Consider the size of the group
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The efficiency problem becomes obvious when you consider a user who accesses a website site many times. For example, imagine using an anonymous credential to replace Google’s session cookies. For most users, this require obtaining and delivering thousands of single-use credentials every single day. You might mitigate this problem by using credentials only for the first registration to a website, after which you can trade your credential for a pseudonym (such as a random username or a normal session cookie) for later accesses. But the downside of this is that all of your subsequent site accesses would be linkable, which is a bit of a privacy tradeoff.