Cyber security is under continuous review across all state agencies in the Republic of Ireland, Taoiseach (Irish prime minister) Micheál Martin has said.
He said such attacks were a very significant threat to both the state and the private sector.
Mr Martin said the government's response would be "steady and methodical".
The Republic's Health Service Executive (HSE) was last week forced to shut down all of its IT systems following the "significant" ransomware attack, which focused on accessing data stored on central servers.
Mr Martin said there was a possibility that patient data had been accessed and could be shared and that security experts were assessing the full impact of this threat and the implications.
"It's a heinous attack, it's a shocking attack on a health service, but fundamentally on the patients and the Irish public," Mr Martin said.
"We're taking advice from the experts in cyber security - the National Cyber Security Centre (NCSC) - and we're also getting very considerable support from experts in the private sector.
"We will deal with this and we will work methodically in response."
'We won't be blackmailed'
The NCSC has identified the gang behind the attacks, according to Irish broadcaster RTÉ.
It is believed to be the 'Wizard Spider' group, from Eastern Europe.
Minister for Justice Heather Humphreys said the government would not be blackmailed into paying the criminals.
"The government will not be paying any money. We will protect our citizens. We will not be blackmailed," she said.
The HSE's chief clinical officer said the ransomware attack had had a profound effect on the entire HSE and the ability to deliver care, and that these challenges would "undoubtedly mount" for most hospitals over this week.
Dr Colm Henry said so much of modern healthcare was heavily reliant on information technology systems for the safe delivery of care.
Speaking on RTÉ's Morning Ireland, Dr Henry said urgent emergency and time critical care was being delivered, but not in the same way as before.
He said the ordering of tests, comparing and writing of results were "completely linked to IT", and there were people in hospitals now delivering results to consultants, while medical teams were phoning GPs directly.
He said the HSE was working with outside agencies and the priority was to re-establish those clinical systems on which critical services depend.
These include maternity, radiology, radiotherapy, newborn and diagnostics.