Fundamentals of Cryptology

Abstract

The protection of sensitive information against unauthorized access or fraudulent changes has been of prime concern throughout the centuries. Modern communication techniques, using computers connected through networks, make all data even more vulnerable for these threats. Also, new issues have come up that were not relevant before, e.g. how to add a (digital) signature to an electronic document in such a way that the signer can not deny later on that the document was signed by him/her. Cryptology addresses the above issues. It is at the foundation of all information security. The techniques employed to this end have become increasingly mathematical of nature. This book serves as an introduction to modern cryptographic methods. After a brief survey of classical cryptosystems, it concentrates on three main areas. First of all, stream ciphers and block ciphers are discussed. These systems have extremely fast implementations, but sender and receiver have to share a secret key. Public key cryptosystems (the second main area) make it possible to protect data without a prearranged key. Their security is based on intractable mathematical problems, like the factorization of large numbers. The remaining chapters cover a variety of topics, such as zero-knowledge proofs, secret sharing schemes and authentication codes. Two appendices explain all mathematical prerequisites in great detail. One is on elementary number theory (Euclid's Algorithm, the Chinese Remainder Theorem, quadratic residues, inversion formulas, and continued fractions). The other appendix gives a thorough introduction to finite fields and their algebraic structure. This book differs from its 1988 version in two ways. That a lot of new material has been added is to be expected in a field that is developing so fast. Apart from a revision of the existing material, there are many new or greatly expanded sections, an entirely new chapter on elliptic curves and also one on authentication codes. The second difference is even more significant. The whole manuscript is electronically available as an interactive Mathematica manuscript. So, there are hyperlinks to other places in the text, but more importantly, it is now possible to work out non-trivial examples. Even a non-expert can easily alter the parameters in the examples and try out new ones. It is our experience, based on teaching at the California Institute of Technology and the Eindhoven University of Technology, that most students truly enjoy the enormous possibilities of a computer algebra notebook. Throughout the book, it has been our intention to make all Mathematica statements as transparent as possible, sometimes sacrificing elegant or smart alternatives that are too dependent on this particular computer algebra package.There are several people that have played a crucial role in the preparation of this manuscript. In alphabetical order of first name, I would like to thank Fred Simons for showing me the full potential of Mathematica for educational purposes and for enhancing many the Mathematica command...