Officials from the National Cyber Security Centre (NCSC) and the Health Service Executive have briefed the Government about last week's cyber attacks on the HSE and the Department of Health.
Taoiseach Micheál Martin, Tánaiste Leo Varadkar, Minister for the Environment Eamon Ryan, Minister for Justice Heather Humphreys, Minister for Health Stephen Donnelly and Minister of State for eGovernment Ossian Smyth attended the briefing.
In a statement this evening, the Government said "its main concern is to secure as speedy a resumption of all medical services as can possibly be achieved" and added, "no effort is being spared to achieve this".
It said: "This attack on Ireland's health care system and its patients was carried out by an international cyber-crime gang.
"It is aimed at nothing other than extorting money and those who carried it out have no concern for the severe impact on patients needing care or for the privacy of those whose private information has been stolen."
It added: "These ransomware attacks are despicable crimes, most especially when they target critical health infrastructure and sensitive patient data. The significant disruption to health services is to be condemned, especially at this time.
"Any public release by the criminals behind this attack of any stolen patient data is equally and utterly contemptible. There is a risk that the medical and other data of patients will be abused.
"Anyone who is affected is urged to contact the HSE and the Garda authorities."
Ossian Smyth rejected claims that the State's computer systems and networks were not sufficiently robust to withstand cyber attacks.
Mr Smyth said over 2,500 attacks are repelled every week but this one was "far beyond anything we've ever had before".
He also said that while the NCSC had been without a director for over a year he was now looking for a higher salary to be offered to attract the right candidate.
Officials from the NCSC will meet with the Oireachtas Transport and Communications Committee tomorrow.
The private meeting will see the NCSC brief politicians on their approach to ransomware attacks and their strategy to address such crises.
Chair of the Committee, Kieran O'Donnell, said that he was eager for the committee to play a constructive role in helping authorities deal with the IT attack.
He expressed hope that a comprehensive, public meeting will be possible with the NCSC in the near future.
Earlier, Minister Humphreys met Garda Commissioner Drew Harris and the head of the Garda Cybercrime Bureau in relation to the ransomware attack.
Mr Harris and Detective Chief Superintendent Paul Cleary updated Ms Humphreys on the criminal investigation into the attacks.
The NCSC has identified the criminal gang behind the cyber attacks as the 'Wizard Spider' group, which is based in Eastern Europe and has been a target of the FBI, the UK's National Crime Agency, Interpol and Europol.
The criminal investigation team is also liaising with international law enforcement partners and has sent digital footprints of the virus found on the computers to the Europol Malware Analysis Centre in the Hague.
A spokesman for the Department of Justice said the minister would remain in close contact with Commissioner Harris and his team.
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Analysis: HSE breach latest in spider's web of cyber attacks
The NCSC said it is working to identify the technical details of the malware used in the cyber attacks.
Four days after the Department of Health and the HSE were attacked, specialists are working to clean infected devices and restore the IT systems.
The NCSC said it first became aware that malware had been inserted into systems at the Department of Health last Thursday, followed by the HSE on Friday.
Meanwhile, the HSE said that work continued today in assessing the impact of the attack and it is beginning to restore HSE IT systems.
It said there are serious concerns about the implications for patient care, arising from the very limited access to diagnostics, lab services and historical patient records.
The shutdown of its systems is having an impact on some health services and this disruption is very likely to go well into this week.
The Chief Executive of the HSE Paul Reid warned it will take weeks to fully repair the damage caused by the attack.
Mr Reid said efforts are continuing to try "to restart 30 years of technology investment over a few days" and he added, "realistically this will work out in weeks rather than just days".
He said steady progress has been made "throughout the last 48 hours in particular, aiming to restore particularly some of the voluntary hospitals" and some of the national systems.
Specialists prevented the ransomware from detonating on the Department of Health's systems, but not at the HSE.
Digital ransom notes have also been received from the cyber crime group seeking engagement on the Darknet in relation to a ransom of millions, but the Government has insisted that no money will be paid to restore the data.
Separately, the Russian Embassy in Ireland said it "condemns in strongest terms any type of criminal activity in cyber space, including this particular incident".
In a statement, it said that Irish authorities have not yet approached the Embassy regarding the attack.
"It's pretty clear that if they do, the Russian Side would be ready to look into the matter, since the Russian government has been consistently promoting initiatives on strengthening international cooperation on the issues of international information security and confronting effectively cyber space crime," a spokesperson said.
Additional reporting Paul Reynolds, Tommy Meskill