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They don't control it and people are quite confused about Secure Boot. Secure Boot does not refer specifically to Microsoft's implementation of it. CPU vendor is core root of trust. Motherboard firmware is a secondary root of trust. Microsoft partners with motherboard vendors.
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Not on x86 Chrome or Android devices. I'm sure there are other x86 devices with other approaches too. They control it for motherboards seemingly only designed to boot Windows. In my experience, they almost always support using custom keys and you have use secure boot without MS.
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It's not generally a good or particularly useful implementation of secure boot though. Also, most Linux distributions only support verifying the kernel and then stop there which is useless and provides no actual useful security properties unlike proper secure/verified boot.
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I don't think I'm missing anything. GPLv3 forbids having an immutable root of trust. It would be a violation of the GPLv3 to distribute software that only works with a specific signature. Microsoft would be distributing it by signing it and sending it back signed.
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Traditional Linux distributions aren't actually providing a real implementation of secure boot anyway. It stops at the bootloader or kernel. Microsoft shouldn't do this at all and when people install other OSes they should add the Linux distribution's key to UEFI keychain.
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Motherboard vendors should support custom keys, and I think most motherboards purchased as a retail product do support this. I don't understand why anyone would want to use a shim signed by Microsoft, or what the point is of this if only a bootloader and the kernel are verified.