NEW YORK — Apple has been ordered to pay $532.9 million after a federal jury found its iTunes software infringed on three patents owned by Smartflash, a patent licensing company based in Texas.

Though Smartflash had been asking for $852 million in damages, the verdict, which came late Tuesday night, was still a costly blow for Apple.

After deliberating for eight hours in the U.S. District Court for the Eastern District of Texas, the jury said that Apple not only had used the Smartflash patents without permission but also had done so willfully.

Apple suggested the outcome was a reason why change is needed in the patent system to curb litigation by companies like Smartflash that do not make products themselves.