are several lines of research in provable security. One is to establish the "correct" definition of security for a given, intuitively understood task. Another is to suggest constructions and proofs based on general assumptions as much as possible, for instance the existence of a one-way function. A major open problem is to establish such proofs based on P ≠ NP, since the existence of one-way functions is not known to follow from the P ≠ NP conjecture. Controversies Several researchers have found mathematical fallacies in proofs that had been used to make claims about the security of important protocols. In the following partial list of such researchers, their names are followed by first a reference to the original paper with the purported proof and then a reference to the paper in which the researchers reported on flaws: V. Shoup; A. J. Menezes; A. Jha and M. Nandi; D. Galindo; T. Iwata, K. Ohashi, and K. Minematsu; M. Nandi; J.-S. Coron and D. Naccache; D. Chakraborty, V. Hernández-Jiménez, and P. Sarkar; P. Gaži and U. Maurer; S. A. Kakvi and E. Kiltz; and T. Holenstein, R. Künzler, and S. Tessaro. Koblitz and Menezes have written that provable security results for important cryptographic protocols frequently have fallacies in the proofs; are often interpreted in a misleading manner, giving false assurances; typically rely upon strong assumptions that may turn out to be false; are based on unrealistic models of security; and serve to distract researchers' attention from the need for "old-fashioned" (non-mathematical) testing and analysis. Their series of papers supporting these claims have been controversial in the community. Among the researchers who have rejected the viewpoint of Koblitz-Menezes is Oded Goldreich, a leading theoretician and author of Foundations of Cryptography. He wrote a refutation of their first paper "Another look at `provable security'" that he titled "On post-modern cryptography." Goldreich wrote: "...we point out some of the fundamental philosophical flaws that underly the said article and some of its misconceptions regarding theoretical research in Cryptography in the last quarter of a century." In his essay Goldreich argued that the rigorous analysis methodology of provable security is the only one compatible with science, and that Koblitz and Menezes are "reactionary (i.e., they play to the hands of the opponents of progress)." In 2007, Koblitz published "The Uneasy Relationship Between Mathematics and Cryptography," which contained some controversial statements about provable security and other topics. Researchers