In the joint declaration, Kim pledged to work towards the complete denuclearization of the Korean Peninsula, while in return, Trump promised to provide certain security guarantees to North Korea.
"If the U.S. fails to keep its promise to the world and misjudge our patience by unilaterally forcing its ways on us and sticks to sanctions and pressure against the republic (North Korea), we can not help but find a new path to protect the country's sovereignty and best interests, and achieve peace and stability on the Korean Peninsula", he said.
Although there were no specifics behind the warning that North Korea may "seek another way" unless the USA makes a move, it's thought it could mean renewed missile testing.
"If the US responds to our active and preemptive efforts with trustworthy steps and corresponding behavior, (the North-US relations) will move forward at an excellent and fast pace in the process of taking concrete and innovative measures", he said, apparently referring to the measures Pyongyang has taken since the June summit, including dismantling a nuclear and missile testing site. "Therefore, we have declared domestically and internationally and took various actions showing our commitment that we will no longer create and test nuclear weapons and will not use or spread them". He said he was willing to resume joint projects with the South, including reopening the Kaesong industrial park and Mount Kumgang resort, without conditions. For either of these to happen, sanctions would have to be removed.
The North Korean leader on Tuesday reiterated his determination to achieve complete denuclearization and also called for economic development in his annual New Year's address, stressing the need to strengthen the North's defense.
Kim's speech also points toward a hard year for the U.S.
That resulted in a dramatic detente past year, including three summits with South Korean President Moon Jae-in, and the historic meeting with Trump in June. If precedent is any indication, the North Korean leader will go into some detail outlining, sector by sector, the country's successes so far and emphasizing what remains to be done.
Seoul and Washington are in a security alliance and the United States stations 28,500 troops in the South to protect it against its neighbour, which invaded in 1950.
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Kim also emphasized the development of the North Korean economy and, without elaborating, mentioned nuclear power as part of the country's plans to boost electricity production.
The North is demanding sanctions relief - it is subject to multiple measures over its banned nuclear weapons and ballistic missile program - and has condemned United States insistence on its nuclear disarmament as "gangster-like".
The North has also bristled at USA demands to provide a detailed account of nuclear and missile facilities that would be inspected and dismantled under a potential deal.
They met another two times after that but the most historic summit of 2018 was the North Korean leader's meeting with US President Donald Trump in Singapore in June.
The hardening stalemate has fueled doubts on whether Kim will ever voluntarily relinquish the nuclear weapons and missiles he may see as his strongest guarantee of survival.
This photo taken on December 31, 2018 shows a general view of revelers during a New Year's eve countdown event and fireworks display at Kim Il Sung Square in Pyongyang, North Korea. The North used a blunt statement last month to reiterate its traditional stance on denuclearization, saying it will never unilaterally give up its weapons unless Washington removes what Pyongyang describes as a nuclear threat.
"Kim's remarks seem to suggest his patience with America is wearing thin".
Pyongyang conducted its latest missile launch in November previous year. There are views that North Korea wants a quick second summit because it thinks it can win major concessions from Trump that they probably couldn't from lower level USA officials, who are more adamant about the North committing to inspections and verification.