A site purporting to speak for the group said newly enacted laws that could mean jail for anyone downloading copyrighted music and movies "will result in scores of unnecessary prison sentences to numerous innocent citizens".
The finance ministry's website came under attack on Tuesday, with a number of webpages defaced, ministry official Takanari Horino told AFP, and websites of the supreme court and intellectual property high court were also down for a short time overnight, he said.
"We are investigating where the illegal item came from," the official said, referring to an unauthorised link posted to the ministry site.
"We are aware of the Anonymous statement referring to the new copyright law, but we don't know at this point if the cyberattacks are linked to the group," he said.
The anonpr.net website said laws passed by both houses of Japan's parliament last week would do "little to solve the underlying problem of legitimate copyright infringement.
"The content industry is now pushing ISPs in Japan to implement surveillance technology that will spy on... every single Internet user in Japan," it said.
"This would be an unprecedented approach and severely reduce the amount of privacy law abiding citizens should have in a free society."
The statement also warned the government and country's Recording Industry Association to "expect us the same way we have come to expect you in violating our basic rights to privacy and to an open Internet".
A Wednesday morning post on Twitter by user @op_japan claiming to speak for Anonymous said: "Good morning #japan. Expect more from us today! #anonymous #opjapan #anonfamily #freeanons."
It gave no further details.
A spokeswoman for the Recording Industry Association said on Wednesday no attacks on its website had been recorded, but that it was investigating the issue.
The new copyright law stipulates that "downloading of copyrighted works knowing that they are not free and that it is illegal" will lead to a prison sentence of up to two years, or a fine of up to two million yen (25,300 dollars), or both.
Some 4.36 billion files, including music and movies, were illegally downloaded in 2010, 10 times the number of legally downloaded music files, the association said.
Anonymous is a "hacker-activist" network that has claimed online attacks on sites ranging from the Vatican to Los Angeles Police Canine Association, but is increasingly the target of police who have arrested dozens of members.
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