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Suppose I have a certificate for *.example.com and privileged access to all example.com websites. Sure I could install the wildcard certificate into https://api.example.com website so that it can prove its identity. What about https://www.cool.example.com?

I'm facing this exact scenario with a fourth level domain name website. IE and Firefox refuse to connect to https://www.cool.example.com which serves *.example.com certificate to prove its identity (however when I request https://cool.example.com serving the same certificate the two browsers are both happy to connect).

Is this the expected behavior?

marked as duplicate by Steffen Ullrich tls Apr 16 '18 at 8:56

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Wildcard certificates only cover one level of subdomains. You can however use subject alternative names to have more leeway.

You can see examples on the Wikipedia page

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