In this breakthrough paper, Shor gives polynomial-time quantum algorithms for integer factoring and for finding discrete logarithms. The implication for quantum computers, hypothetical devices first studied by Benioff and Feynmann, is that, if they could really be built and used, their impact on real-world computation would be revolutionary. The assumption that there are no polynomial-time factoring or discrete-log algorithms for classical computers is needed to prove the security of all of today's widely deployed public-key cryptosystems. These two problems have been extensively studied, and indeed the fastest known algorithms that solve them on classical computers require much more than polynomial time. If quantum computers could be built and used, cryptanalysts could use Shor's algorithms to break all of the public-key cryptosystems in wide use today and all of the electronic-commerce applications that depend on them. Moreover, Shor's quantum algorithms for factoring and discrete logarithms have been profoundly influential outside of the study of public-key cryptography and even outside of computer science.