Plaintiff Anthony Amaker appeals from a judgment of the district court (Seybert, J.) dismissing his amended complaint. In substance, the amended complaint alleges that police, prosecutors, plaintiffs defense attorneys, the trial judge, an eyewitness, and various court personnel conspired to secure plaintiffs conviction on charges of murder by manufacturing inculpatory evidence and subsequently suppressing evidence probative of their misconduct, all in violation of 42 U.S.C. §§ 1981, 1983, 1985(3) and 1986. Plaintiff ultimately was convicted in New York state court on two counts of second degree murder and is currently serving his sentence. The Ap*50pellate Division affirmed the conviction, People v. Amaker, 195 A.D.2d 605, 602 N.Y.S.2d 546 (2d Dept.1993), and the Court of Appeals denied leave to appeal. People v. Amaker, 82 N.Y.2d 804, 604 N.Y.S.2d 940, 624 N.E.2d 1035 (1993).
The district court dismissed the amended complaint in an unpublished opinion, holding primarily that all of plaintiffs claims were barred by Heck v. Humphrey, 512 U.S. 477, 114 S.Ct. 2364, 129 L.Ed.2d 383 (1994) (§ 1983 claim necessarily implicating the invalidity of the plaintiffs underlying conviction or sentence does not accrue unless and until the underlying conviction otherwise is invalidated). The court held in the alternative that (1) the claims are barred by the three-year statute of limitations applicable to § 1983 actions in New York; (2) the prosecutors are entitled to absolute immunity; (3) the trial judge is entitled to absolute immunity; (4) the private attorney defendants were not state actors for purposes of § 1983; and (5) plaintiffs conspiracy allegations were insufficiently particularized. The court dismissed the amended complaint with prejudice.
On appeal, plaintiff contends that (1) the district court was obliged to convert the motions to dismiss to motions for summary judgment because an affidavit was attached to one of the motions; (2) not all of plaintiffs claims implicate the invalidity of his conviction and, in any event, Heck applies only to the § 1983 cause of action; (3) the statute of limitations should have been equitably tolled under the doctrine of fraudulent concealment; and (4) no one was entitled to absolute immunity.
Plaintiffs initial contention, that the district court erred by failing to apply Fed.R.Civ.P. 12(b) to convert the motions to dismiss to motions for summary judgment, is without merit. Fed.R.Civ.P. 12(b) provides in relevant part:
If, on a motion asserting the defense numbered (6) to dismiss for failure of the pleading to state a claim upon which relief can be granted, matters outside the pleading are presented to and not excluded by the court, the motion shall be treated as one for summary judgment and disposed of as provided in Rule 56, and all parties shall be given reasonable opportunity to present all material made pertinent to such a motion by Rule 56.
According to plaintiff, Rule 12(b) conversion was required in this case because the motion to dismiss on behalf of defendant Modest included an affidavit from her attorney.
This Court does strictly enforce the conversion requirement of Rule 12(b) where there is a legitimate possibility that the district court relied on inappropriate material in granting the motion. See Kopec v. Coughlin, 922 F.2d 152, 154-55 (2d Cir.1991) (reversing a Rule 12(b)(6) dismissal where district court had not converted motion to one for summary judgment but nonetheless had relied in part on information contained only in the extrinsic materials attached to the motion); Fonte v. Board of Managers of Continental Towers Condominium, 848 F.2d 24, 25 (2d Cir.1988) (reversing where district court’s opinion referred to factual matter contained only'in movant’s memorandum of law, which raised “possibility that [the court] improperly relied on matters outside the pleading”). Such vigorous enforcement of the conversion requirement helps ensure that courts will refrain from engaging in fact-finding when considering a motion to dismiss, and also that plaintiffs are given a fair chance to contest defendants’ evidentiary assertions where a court nonetheless does consider evidence extrinsic to the complaint in that context.
Attachment of an affidavit or exhibit to a Rule 12(b)(6) motion, however, does not without more establish that conversion is required. In Pani v. Empire Blue Cross Blue Shield, 152 F.3d 67 (2d Cir.1998), cert. denied, — U.S. —, 119 S.Ct. 868, 142 L.Ed.2d 770 (1999), the defendant had attached a copy of the Medi*51care Carriers’ Manual as an exhibit to its motion to dismiss, and the district court had cited to the manual in the course of reaching a particular conclusion supporting its ruling. See id. at 75. Because the district court also relied upon a prior decision to establish the same proposition for which the manual was cited, however, we declined to reverse for lack of conversion. See id. Pani thus establishes that reversal for lack of conversion is not required unless there is reason to believe that the extrinsic evidence actually affected the district court’s decision and thus was not at least implicitly excluded.
There is no reason to believe that the court below relied on the Modest affidavit to decide the motion to dismiss. The information conveyed in that affidavit is limited in scope: It sets forth facts about plaintiffs ongoing efforts to compel the production of certain documents under New York’s Freedom of Information Law, and Modest’s responses to those efforts. Because it is plain that this information had no impact whatsoever on the district court’s analysis, at least with respect to those aspects of the decision upon which our review turns, we reject plaintiffs first argument.
Plaintiffs second contention, that not all of his claims are barred by Heck, also fails. Heck confronted the question of whether, given the overlap between § 1983 and the federal habeas corpus statute, a prisoner seeking civil damages may proceed with a § 1983 claim where success on the claim necessarily would implicate the unconstitutionality of the prisoner’s conviction or sentence. See 512 U.S. at 480-90, 114 S.Ct. 2364; cf. Jenkins v. Haubert, 179 F.3d 19 (2d Cir.1999) (summarizing the holding in Heck). The Court resolved the question by importing from the common law tort of malicious prosecution “the hoary principle that civil tort actions are not appropriate vehicles for challenging the validity of outstanding criminal judgments.” Heck, 512 U.S. at 486, 114 S.Ct. 2364. Accordingly, the Court adopted the rule that
when a state prisoner seeks damages in a § 1983 suit, the district court must consider whether a judgment in favor of the plaintiff would necessarily imply the invalidity of his conviction or sentence; if it would, the complaint must be dismissed unless the plaintiff can demonstrate that the conviction or sentence has already been invalidated.
Id. at 487, 114 S.Ct. 2364.
Plaintiff argues that his amended complaint includes at least one claim that does not implicate the invalidity of his conviction and therefore does not trigger the Heck rule: a claim that his right to meaningful court access has been denied by the withholding of exculpatory evidence. In substance, however, this claim sounds under Brady v. Maryland, 373 U.S. 83, 87, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963) (Fourteenth Amendment’s Due Process Clause requires prosecution to turn over exculpatory evidence), and therefore does indeed call into question the validity of his conviction. Accordingly, it is barred by Heck.
Plaintiff argues also that, regardless of whether Heck applies to bar his § 1983 cause of action, it does not apply to his §§ 1981, 1985(3), and 1986 causes of action. The principle established in Heck — that a prisoner-plaintiff may not assert a civil damages claim that necessarily challenges the validity of an outstanding criminal conviction' — -is not so limited, however. See, e.g., Candelaria v. Greifinger, No. 96-CV-0017 (RSP), 1997 WL 642464, at *1-2 (N.D.N.Y. Oct.14, 1997) (Heck applies to §§ 1985 and 1986); Martinez v. Ensor, 958 F.Supp. 515, 516-18 (D.Col.1997) (§§ 1981 and 1985); Duamutef v. Morris, 956 F.Supp. 1112, 1117-18 (S.D.N.Y.1997) (§ 1985); cf. Channer v. Mitchell, 43 F.3d 786 (2d Cir.1994) (per curiam) (implicitly affirming application of Heck to §§ 1985 and 1986). Each of plaintiffs causes of action here turns on the *52asserted existence of a conspiracy to frame plaintiff for murder, and thus deciding in his favor on any of them plainly would call into question the validity of his conviction. Heck therefore applies with respect not only to plaintiffs § 1983 claim but also to his §§ 1981,1985(3) and 1986 claims.
Disposition of the ease on Heck grounds, however, warrants only dismissal without prejudice, because the suit may be reinstituted should plaintiffs conviction be “expunged by executive order, declared invalid by a state tribunal authorized to make such determination, or called into question by a federal court’s issuance of a writ of habeas corpus.” Heck, 512 U.S. at 487, 114 S.Ct. 2364. See also id. at 479-80, 489-90, 114 S.Ct. 2364; White v. Gittens, 121 F.3d 803, 807 (1st Cir.1997); Fottler v. United States, 73 F.3d 1064, 1065-66 (10th Cir.1996); Perez v. Sifel, 57 F.3d 503, 505 (7th Cir.1995) (per curiam); Trimble v. City of Santa Rosa, 49 F.3d 583, 585 (9th Cir.1995) (per curiam); Schafer v. Moore, 46 F.3d 43, 45 (8th Cir.1995); but see Boyd v. Biggers, 31 F.3d 279, 283-84 (5th Cir.1994) (dismissing complaint, pursuant to Heck, with prejudice); Stephenson v. Reno, 28 F.3d 26, 27-28 (5th Cir.1994) (same).
In the wake of its Heck analysis, the district court went on to consider several alternative bases for dismissing plaintiffs amended complaint, and plaintiffs remaining contentions on appeal all address the conclusions resulting therefrom. While we recognize that it may be appropriate in some circumstances to consider alternative bases for dismissal notwithstanding a finding that Heck is applicable, cf. Boyd, 31 F.3d at 284 (conducting absolute immunity analysis despite applicability of Heck), we conclude that such consideration was unwarranted in this case. The question of whether certain defendants were entitled to absolute immunity, for example, requires particularized inquiry into the circumstances relevant to those defendants and, in some cases, this inquiry involves questions more appropriately resolved on a motion for summary judgment than on a motion to dismiss. All claims of immunity therefore cannot be resolved at this stage and, in any event, other defendants would remain subject to the possibility that plaintiffs action might be reinsti-tuted at some future date. Accordingly, rather than consider piecemeal the liability of particular defendants in the context of a complex, unaccrued action, we limit our affirmance to the Heck analysis set forth above.
We also find that in cases such as this, where the viability of the plaintiffs claim depends on his conviction being invalidated, the statute of limitations begins to run upon the invalidation, not the time of the alleged government misconduct. See Heck, 512 U.S. at 489-90, 114 S.Ct. 2364; see also Covington v. City of New York, 171 F.3d 117 (2d Cir.1999). Inasmuch as plaintiffs conviction has not been overturned, the statute of limitations has not begun to run. The district court’s dismissal of the entire complaint with prejudice on these grounds was thus also unwarranted.
Accordingly, we remand with instructions to the district court to modify the judgment to reflect that the amended complaint is dismissed without prejudice.