Snowden's statement came as he met with human rights groups and lawyers at Moscow's Sheremetyevo International Airport, RIA Novosti reported.
"I ask for your assistance in requesting guarantees of safe passage from the relevant nations in securing my travel to Latin America, as well as requesting asylum in Russia until such time as ... my legal travel is permitted," Snowden said in the statement posted on the WikiLeaks website. "I will be submitting my request to Russia today, and hope it will be accepted favorably."
President Vladimir Putin has said Russia offered Snowden asylum, provided he stop activities that are "damaging our American partners."
Following Friday's meeting, Russian lawmaker Vyacheslav Nikonov told reporters Snowden has accepted Putin's conditions because he has "no other choice."
Putin has said Russia will not extradite Snowden to the United States but RIA Novosti noted the Kremlin has emphasized the human rights aspect of the case.
Snowden, who has been stuck in the airport's transit zone since arriving in Moscow from Hong Kong June 23, asked the activists and lawyers at Friday's meeting, to lobby the Russian government to grant him temporary asylum, CNN reported.
Russia's human rights ombudsman and representatives of Amnesty International, Human Rights Watch and Russian human rights groups were in attendance at the meeting.
Those who were invited to the meeting said they received an email Thursday, asking them to join Snowden at the airport "for a brief statement and discussion regarding the next steps forward in my situation."
Snowden is wanted by the United States on charges of espionage and property theft after he leaked details of the National Security Agency's cellphone and Internet monitoring. The former NSA contract employee has not been seen in public since his arrival in Moscow.
The text of the invitation was posted on Facebook by Tatyana Lokshina, the deputy director of the Moscow branch of Human Rights Watch, who said she received it Thursday from the email address email@example.com, RIA Novosti said.
The email's sender said he had been "extremely fortunate to enjoy and accept many offers of support and asylum from brave countries around the world."
"Unfortunately, in recent weeks, we have witnessed an unlawful campaign by officials in the U.S. government to deny my right to seek and enjoy this asylum," the email said.
Snowden, whose passport has been revoked by the United States, has submitted more than 20 asylum applications to countries across the world. Bolivia, Nicaragua and Venezuela have indicated they would be prepared to offer him refuge.
The British newspaper The Guardian reported Snowden leaked it documents indicating Microsoft Corp. helped the NSA dodge software encryption to intercept user emails and Web chats. The collaboration also let the agency's formerly secret PRISM electronic-spying program collect video and audio of conversations of people using Skype, an online chat and audio-video conversation service Microsoft bought 21 months ago, the documents said.
The NSA boasted its new video access let it triple the number of Skype video calls it could collect through PRISM, the newspaper said.
"The audio portions of these sessions have been processed correctly all along, but without the accompanying video. Now, analysts will have the complete 'picture,'" a July 14, 2012, NSA document says.
Information collected through PRISM is routinely shared with the FBI and CIA, The Guardian said. One NSA document describes the collaboration as a "team sport," the newspaper reported.
Microsoft's decision to work around its own encryption helped the NSA intercept emails and Windows Live Messenger Web chats through Microsoft's new Outlook.com portal.
Outlook.com is a free Web-based email service that mimics the interface of the Microsoft Outlook personal information manager. It supplanted Microsoft's Hotmail in February, but the cooperation began a year ago, The Guardian said.
Microsoft also gave PRISM access to Microsoft SkyDrive, a file hosting service that lets users upload and sync files to a "cloud storage" elsewhere and then access them from a Web browser or their local device.
SkyDrive is part of the Windows Live online services that Microsoft says lets users keep files private, share them with contacts or make them public.
Microsoft, of Redmond, Wash., has said it provided the NSA with no direct access to user information.
It said in a website statement Thursday it provided the information only because the government lawfully asked for it.
"To be clear, Microsoft does not provide any government with blanket or direct access to SkyDrive, Outlook.com, Skype or any Microsoft product," the software maker said.
In a statement to The Guardian, it said, "We take our commitments to our customers and to compliance with applicable law very seriously, so we provide customer data only in response to legal processes."
Microsoft added: "There are aspects of this debate that we wish we were able to discuss more freely. That's why we've argued for additional transparency that would help everyone understand and debate these important issues."
Microsoft, along with Facebook Inc., Google Inc. and other Web companies, have urged Washington to let them reveal more information about government surveillance requests after The Guardian and The Washington Post started publishing Snowden's leaked documents.
The NSA and the office of National Intelligence Director James R. Clapper said in a joint statement to The Guardian Microsoft complied with "legally mandated requirements."
"The U.S. operates its programs under a strict oversight regime, with careful monitoring by the courts, Congress and the director of national intelligence," the statement said. "Not all countries have equivalent oversight requirements to protect civil liberties and privacy."
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