At present, little can be proven under the definitions of modern cryptography. To prove that a cryptosystem is secure, one would first have to prove that P does not equal NP. This book is devoted to cryptographic constructions that can be proven secure in a weaker sense. We cover three topics in the book. Complete one-way functions are one-way if there are any one-way functions at all. Feebly secure cryptographic primitives can be proven secure in the strongest classical model of computation, namely general circuit complexity, but security guarantees are only constant. Finally, algebraic cryptography provides examples of noncommutative constructions that are secure against provable break, i.e., against an adversary who can present encoding examples for the messages he deciphers.