The components of the global cyberattack that seized hundreds of thousands of computer systems last week may be more complex than originally believed, a Trump administration official said Sunday, and experts warned that the effects of the malicious software could linger for some time.

As a new workweek started Monday in Asia, there were concerns the malicious software could spread further and in different forms, with new types of ransomware afflicting computers around the globe.

There were initial reports of new cases found over the weekend in Japan, South Korea and Taiwan.

President Trump has ordered his homeland security adviser, Thomas P. Bossert, who has a background in cyberissues, to coordinate the government’s response to the spread of the malware and help organize the search for who was responsible, an administration official said Sunday.

The attack is more complicated because “the experts tell us that this code was cobbled together from many places and sources,” according to an administration official who insisted on anonymity to discuss the government’s cybersecurity plans. The more potential sources of the malicious code, the harder it is for investigators to run down the trail of possible perpetrators.