In June, many Google users were surprised to see an unusual greeting at the top of their Gmail inbox, Google home page or Chrome browser. “Warning: We believe state-sponsored attackers may be attempting to compromise your account or computer.”

On Tuesday, tens of thousands more Google users will begin to see that message. The company said that since it started alerting users to malicious — probably state-sponsored — activity on their computers in June, it has picked up thousands more instances of cyberattacks than it anticipated.