Note that the sum for n=[0, infinity] of 1/n^s does not actually converge (is not a complex number) if s is not positive real, if I remember correctly. Fortunately, the sum does behave nicely (is an analytic function) when s is positive real, and you can thus use techniques from complex analysis to extend it uniquely over the entire complex plane.
Chris's statement about the Riemann Hypothesis is correct; what we do have are a set of bounds which say that the (nontrivial) zeros of the Riemann zeta function can only occur in certain regions of the complex plane. (re: nontrivial: there are a few zeros of the Riemann zeta function on the real line as well; they are trivial zeros).
Though no encryption agorithm other than a one time pad has been proven unbreakable,
This is true.
the foundations of computer science are based on the ability to calculate (for some problems) with 100% certainty that "you have to do operation o at least f(n) times to solve a this problem", and that certain problems (ie, the halting problem) cannot be solved by computers.
Lower bounds are known for some questions, like sorting, but most of the time they're just too hard to prove.
Even so, lower bounds and impossibility proofs only apply to the general case of a problem. Sorting a general list of n numbers takes O(n log n) time, but sorting the list [1 2 3 4 5 6 8 7] can be done a lot faster. Similarly, the halting problem is undecidable in general, but you can decide a lot of specific problems.
There are NP-complete problems which are strongly believed to be intractable in practice (exponential-time), like satisfiability of boolean equations. Again, though, basing a cryptosystem on NP-complete problems is a bad bet, because a lot of specific instances are easy to solve.
I don't think that any wide spread encryption algorithm falls into either the "unsolvable" or "known scale super-polynomically", and I don't expect to see any of the former (that would make it kind of hard to decrypt), but super-polynomic encryption algorithms are certainly possible. That kind of algorithm, while crackable, can be made arbitrairly hard to crack, at much lower cost to the encryptor (assuming the actual alg. runs as a polynomial of the key size).
You don't want your encryption or decryption to be anything but polynomial in both the key size and the input data. On the other hand, the cryptosystems in wide use today have the property that breaking the key appears to require time exponential in the key size. There are no known proofs that these algorithms are hard to break, though.
Quantum Encryption (which really isn't an encryption algorithm, but a protocol for securely exchanging one time pads) looks like it is provably secure. It is based on the principle that it is impossible to duplicate a 2-state system exactly.
Quantum encryption is provably secure. The one time pads, though, are generated by quantum mechanics, taking care of one of the big problems with otp encryption. You make two particles share some quantum state (at the time of creation, you don't know what the state is, just that the particles share it), move them apart and read off the state. There's your one-time pad.
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Note that the sum for n=[0, infinity] of 1/n^s does not actually converge (is not a complex number) if s is not positive real, if I remember correctly. Fortunately, the sum does behave nicely (is an analytic function) when s is positive real, and you can thus use techniques from complex analysis to extend it uniquely over the entire complex plane. Chris's statement about the Riemann Hypothesis is correct; what we do have are a set of bounds which say that the (nontrivial) zeros of the Riemann zeta function can only occur in certain regions of the complex plane. (re: nontrivial: there are a few zeros of the Riemann zeta function on the real line as well; they are trivial zeros).
By the way, said police cruiser replica is on display at the MIT museum.
English phrases tend to have about 1.1 bits of entropy per letter; see, for instance, Refining the Estimated Entropy of Engligh by Shannon Game Simulation. So guessing an English phrase (for instance, someone's password) tends to be easy.
Looks like the article is about free net access, not email, two rather different things.