While encryption algorithms like 3-DES and RSA have yet to be proven secure they have additionally yet to be proven crackable, which, given the number of researchers testing them and the somewhat open nature of the security community indicates that they are probably not at this time. Mathematical proofs of complex systems can be prohibitively difficult verging on impossibility (Gödel, 1931), especially if they involve nondeterministic polynomials (RSA, 2010). Factoring large numbers into primes is currently such a problem (Hohenberger, 2007). Key permutations may also be one. NP space may be shrinking as new algorithms are designed and discovered, but neither the DES nor RSA algorithm appears to be at threat at this time. If there is a flaw in either or trivial solutions to the problems on which they are based it is not in public knowledge today.
Kurose and Ross (2004) note that 'it is not known whether or not there exist fast algorithms for factoring a number, and in this sense the security of RSA is not "guaranteed."' The factoring of large numbers into primes is fundamental to several encryption methods. But the fact that “we can not even prove the almost certain truth that the density of integers n with P(n)>P(n+1) is ½” (Erdös & Pomerance, 1978) is indicative of the difficulty inherent in formal proofing of some mathematical constructs. Still, it indicates that a construct may be almost certainly true yet unprovable. So we can choose to be satisfied with the anecdotal evidence that no one has yet come forward with a successfully demonstrated exploit of these encryptions.
Erdös, P. & Pomerance, C. (1978) On the largest prime factors of n and n+1 [Online]. Available from: http://www.math.dartmouth.edu/~carlp/PDF/paper17.pdf (Accessed: 21 November, 2010)
Gödel, K. (1931) On formally undecidable propositions of Principia Mathematica and related systems I [Online]. Available from: http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf (Accessed: 21 November, 2010)
Hohenberger, S. (2007) Special Topics in Theoretical Cryptography – Lecture 1: Introduction to Cryptography [Online]. Available from:http://www.cs.jhu.edu/~susan/600.641/scribes/lecture1.pdf (Accessed: 21 November, 2010)
Kurose, J. & Ross, K. (2004) Computer Networking A Top Down Approach, 4th Ed. Pearson Education