Hillary Clinton on Monday accused Donald Trump of inviting Russian President Vladimir Putin to "hack into Americans," with Trump firing back that American cybercapacities have declined under President Obama.

"Increasingly we are seeing cyberattacks coming from states [and] organs of states," Clinton said in the first presidential debate between the two candidates. "The most recent and troubling of these has been Russia. There is no doubt now that Russia has used cyberattacks against all kinds of organizations in our country.

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"I am deeply concerned about this … Putin is playing a really tough long game here. One of the things he's done is to let loose cyberattackers to hack into government files, to hack into personal files, to hack into the Democratic National Committee. And we recently have learned that this is one of their preferred methods of trying to wreak havoc and collect information," Clinton said.

"We're going to have to make it clear we don't want to have to use the kinds of tools that we have, we don't want to engage in a different kind of warfare, but we will defend the citizens of this country," Clinton added. "And the Russians need to understand that. I think they've been treating it as almost a probing of how far can we go, how much can we do. That's [why] I was so shocked when Donald publicly invited Putin to hack into Americans."

Trump in July said "Russia, if you're listening, I hope you're able to find the 33,000 emails that are missing. I think you will probably be rewarded mightily by our press. Let's see if that happens."

Responding in Monday's debate, Trump pointed out that Russia has not been formally declared as the culprit behind a spate of hacks this year, and said that American cybercapacities had slipped as a result of Democratic policies. "As far as the cyber, I agree to parts of what Secretary Clinton said. We should be better than anybody else, and perhaps we're not.

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"I don't think anybody knows it was Russia that broke into the DNC," Trump said. "Maybe it was. It could be Russia, it could also be China, it could also be somebody sitting on their bed who weighs 400 pounds.

"What did we learn with DNC? We learned that Bernie Sanders was taken advantage of by your people, by Debbie Wasserman Schultz," Trump said. "Look at what happened to her. Bernie Sanders was taken advantage of. Whether that was Russia, whether that was China, whether it was another country, we don't know, because the truth is that under President Obama, we've lost control of things we used to have control of.

"They're beating us at our own game," Trump added. "And so we have to get very, very tough on cyber and cyberwarfare."

Both candidates have been accused of having ties with Russia.

Trump praised Putin for being "a leader far more than our president has been a leader" and appearing in a TV interview with RT America, a network owned by the Russian government.

The appearance on RT prompted Clinton to say that Trump must not actually understand all of Putin's actions, including the annexation of Crimea in 2014.

Trump's Russian ties that make some uncomfortable extend beyond the businessman himself. Paul Manafort, Trump's campaign chairman, said there was no secret relationship between Trump and Putin, though he himself previously worked for a Ukrainian political group loyal to Russia to send money to Washington lobbying firms.

Manafort resigned from Trump's campaign in August. And more recently, Trump adviser Carter Page has come under suspicion for holding secret conversations with Russian officials. He is no longer a part of the campaign.

Putin has said that he "welcomes" working with Trump as commander in chief, but Trump said the Russian leader's compliments will not impact his ability to negotiate.

On the other side, Clinton is also familiar with the Russians, having a leading role in the Obama administration's failed effort to reset relations with Russia in 2009. She presented Russian Foreign Minister Sergey Lavrov with a red "reset" button in March 2009, but learned that the Russian word on the button actually meant "overcharged."

Clinton's ties to the Kremlin have been scrutinized this year as well, beginning with her campaign chairman, John Podesta, whose Washington-based lobbying firm in March signed up to lobby for Russia's largest state-backed financial institution, Sberbank of Russia. The Washington-based Podesta Group signed up to lobby both for the bank and several of its affiliates around the globe.

Clinton has also received criticism for her work as secretary of state in funneling U.S. dollars toward Skolkovo, the city known as the "Silicon Valley" of Russia and the epicenter of technological development for the country's military. That work was part of an initiative to encourage investment in American companies by allies of the Kremlin.

Rudy Takala contributed to this report.