Provable security is another feature that is sought -- and even claimed -- fairly frequently. It turns out that there actually are some very interesting proofs for security properties of some algorithms; however, these proofs must be regarded within their precise mathematical context, and what they prove is contingent upon all sorts of assumptions and limitations. Further, most algorithms that have provable properties like this are developed for academic research purposes. In general, none of the algorithms in widespread use (whether public-key or symmetric) has rigorously proven mathematical properties. Instead, we settle for algorithms that have stood up well to years of attack efforts by the best cryptanalysts. This is not the certainty of a mathematical proof, but it is pretty good.

The point of these observations is that you should look with suspicion upon vendors or amateurs who claim to have proven the security of their algorithms. Most likely they have not, except perhaps in highly constrained and circumscribed ways. Unless you are the type of expert cryptanalyst who is able to evaluate such alleged proofs (and if you are, this tutorial is way too basic for you), you should take claims about provable security with a grain of salt.