Over the past few years, as many people rushed to trade in their old phones for smartphones, Japan's philanderers have remained faithful to one particular brand: Fujitsu Ltd.'s older "F-Series" phones, which feature some attractive stealth privacy features. The aging flip-phone—nicknamed the "uwaki keitai" or "infidelity phone"—owes its enduring popularity to customers who don't believe newer smartphones are as discreet at hiding their illicit romances. Fujitsu's "privacy mode" is a layer of nearly invisible security that hides missed calls, emails and text messages from contacts designated as private. If one of those acquaintances gets in touch, the only signal of that communication is a subtle change in the color or shape of how the battery sign or antenna bars are displayed. If ignored, the call doesn't appear in the phone log. "If Tiger Woods had this Japanese feature in his phone, he wouldn't have gotten in trouble," said Mr. Natsuno, now a professor at Keio University's Graduate School of Media and Governance.