BOSTON—A month before the Boston Marathon explosions, accused bomber Dzhokhar Tsarnaev told a college friend that he knew how to build a bomb, a federal prosecutor alleged Monday in the opening of the first trial linked to last year’s deadly attack.

Prosecutor Stephanie Siegmann told jurors that Mr. Tsarnaev also shared his views on martyrdom with his friend, Azamat Tazhayakov, in March 2013, a month before U.S. authorities believe Mr. Tsarnaev and his older brother planted two bombs that killed three people and wounded more than 260.

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