1 IntroductionReferences 2 Multivariate Cryptography 2.1 Multivariate Polynomials 2.2 Construction Methods for MPKC's2.3 Underlying Problems2.4 Security and Standard Attacks 2.5 Advantages and Disadvantages References 3 The Matsumoto-Imai Cryptosystem 3.1 The Basic Matsumoto-Imai Cryptosystem3.2 The Linearization Equations Attack3.3 Encryption Schemes Based on MI3.4 Signature Schemes Based on MI References 4 Hidden Field Equations 4.1 The Basic HFE Cryptosystem 4.2 Attacks on HFE 4.3 Encryption Schemes Based on HFE4.4 Signature Schemes Based on HFE References5 Oil and Vinegar 5.1 The Oil and Vinegar Signature Scheme5.2 Cryptanalysis of Balanced Oil and Vinegar and UOV5.3 Other Attacks on UOV 5.4 The Rainbow Signature Scheme5.5 Attacks on Rainbow 5.6 Reducing the Public Key Size5.7 Efficient Key Generation of Rainbow References 6 MQDSS 6.1 The MQ Based Identification Scheme 6.2 The Fiat-Shamir Transformation 6.3 The MQDSS Signature Scheme References7 The SimpleMatrix Encryption Scheme 7.1 The Basic SimpleMatrix Encryption Scheme 7.2 The Rectangular SimpleMatrix Encryption Scheme 7.3 Attacks on SimpleMatrix References 8 Solving Polynomial Systems 8.1 History of Solving Polynomial Equations 8.2 Solving Univariate Polynomials of High Degree 8.3 The XL-Algorithm8.4 Gr]obner Bases 8.5 Buchberger's Algorithm and F4 8.6 Estimating the Degree of Regularity8.7 Solving Over- and Underdetermined Systems ReferencesIndex List of Toy Examples Software
About the Author: Jintai Ding is a Charles Phelps Taft professor at the Department of Mathematical Sciences at the University of Cincinnati. He received B.A. from Xian Jiao tong University in 1988, M.A. from the University of Science and Technology of China in 1990 and PhD from Yale in 1995. He was a lecturer at the Research Institute of Mathematical Sciences of Kyoto University from 1995 to 1998. He has been at the University of Cincinnati since 1998. In 2006-2007, he was a visiting professor and Alexander von Humboldt Fellow at TU Darmstadt. He received the Zhong Jia Qing Prize from the Chinese Mathematical Society in 1990 for his Master Thesis on proving a conjecture by C. L. Siegel. His research was originally in quantum affine algebras and its representation theory, where he was credited for the invention of the Ding-Iohara-Miki algebra. His current interest is in post-quantum cryptography, in particular, multivariate cryptography, latticed-based cryptography and quantum-proof blockchain. He was a co-chair of the 2nd, 10th and 11th international conference on post-quantum cryptography. He and his colleagues developed the Rainbow signature, the GUI HFEv- signature, the Simple Matrix encryption and the LWE-based key exchange schemes. Rainbow is a second round candidate for the NIST post-quantum standardization process. He and his students completely broke a NIST second round post-quantum signature candidate LUOV.
Albrecht Petzoldt received a diploma in mathematics in 2009 from FAU Erlangen-Nuremberg and a PhD in Computer Science in 2013 from Technische Universität Darmstadt / Germany. Since then he worked for several academic and non academic institutions, including Kyushu University / Japan and the National Institute of Standards and Technology (NIST) / USA. Currently, he works as a lecturer at FAU Erlangen-Nuremberg / Germany.His main research interests are located in the field of multivariate cryptography, and in particular in the development and improvement of multivariate signature schemes such as UOV and Rainbow.
In 1966 Dieter Schmidt received his "Diplom in Mathematik" from the Technische Hochschule Stuttgart, Germany. He then went to the University of Minnesota, where he received his PhD in Mathematics in 1970. During that time he also worked for Univac and gained valuable experience in computer programming. After an initial appointment at the University of Maryland, he accepted a position in the Department of Mathematical Sciences at the University of Cincinnati. The department started offering courses in Computer Science in the late 1970's. It was natural for him to teach some of these courses and then to join the Department of Computer Science when it was formed in 1984. In 2002 he started his collaboration with Jintai Ding. He offered his expertise in programming in order to create the software for cryptographic schemes or the code to attack them. Although Dieter Schmidt retired from the University of Cincinnati in 2011, he has continued the collaboration with Jintai Ding.