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The peer, who was appointed to Boris Johnson's cabinet last month, said in an article for the Sunday Telegraph he wanted Brussels to "build a friendly relationship, between sovereign equals".
However, he also blamed the EU for undermining the Northern Ireland protocol by threatening to override the Brexit deal to restrict exports of Covid-19 vaccines.
Lord Frost claimed that the move created a "fragile" situation which led to the government extending the grace period on checks on the movement of goods between Northern Ireland and Great Britain.
"We have had to take some temporary operational steps to minimise disruption in Northern Ireland," he wrote.
"They are lawful and are consistent with a progressive and good faith implementation of the protocol. They are about protecting the everyday lives of people in Northern Ireland, making sure they can receive parcels and buy the usual groceries from the supermarket.
"Without this threat of disruption, we can continue our discussions with the EU to resolve difficulties arising from the protocol constructively – and we aim to do so."
The European Commission described the extension of the grace period as a "violation" of the Brexit agreement and pledged legal action.
Lord Frost, who is taking over from Michael Gove as London’s representative on the Joint Committee with Brussels on implementation of the Brexit divorce agreement, also highlighted the UK’s vaccine rollout as one of the benefits of leaving the EU, along with the ability to “enact independent national sanctions”.
And he rejected criticism of his Brexit deal as prioritising sovereignty over the economy. "This is a false choice," he wrote. "Sovereignty and democracy are vital to economic success. Sovereignty is meaningful because it enables us to set our own rules democratically for our own benefit, and thereby become more prosperous.
"It is a conviction that we, the British people, will make better decisions for ourselves than others will on our behalf."
Lord Frost concluded: "With Boris Johnson as prime minister, our agenda is one of an outward-looking country, confident we can work with others towards common goals.
"That is our hope for our ties with our European friends and allies too. I hope they will shake off any remaining ill will towards us for leaving, and instead build a friendly relationship, between sovereign equals.
"That is what I will be working towards, acting constructively when we can, standing up for our interests when we must – as a sovereign country in full control of our own destiny."