The global crime-fighting organisation has also revealed there were 200,000 victims in 150-plus countries in Friday's attack.
"We are in the face of an escalating threat", Wainwright added. "The numbers are going up, I am anxious about how the numbers will continue to grow when people go to work and turn (on) their machines on Monday morning".
He warned that more people may be hit by the virus Monday when they return to work and switch on their computers. "They're processing a lot of sensitive data", he said.
Avast said the majority of the attacks targeted Russia, Ukraine and Taiwan - though exactly which computers in those areas were targeted remains fuzzy.
The worldwide effort to extort cash from computer users is the first widely successful example of ransomware that self-replicates like a virus, and it prompted Microsoft to quickly change its policy, announcing free security patches to fix this vulnerability in the older Windows systems still used by millions of individuals and smaller businesses.
On Sunday MalwareTech issued a warning that hackers could upgrade the virus to remove the kill switch.
"The global reach is unprecedented".
This one worked because of a "perfect storm" of conditions, including a known and highly unsafe security hole in Microsoft Windows, tardy users who didn't apply Microsoft's March software fix, and malware created to spread quickly once inside university, business or government networks.
The Beijing News said that students at several universities around the country reported blocked access to their thesis papers and dissertation presentations.
Soon he and MalwareTech were communicating about what they'd found: That registering the domain name and redirecting the attacks to MalwareTech's server had activated the kill switch, halting the ransomware's infections.
"I think it is concerning that we could definitely see a similar attack occur, maybe in the next 24 to 48 hours or maybe in the next week or two", Huss said.
"Remarkably few payments" had so far been made in response to this attack, he added.
The ransomware is spread by taking advantage of a Windows vulnerability that Microsoft released a security patch for in March.
Britain's official emergency committee, known as Cobra, met in London Saturday afternoon to discuss the cyber attack that has caused widespread disruption to the country's National Health Service (NHS). Several cybersecurity firms said they had identified the malicious software behind the attack, which has apparently hit Russian Federation the hardest.
Cybersecurity experts said the spread of the virus dubbed WannaCry - "ransomware" which locked up more than 200,000 computers - had slowed, but the respite might only be brief.
However, it seems that many NHS trusts had not applied it or were using an older version of the operating system which is no longer supported - Windows XP.
An Garda Síochána said early Sunday there had been no identified cyber attack on any Irish state computer system from the wave of global ransomware attacks, although RTÉ reported a suspected attack on healthcare facility in the south-east.
UNITED STATES In the U.S., FedEx Corp. reported that its Windows computers were "experiencing interference" from malware, but wouldn't say if it had been hit by ransomware.
Among those affected by the virus was Nissan UK, but the auto manufacturer said there had been no major impact.
Account addresses hardcoded into the malicious WannaCry software code appear to show the attackers had received just under $32,500 in anonymous bitcoin currency as of 1100 GMT on Sunday, but that amount could rise as more victims rush to pay ransoms of $300 or more to regain access to their computers, just one day before the threatened deadline expires.
Dr Jakobsson said: "There's absolutely no excuse for any business not to have up to date systems".
Experts believe the attack is most likely to be criminal in nature, rather than connected to a state actor.
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