While overseeing a ballistic missile launch yesterday, Kim stressed the importance of conducting "more nuclear explosion tests to estimate the destructive power of the newly produced nuclear warheads", the North's official KCNA news agency said.
State media on Wednesday had published photos of Kim visiting nuclear technicians and standing next to what was claimed to be a miniaturised nuclear warhead.
During the visit, he stated that his scientists had mastered the process of shrinking warheads to a size capable of fitting on a ballistic missile -- a step that would present a heightened threat to South Korea, other countries in the region and, eventually, the US mainland.
The exercise was clearly a response to ongoing large-scale, South Korea-US military drills that Pyongyang views as provocative rehearsals for invasion.
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The aim was to simulate conditions for "exploding nuclear warheads from the preset altitude above targets in the ports under enemy control of foreign aggressor forces", the agency said.
Watching the exercise, Kim reiterated an earlier threat to launch an immediate nuclear attack if the "sabre-rattling" South Korea-US drills should harm "even a single tree or a blade of grass" on North Korean territory.
Military tensions on the divided Korean peninsula have been on the rise since the North carried out its fourth nuclear test in January, followed by a long-range rocket launch last month.
The UN Security Council responded with tough, new sanctions, which Pyongyang condemned as a "gangster-like" provocation orchestrated by the United States.
UN Secretary-General Ban Ki-moon on yesterday voiced grave concern over the tensions, and urged North Korea to avoid any further "destabilising acts" including missile launches.
Kim, however, highlighted the need to develop a diversified nuclear strike force, capable of delivering warheads from the ground, air, sea and underwater.
Outside experts have questioned the results of those tests, suggesting Pyongyang had gone little further than a "pop-up" test from a submerged platform.