The best replacement for open authentication is shared.

The best alternatives to open authentication are rated as following:
★★★★★  shared
★★★★  shared key
★★  shared authentication
★  shared key authentication

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Speaking about open authentication, one probably shouldn't forget about wep.

Other search terms related to open authentication and their ratings are:
★★★★★ wep
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Google Trends

Scale is based on the average worldwide traffic of open authentication in all years. Click on graph for more.
open authentication
shared
shared key