Thousands of people have taken to the streets in Berlin in protests against Internet surveillance activities by the US National Security Agency (NSA) and other intelligence agencies, and the German government's perceived lax reaction to them.

Organisers, among them the opposition Greens, The Left and Pirates parties, said 20,000 people turned out in Saturday’s demonstration.

Police would not confirm the figure, saying only their "tally differs from that of the organisers".

The protest was organised under the slogan "Freedom Rather Than Fear" and demonstrators carried banners saying: "Stop spying on us" and, more sarcastically: "Thanks to PRISM (the US government's vast data collection programme) the government finally knows what the people want".

"Intelligence agencies like the NSA shamelessly spy on telephone conversations and Internet connections… our government, one of whose key roles is the protection from harm, sends off soothing explanations," said one protester, Kai-Uwe Steffens.

On Thursday, newly leaked documents alleged that US and British intelligence agencies have cracked the encryption that secures a wide range of online communications, including emails, banking transactions and phone conversations.

The documents provided by former US intelligence contractor Edward Snowden to The New York Times, ProPublica and The Guardian newspaper suggest that the spy agencies are able to decipher data even with the supposedly secure encryption designed to make it private.