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Modern cryptography libraries make sure that the execution path (and memory access patterns) do not depend on sensitive data. Usually this is what is meant with 'constant-time'.

If a debugger can read out registers or memory you can just read out the sensitive material of course.

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Whoops... Yeah, you're right. You wouldn't expect to see instructions like bne, bge, etc. that depend on key material directly, so you wouldn't be able to rely the instruction pointer alone.

Instead, you might see instructions like addlt, so you'd also need to inspect the value before and after, which, as you correctly state, the debugger will happily let you do.

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