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comment Why is the permutation in AES (and other ciphers) not random?
The problem with just a key dependency is that the permutation is invariant to the input, and thus still susceptible to differential cryptanalysis (same key, different inputs). And obviously you can't use the full plaintext during decryption. However, you could permute the permutation using the decrypted input so far. This is conceptually comparable to CBC mode.
comment Hash function that allows to decide if A > B if you only have hash(A) and hash(B)?
This does assume a full ordering of the input domain, though. If the number of equivalence classes is no more than the number of possible outputs, then the pigeon hole principle does not hold. But here it's given that A and B are integers, where the number of equivalence classes is (literally!) infinite.
comment Why can't I reverse a hash to a possible input?
And note that cryptographic hashes are designed to have a high degree of mixing, which is necessarily achieved by designing circuits with high fan-out.
comment How do we know a cryptographic primitive won't fail suddenly?
@user3491648: You're clearly unhappy with this answer, but that doesn't make it in any way wrong. Basic logic dictates that statement like these do not require proof, as such proof is generally impossible and yet they're refuted by a single counter-example (In this case, such a counter-example would have the form "we know because X").
comment Voting scheme where the votes become public when a threshold is reached
@mikeazo: Under the weak assumption of secret voting, your observation does prove that you can't have any further votes. But this is no issue e.g. when the threshold is set to 51% of voters.
comment Is it possible to create an easy to use encryption/decryption method that will never be comprimised?
Actually, there's an infinite set of ciphers that are cannot be broken, but they all share the 4 points you list. Trivially, for any bijective function f, f(OTP(x)) is such a cipher.