WASHINGTON — Line-drawing, a typical task in resolving First Amendment disputes at the Supreme Court, seemed to elude the justices yesterday as they heard arguments in a challenge to a key anti-terrorism law.

At issue in Holder v. Humanitarian Law Project was a law that makes it a crime to give “material support,” including things like “expert advice” and training, to groups that have been designated by the secretary of state as terrorist organizations.

Critics of the law say its terminology defining prohibited support is so vague that it would cover even the most benign forms of speech unrelated to terrorism — such as training in nonviolence or help in petitioning the United Nations.

But that was Congress’ point in enacting the law, said Solicitor General Elena Kagan. Even seemingly unrelated aid boosts a group’s terrorist activities, she said.

“Hezbollah builds bombs,” she told the Court. “Hezbollah also builds homes. What Congress decided was that when you help Hezbollah build homes you are also helping Hezbollah build bombs … . It’s a reasonable theory.”

But that broad view of the law’s sweep seemed to draw a line too far for several justices, who came up with examples of speech or activities they would regard as difficult to ban under the First Amendment in almost any circumstance.

What seemed to bother the justices most was the assertion by Kagan that the law would bar a lawyer from writing an amicus curiae or "friend of the court" brief on behalf of such a group in U.S. courts.

That statement came as Kagan defended the law as a “vital weapon in this nation’s continuing struggle against international terrorism.” When Justice Ruth Bader Ginsburg told her that “I am still having trouble with the line” between what kinds of communications the law allows and forbids, Kagan said mere “discussion of ideas” is not barred. But Kagan said the law does criminalize “the provision of actual support — services to the organization that the organization can use in its activities, both legal and illegal.”

Justice Anthony Kennedy, who said he found the case “difficult,” then asked Kagan whether, as the government had conceded at an earlier stage, the law barred filing an amicus brief in a case on behalf of designated groups.

Kagan said yes. “I think that would be a [prohibited] service ... . To the extent that a lawyer drafts an amicus brief [for designated terrorist groups] ... then that indeed would be prohibited.”

At which point Justice John Paul Stevens said, “Then it says to me that your opponent’s argument here today is prohibited.” Stevens was referring to Georgetown University Law Center professor David Cole, who argued for the Humanitarian Law Project on behalf of people who wanted to teach international law and nonviolence to designated groups, but feared prosecution under the law.

“No, no, no,” exclaimed Kagan, saying that Cole was representing entities wanting to aid the groups, not the groups themselves. But the damage seemed done.

“Under the definition of the statute” offered by Kagan, said Justice Sonia Sotomayor, “teaching these members to play the harmonica would be unlawful.”

That produced laughs, and more laughs followed when Justice Antonin Scalia posited that a terror group’s harmonica quartet “might tour the country and make a lot of money.”

After some further discussion of the harmonica hypothetical, Justice Stephen Breyer snapped, “Just forget the harmonica for a second; I’m more worried about the lawyer.” Breyer asked incredulously whether the law at issue would bar a terror group that might have American members from being “entitled under the Constitution to have a lawyer in the United States who does legal work like filing amicus briefs.”

Kagan appeared to back off her earlier stance somewhat, suggesting that if Sixth Amendment or due-process issues were involved, such aid might not be covered by the law.

Kagan also said individuals were free to speak out about designated groups or even meet with or join the groups. “The discussion must stop when you go over the line into giving valuable advice, training, support to these organizations.”

Cole also received pointed questions about his view that government cannot prohibit speech “when that speech advocates solely lawful, peaceable activities.”

Both Stevens and Justice Samuel Alito Jr. pressed Cole to see if he thought that, under the First Amendment, any speech or training given to designated groups could be prohibited.

“It depends on the speech,” Cole said to Alito. “There may be some forms of training that are so closely connected that Congress legitimately seeks to proscribe, like training in bomb making.”

Cole also said that if the law pertained only to aiding the Taliban or al Qaida during wartime, then wartime-treason laws could trump free-speech protections. Otherwise, Cole said, “There’s no dispute … that the government has a compelling interest in cutting off aid to terrorism. The question is whether it can do so by criminalizing pure speech.”

When Cole drew parallels between the law at issue and laws restricting association with the Communist Party during the Cold War, Scalia said, “I think it’s very unrealistic to compare these terrorist organizations with the Communist Party.” Whereas communism drew philosophical adherents, Scalia said, “I don’t think that Hamas or any of these terrorist organizations represent such a philosophical organization.”

Apart from Scalia, the government’s support on the Court seemed weak. Even Chief Justice John Roberts Jr., ordinarily a predictable supporter of government anti-terrorism measures, said it was “kind of hard to decide” the meaning of expert advice and other terms in the law. Roberts raised the possibility of remanding the case to lower courts to re-examine the law under a strict-scrutiny standard. Strict scrutiny in constitutional law means the state must show it has a compelling interest, such as national security, in imposing a restriction on a fundamental right, such as speech.

The Holder case has its roots in the Antiterrorism and Effective Death Penalty Act of 1996, which gave the secretary of state the power to designate organizations as terrorist, and banned material support to those groups. Soon after the law passed, 30 organizations were tagged as terrorists, including the groups involved in the current case: the Kurdistan Workers Party, which advocates establishing a Kurdish state in Turkey, and the Tamil Tigers, which seeks creation of an independent Tamil state in Sri Lanka.

After the law was found unconstitutionally vague in 1998, Congress amended it to clarify or expand its provisions, including in the 2001 USA Patriot Act, which added “expert advice or assistance” to the list of prohibited activities. The law was amended again in 2004, but each time it has been changed, judges have struck down parts of the law.