The government has revealed details of its proposed new security treaty between the UK and the EU after Brexit.
Ministers hope the treaty will provide a legal basis for co-operating on law enforcement, security and criminal justice, but did not outline any costs.
Whitehall officials are understood to be optimistic the plans will be agreed.
Both Labour and the Lib Dems criticised a specific proposal to end the direct jurisdiction of the European Court of Justice (ECJ) after the UK's departure.
The government document outlining the plans said the new security treaty would need to be underpinned by a new legal agreement since the UK would leave the ECJ.
But it said it should be possible for the UK to secure an agreement with Europol - the EU intelligence agency - that provides the same benefits as now.
'Really dangerous'
Theresa May has previously said membership of the ECJ, the EU's highest legal authority, was "not going to happen" after Brexit.
But Labour's Yvette Cooper, who chairs the home affairs select committee, said the paper failed to "answer the crucial question" of what could replace the court.
"Where are the proposals for an alternative model of dispute resolution?" she said, adding: "As the paper makes clear, it would be really dangerous to end up with operational gaps in law enforcement and justice."
Lib Dem shadow home secretary, Ed Davey, said the paper was based on "delusion", insisting that leaving the ECJ would be a "major stumbling block" to thrashing out a security agreement.
"Instead of accepting a role for the ECJ, the paper repeats Theresa May's ridiculous red line," he said.
However, the government described the plans as a new, "ambitious" model of co-operation - rejecting the idea of negotiating a number of separate agreements covering each area of law enforcement.
The National Crime Agency (NCA) said the paper was an "important milestone" in tackling pan-European threats such as organised crime, child sex abuse, cyber-attacks and terrorism.
In a statement, the agency said there was "broad consensus on the need to retain our ability to share intelligence, biometrics and other data at speed".
Officials say the new treaty would aim to replicate the provisions of the European Arrest Warrant system, under which suspects can be speedily extradited between member states, but it would not necessarily mean Britain belonging to the EAW.
Brexit Secretary David Davis said: "Together with the EU we have developed some of the world's most sophisticated systems in the fight against crime, because cross-border co-operation is absolutely crucial if we're to keep our citizens safe and bring criminals to justice.
"That is why we want to build a new partnership with the EU that goes beyond any existing relationship it has with non-member states, so we can continue countering these cross-border threats together."
Other areas listed in the document that the government wishes to continue to contribute to and benefit from are:
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Schengen Information System II, which provides real-time alerts of suspects and vehicles
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European Investigation Order, which allows member states to conduct inquiries in other countries
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Prum agreement, which allows countries to share DNA , fingerprint and vehicle registration data
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Joint investigation teams to help tackle crime
The UK has the largest defence budget in the EU and, along with France, is one of only two countries in the bloc with permanent membership of the United Nations Security Council.
German Chancellor Angela Merkel has previously said Europe can no longer "completely depend" on the US and UK following the election of President Trump and the Brexit vote, while European Commission President Jean-Claude Juncker recently called on EU member states to step up their military co-operation.
In her letter formally triggering Brexit in March, Prime Minister Theresa May warned that failure to reach a deal with Brussels would mean "co-operation in the fight against crime and terrorism would be weakened".
(Published by BBC - September 18, 2017)