The Supreme Court on Tuesday explored the tension between Americans' right to free speech and a federal law that prohibits aid to terrorist groups, and hardly anyone seemed clear about the lines of demarcation.

The case stems from a challenge to an antiterrorism act by American advocates who say they want to support only the peaceful efforts of groups that the State Department has deemed to be terrorist organizations.

"This is a difficult case for me," allowed Justice Anthony M. Kennedy, whose vote often is the one that decides closely divided cases.

Georgetown law professor David D. Cole, who represents the Humanitarian Law Project, said his clients do not want to provide material support to the groups, but only to help them pursue peaceful ways to end conflict. "The government has spent a decade arguing that our clients cannot advocate for peace, cannot inform about international human rights," Cole told the court.

The project wants to support the lawful activities of the Kurdistan Workers' Party, a militant group in Turkey known as the PKK, and a Sri Lankan group known as the Liberation Tigers of Tamil Eelam. Both are among dozens of organizations on the State Department list, along with better-known groups such as al-Qaeda, Hamas and Hezbollah.

Solicitor General Elena Kagan countered that, as individuals, Cole's clients can advocate whatever they want. But Congress was within its rights, she said, to determine that it was impossible to separate support of any terrorist group's peaceful activities from its violent goals.