This book discusses the current research concerning public key cryptosystems. It begins with an introduction to the basic concepts of multivariate cryptography and the history of this field. The authors provide a detailed description and security analysis of the most important multivariate public key schemes, including the four multivariate signature schemes participating as second round candidates in the NIST standardization process for post-quantum cryptosystems. Furthermore, this book covers the Simple Matrix encryption scheme, which is currently the most promising multivariate public key encryption scheme. This book also covers the current state of security analysis methods for Multivariate Public Key Cryptosystems including the algorithms and theory of solving systems of multivariate polynomial equations over finite fields. Through the book's website, interested readers can find source code to the algorithms handled in this book. In 1994, Dr. Peter Shor from Bell Laboratories proposed a quantum algorithm solving the Integer Factorization and the Discrete Logarithm problem in polynomial time, thus making all of the currently used public key cryptosystems, such as RSA and ECC insecure. Therefore, there is an urgent need for alternative public key schemes which are resistant against quantum computer attacks. Researchers worldwide, as well as companies and governmental organizations have put a tremendous effort into the development of post-quantum public key cryptosystems to meet this challenge. One of the most promising candidates for this are Multivariate Public Key Cryptosystems (MPKCs). The public key of an MPKC is a set of multivariate polynomials over a small finite field. Especially for digital signatures, numerous well-studied multivariate schemes offering very short signatures and high efficiency exist. The fact that these schemes work over small finite fields, makes them suitable not only for interconnected computer systems, but also for small devices with limited resources, which are used in ubiquitous computing. This book gives a systematic introduction into the field of Multivariate Public Key Cryptosystems (MPKC), and presents the most promising multivariate schemes for digital signatures and encryption. Although, this book was written more from a computational perspective, the authors try to provide the necessary mathematical background. Therefore, this book is suitable for a broad audience. This would include researchers working in either computer science or mathematics interested in this exciting new field, or as a secondary textbook for a course in MPKC suitable for beginning graduate students in mathematics or computer science. Information security experts in industry, computer scientists and mathematicians would also find this book valuable as a guide for understanding the basic mathematical structures necessary to implement multivariate cryptosystems for practical applications.

This IMA Volume in Mathematics and its Applications COMPUTER AIDED PROOFS IN ANALYSIS is based on the proceedings of an IMA Participating Institutions (PI) Conference held at the University of Cincinnati in April 1989. Each year the 19 Participating Institutions select, through a competitive process, several conferences proposals from the PIs, for partial funding. This conference brought together leading figures in a number of fields who were interested in finding exact answers to problems in analysis through computer methods. We thank Kenneth Meyer and Dieter Schmidt for organizing the meeting and editing the proceedings. A vner Friedman Willard Miller, Jr. PREFACE Since the dawn of the computer revolution the vast majority of scientific compu tation has dealt with finding approximate solutions of equations. However, during this time there has been a small cadre seeking precise solutions of equations and rigorous proofs of mathematical results. For example, number theory and combina torics have a long history of computer-assisted proofs; such methods are now well established in these fields. In analysis the use of computers to obtain exact results has been fragmented into several schools."

Multivariate public key cryptosystems (MPKC) is a fast-developing area in cryptography. This book systematically presents the subject matter for a broad audience and is the first book to focus on this exciting new topic. Information security experts in industry can use the book as a guide for understanding what is needed to implement these cryptosystems for practical applications, and researchers in both computer science and mathematics will find it a good starting point for exploring this new field. It is also suitable as a textbook for advanced-level students.