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Notes -
I think that in theory, you are correct. Software could be designed so that it is provable that e.g. there is no possible TLS traffic which will allow an attacker to reconstruct the server's private key any faster than just cracking the public key.
For the moment, just about none of the software we use has such proofs, however. As you mention, even the mathematical foundations of existing crypto primitives rarely offer such guarantees -- more often it is just "we looked into that problem for three decades, and it seems really hard" (e.g. integer factorization). Concrete implementations without formal verification likely have implementation bugs as well. Mythos did not discover any exploits which would have been impossible for humans to discover, it was just that nobody had spent that much human eyeball time on auditing the software. This is what I meant by "token pissing contest".
Furthermore, actually specifying what theorems should hold to keep your system secure is itself hard. If you have a TLS server which will provably never leak its key, but is happy to use it to sign and decrypt on behalf of the attacker, that is still a broken system. If you have a larger system, then completely specifying what you do not want an attacker to be able to do seems difficult.
Also, the software we care about does not run on Turing machines, it runs on physical hardware. In everyday use, most computer hardware behaves as an idealized model. Billions of people use DRAM every day, and it just works. Except that there are corner cases where it will not behave as advertised.
I can not speculate if an ASI could build a system which even a much stronger ASI could not penetrate, and what the performance costs would be. But I think it is unlikely that human-level intelligences subject to design pressures besides security will build hardware and software systems which ASI's will not be able to penetrate.
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