The former intelligence contractor with the US National Security Agency (NSA) said that the Government Communications Headquarters (GCHQ), British intelligence and security organisation responsible for providing signals intelligence, has been monitoring Pakistan's communication data.
To access the data, the GCHQ hacked routers made by Cisco, Snowden said in an interview with the BBC.
He also stated that the spying was done with the permission of the British government and the apparent aim of such hacking was to aid in the identification of terrorists.
Snowden also said that British spies can hack into phones remotely with a simple text message and make audio recordings or take photos without owners knowing.
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"They want to own your phone instead of you," he said.
The GCHQ had the power to hack into phones without their owners' knowledge, he said.
Snowden said GCHQ could gain access to a handset by sending it an encrypted text message and use it for such things as taking pictures and listening in.
He said both GCHQ and the NSA had invested heavily in technology allowing them to hack smartphones.
Snowden talked about GCHQ's "Smurf Suite", a collection of secret intercept capabilities individually named after the little blue imps of Belgian cartoon fame.
"Dreamy Smurf is the power management tool which means turning your phone on and off with you knowing," he said.
"Nosey Smurf is the 'hot mic' tool. For example if it's in your pocket, [GCHQ] can turn the microphone on and listen to everything that's going on around you - even if your phone is switched off because they've got the other tools for turning it on.
Once GCHQ had gained access to a user's handset, Snowden said the agency would be able to see "who you call, what you've texted, the things you've browsed, the list of your contacts, the places you've been, the wireless networks that your phone is associated with.
"And they can do much more. They can photograph you".
Snowden also explained that the SMS message sent by the agency to gain access to the phone would pass unnoticed by the handset's owner.