Everything in the Infinity Machine is at most countably infinite (aleph-0). Memory is countable since memory locations are directly bijectable on to the natural numbers; processing power is countable too, since every instruction occupies a non-zero time interval, and we can therefore show that at most countably many instructions are ever executed by observing that each such interval contains at least one rational.
There's no difference between real numbers and "infinite floating point" (modulo the occasional number with redundant encodings, such as 0.9 recurring being equal to 1). The reals are the same size as 2^aleph-0. (Be careful saying "aleph-1"; that's not a terribly useful concept owing to the undecidability of the Continuum Hypothesis. For the infinity of the reals we generally say C.)
A signature algorithm generating an infinitely long signature is not impossible (or at least can't be shown to be impossible by this type of argument), since there are uncountably many such signatures and hence even the Machine can't generate and test them all. Likewise a public/private key scheme and a Diffie-Hellman-like key exchange: anything which involves generating an infinitely long bit string using a secret key is secure in principle, because another Machine can't possibly generate all the possibilities to reverse the algorithm.
I agree that a finitely long hash wouldn't be secure for most purposes, but even an infinitely long one would have uses. For example, consider the cryptographic scissors-paper-stone protocol: I tell you the hash of my move, you tell me your move, I reveal my move and you check the hash. The virtue of a hash is not just that it reduces the size of the data; its trapdoor-ness is also useful.
![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
There's no difference between real numbers and "infinite floating point" (modulo the occasional number with redundant encodings, such as 0.9 recurring being equal to 1). The reals are the same size as 2^aleph-0. (Be careful saying "aleph-1"; that's not a terribly useful concept owing to the undecidability of the Continuum Hypothesis. For the infinity of the reals we generally say C.)
A signature algorithm generating an infinitely long signature is not impossible (or at least can't be shown to be impossible by this type of argument), since there are uncountably many such signatures and hence even the Machine can't generate and test them all. Likewise a public/private key scheme and a Diffie-Hellman-like key exchange: anything which involves generating an infinitely long bit string using a secret key is secure in principle, because another Machine can't possibly generate all the possibilities to reverse the algorithm.
I agree that a finitely long hash wouldn't be secure for most purposes, but even an infinitely long one would have uses. For example, consider the cryptographic scissors-paper-stone protocol: I tell you the hash of my move, you tell me your move, I reveal my move and you check the hash. The virtue of a hash is not just that it reduces the size of the data; its trapdoor-ness is also useful.