA * yB * yB * yB * yB * yC * a * xB * xB * xB * xC * xC * 1 * d * d * d * xA * xA * xA -
yA * yA * yB * yB * yB * yC * yC * a * xB * xB * xB * xB * xC * 1 * d * d * d * xA * xA * xA -
yA * yA * yB * yB * yB * yC * yC * xC * 1 * d * xA +
yA * yA * yB * yB * yB * a * xB * xB * xC * 1 * 1 * d * d * xA * xA * xA +
yA * yA * yB * yB * yC * a * xB * xB * xB * 1 * 1 * d * d * xA * xA * xA +
(1 + 1) * yA * yA * yB * yB * yC * a * xB * xC * xC * 1 * d * xA -
yA * yA * yB * yC * yC * a * a * xB * xB * xB * xB * xC * xC * xC * d * d * xA +
(1 + 1) * yA * yA * yB * yC * yC * a * xB * xB * xC * 1 * d * xA -
yA * yA * yC * a * a * xB * xB * xB * xC * xC * 1 * d * xA -
yA * yB * yB * yB * yB * yC * yC * a * xB * xC * xC * xC * d * d * xA * xA +
yA * yB * yB * yB * yC * a * xC * xC * 1 * d * xA * xA +
(1 + 1) * yA * yB * yB * yC * yC * a * xB * xC * 1 * d * xA * xA +
yA * yB * yB * yC * yC * xB * xC * 1 * 1 * 1 * d +
yA * yB * yC * yC * yC * a * a * xB * xB * xB * xB * xC * xC * d * d * xA * xA -
(1 + 1) * yA * yB * yC * a * a * xB * xB * xC * xC * 1 * d * xA * xA -
yA * yB * yC * a * xB * xB * xC * xC * 1 * 1 * 1 * d +
yA * yB * yC * 1 * 1 * 1 * 1 - yA * yC * yC * a * a * xB * xB * xB * xC * 1 * d * xA * xA -
yA * a * xB * xC * 1 * 1 * 1 * 1 +
yB * yB * yB * yC * yC * a * a * xB * xB * xC * xC * xC * d * d * xA * xA * xA +
yB * yB * yC * yC * yC * a * a * xB * xB * xB * xC * xC * d * d * xA * xA * xA -
yB * yB * yC * a * a * xB * xC * xC * 1 * d * xA * xA * xA +
yB * yB * yC * a * xB * xC * xC * 1 * 1 * 1 * d * xA -
yB * yC * yC * a * a * xB * xB * xC * 1 * d * xA * xA * xA +
yB * yC * yC * a * xB * xB * xC * 1 * 1 * 1 * d * xA -
yB * a * xC * 1 * 1 * 1 * 1 * xA - yC * a * xB * 1 * 1 * 1 * 1 * xA) -
(- yA * yA * yA * yB * yB * yB * yB * yC * yC * xB * xC * d * d * xA * xA -
yA * yA * yA * yB * yB * yB * yC * yC * yC * xB * xB * d * d * xA * xA +
yA * yA * yA * yB * yB * a * a * xB * xB * xB * xC * xC * xC * d * d * xA * xA +
yA * yA * yA * yB * yC * a * a * xB * xB * xB * xB * xC * xC * d * d * xA * xA -
yA * yA * yB * yB * yB * yB * yC * yC * yC * xB * xB * xB * xC * xC * 1 * d * d * d * xA -
yA * yA * yB * yB * yB * yB * yC * a * xB * xC * xC * d * d * xA * xA * xA -
yA * yA * yB * yB * yB * yC * yC * a * xB * xB * xB * xB * xC * xC * xC * 1 * d * d * d * xA -
yA * yA * yB * yB * yB * yC * yC * xC * 1 * d * xA +
yA * yA * yB * yB * yB * a * a * xB * xB * xC * xC * xC * d * d * xA * xA * xA +
yA * yA * yB * yB * yC * yC * yC * a * xB * xB * xB * d * d * xA * xA * xA -
yA * yA * yB * yB * yC * yC * yC * xB * 1 * d * xA +
(1 + 1) * yA * yA * yB * yB * yC * a * xB * xC * xC * 1 * d * xA +
yA * yA * yB * yB * yC * xB * 1 * 1 * 1 * d * xA -
yA * yA * yB * yC * yC * a * a * xB * xB * xB * xB * xC * d * d * xA * xA * xA +
(1 + 1) * yA * yA * yB * yC * yC * a * xB * xB * xC * 1 * d * xA -
yA * yA * yB * a * a * xB * xB * xC * xC * xC * 1 * d * xA +
yA * yA * yB * a * xB * xB * xC * 1 * 1 * 1 * d * xA -
yA * yA * yC * a * a * xB * xB * xB * xC * xC * 1 * d * xA -
yA * yB * yB * yB * yB * yC * yC * a * xB * xB * xB * xC * xC * xC * 1 * d * d * d * xA * xA +
yA * yB * yB * yB * yC * yC * yC * a * xB * xB * xB * xB * xC * xC * 1 * d * d * d * xA * xA -
yA * yB * yB * yB * yC * yC * yC * xB * xB * xC * xC * 1 * 1 * d * d +
yA * yB * yB * yB * yC * a * xC * xC * 1 * d * xA * xA +
yA * yB * yB * yC * yC * a * xB * xB * xB * xC * xC * xC * 1 * 1 * d * d +
(1 + 1) * yA * yB * yB * yC * yC * a * xB * xC * 1 * d * xA * xA -
yA * yB * yB * a * a * xB * xC * xC * xC * 1 * d * xA * xA +
yA * yB * yB * a * xB * xC * 1 * 1 * 1 * d * xA * xA +
yA * yB * yC * yC * yC * a * xB * xB * 1 * d * xA * xA -
(1 + 1) * yA * yB * yC * a * a * xB * xB * xC * xC * 1 * d * xA * xA -
yA * yB * yC * a * xB * xB * 1 * 1 * 1 * d * xA * xA +
yA * yB * yC * 1 * 1 * 1 * 1 - yA * yC * yC * a * a * xB * xB * xB * xC * 1 * d * xA * xA -
yA * a * xB * xC * 1 * 1 * 1 * 1 +
yB * yB * yB * yC * yC * a * xB * xB * xC * xC * xC * 1 * 1 * d * d * xA +
yB * yB * yC * yC * yC * a * xB * xB * xB * xC * xC * 1 * 1 * d * d * xA -
yB * a * xC * 1 * 1 * 1 * 1 * xA - yC * a * xB * 1 * 1 * 1 * 1 * xA)) in I)"
*)
- admit. (* nonzero demonimator conditions, to be resolved as in ExtendedCoordinates.v *)
Admitted.
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.
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.
|
