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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").
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.
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,