SAN FRANCISCO — A confounding computer bug called “Heartbleed” is causing major security headaches across the Internet as websites scramble to fix the problem and Web surfers wonder whether they should change their passwords to prevent theft of their email accounts, credit card numbers and other sensitive information.

The breakdown revealed this week affects a widely used encryption technology that is supposed to protect online accounts for a variety of online communications and electronic commerce.

Security researchers who uncovered the threat are particularly worried about the lapse because it went undetected for more than two years. They fear the possibility that computer hackers may have been secretly exploiting the problem before its discovery. It’s also possible that no one took advantage of the flaw before its existence was announced late Monday.

Although there is now a way to close the security hole, there are still plenty of reasons to be concerned, said David Chartier, CEO of Codenomicon. A small team from the Finnish security firm diagnosed Heartbleed while working independently from another Google Inc. researcher who also discovered the threat.

Canada’s tax agency isn’t taking any chances. Citing the security risks posed by Heartbleed, the Canada Revenue Agency shut off public access to its website “to safeguard the integrity of the information we hold,” according to a notice posted on its website Wednesday. The agency said it hopes to re-open its website this weekend. The lockdown comes just three weeks from Canada’s April 30 deadline for filing 2013 tax returns.

The U.S. Internal Revenue Service said in a statement Wednesday that it’s not affected by the security hole. “The IRS advises taxpayers to continue filing their tax returns as they normally would in advance of the April 15 deadline,” the agency said.

TurboTax, the most popular tax preparation software, also issued a Wednesday statement reassuring people that its website is now protected against Heartbleed.

Computer security experts are still advising people to consider changing all their online passwords.

“I would change every password everywhere because it’s possible something was sniffed out,” said Wolfgang Kandek, chief technology officer for Qualys, a maker of security-analysis software. “You don’t know because an attack wouldn’t have left a distinct footprint.”

Google is so confident that it inoculated itself against the Heartbleed bug before any damage could be done that the Mountain View, Calif., company is telling its users they don’t have to change the passwords they use to access Gmail, YouTube and other product accounts. More than 425 million Gmail accounts alone have been set up worldwide.

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