DEAUVILLE, France — President Obama and President Dmitry Medvedev of Russia emerged from private talks yesterday unable to show progress on the contentious issue of missile defense, underscoring an enduring mistrust underlying the US-Russian relationship.

Mike McFaul, Obama’s top adviser on Russia, put the problem plainly after the meeting with the Russians: “They don’t believe us.’’

At issue is Washington’s plan to site missile interceptors in Central and Eastern Europe in phases through 2020. Despite repeated assurances, Russia has not let go of the fear that the United States would end up threatening Russia’s own missile arsenal, something US officials say will not happen.

Obama said that they had committed to working together on missile defense to find an approach that is “consistent with the security needs of both countries, that maintains the strategic balance and deals with potential threats that we both share.’’

Medvedev, however, suggested that the problem would not be solved anytime soon.

He has warned that failure to cooperate with Moscow on the missile shield could spark a new arms race.

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