Bent Functions and Permutation Methods
Description
This book discusses in a uniform way binary, ternary, and quaternary bent functions, while most of the existing books on bent functions refer to just binary bent functions. The authors describe the differences between binary and multiple-valued cases and the construction methods for bent functions are focused on the application of two types of permutation matrices. These matrices are derived from a class of differential operators on finite groups and Fast Fourier transform algorithms, respectively. The approach presented is based on the observation that given certain bent functions, many other bent functions can be constructed by manipulating them. Permutations are possible manipulations that are easy to implement. These permutations perform spectral invariant operations which ensure that they preserve bentness.
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Radomir S. Stankovic received the B.Sc. degree in electronic engineering from the Faculty of Electronics, University of Nis, Serbia, in 1976, and M.Sc. and Ph.D. degrees in applied mathematics from the Faculty of Electrical Engineering, University of Belgrade, Serbia, in 1984 and 1986, respectively. He was a Professor at the Department of Computer Science, Faculty of Electronics, University of Nis, Serbia, until 2017, when he moved at the Mathematical Institute of SASA, Belgrade, Serbia, working there until 2019, when he is retired. In 1997, he was awarded by the Kyushu Institute of technology Fellowship and worked as a visiting researcher at the Department of Computer Science and Electronics, Kyushu Institute of Technology, Iizuka, Fukuoka, Japan. In 2000 he was awarded by the Nokia Professorship by Nokia, Finland. From 1999 until 2017, he worked in part at the Tampere International Center for Signal Processing, Department of Signal Processing, Faculty of Computing and Electrical Engineering, Tampere University of Technology, Tampere, Finland, first as a visiting professor and from 2009 until 2017 as an adjunct professor. His research interests include switching theory, multiple-valued logic, spectral techniques, and signal processing. He is an author of a few books, three of them, with Jaakko Astola, published by Springer, and another two with Jaakko Astola and Claudio Moraga, published by Wiley/IEEE Press, and Claypool & Morgan.
Milena Stankovic received the B.Sc. degree in electronic engineering from the Faculty of Electronics, University of Nis, Serbia, in 1976, and M. Sc., and Ph. D. degrees in Computing by the same Faculty in 1982 and 1988, respectively. She was a Professor at the Department of Computer Science, Faculty of Electronics, University of Nis, Serbia, until 2018, when she is retired. Milena Stankovic served as the Head of the Department of Computing in 2003 to 2005, and again in 2016 to 2018. Her research interests include switching theory, multiple-valued logic, programming languages. She is author of five books in Serbian, with a book coauthored by Radomir S. Stankovic.
Jaakko T. Astola received B.Sc., M.Sc., Licenciate and Ph.D. degrees in mathematics (specializing in error correcting codes) from Turku University, Finland in 1972, 1973, 1975, and 1978, respectively. From 1976 to 1977 he was with the Research Institute for Mathematical Sciences of Kyoto University, Kyoto, Japan. Between 1979 and 1987 he was with the Department of Information Technology, Lappeenranta University of Technology, Lappeenranta, Finland, holding various teaching positions in mathematics, applied mathematics, and computer science. In 1984, he worked as a visiting scientist in Eindhoven University of Technology, The Netherlands. From 1987 to 1992 he was Associate professor in Applied Mathematics at Tampere University, Tampere, Finland. From 1993, he has been a Professor of Signal processing and the Director of Tampere International center for Signal Processing leading a group of about 60 scientists and was nominated Academy Professor by Academy of Finland (2001-2006). He was the Director of the Centre of Excellence of the Academy of Finland in Signal Processing. His research interests include signal processing, coding theory, spectral techniques, and statistics. He is a Fellow of IEEE. He is an author of several books, published with Radomir S. Stankovic, and Claudio Moraga.
Claudio Moraga received his B.Sc. in E.E. from the Catholic University of Valparaiso (UCV), Chile, in 1961 and M.Sc. in E.E., Massachusetts Institute of Technology (MIT), USA, in 1962, and Ph.D. in E.E. (Summa cum Laude) Technical University "Federico Santa Maria" (UTFSM), Valparaiso, Chile, in 1972. From 1963 to 1970 he was a lecturer at Dept. E.E., UCV, where he served as the Head Dept. E.E. from 1964-1970. From 1970 to 1973 he worked at the Technical University "Federico Santa Maria" (UTFSM), as the Associate Professor of E.E., and become promoted into a Professor of Computer Science in 1973. In the period 1972-1973, Claudio Moraga served as the Academic Vice President of the same University. In November 1974 to August 1976, Claudio Moraga was awarded by the Alexander von Humboldt Research Fellow at the Dept. Computer Science, University of Dortmund, FRG. He continued working at the same University as a lecturer at the Dept. Computer Science, from September 1976 to March 1985. In April 1985 to September 1986, he was a Professor of Computer Science, in the area of Computer Architecture, at the Dept. Mathematics and Computer Science, University of Bremen, FRG. From October 1986 to February 2002, he was a Professor of Computer Science, in the area Theory of Automata at the Dept. Computer Science, University of Dortmund, FRG. From March 2002 to date, Claudio Moraga is a Professor Emeritus at the Dept. Computer Science, University of Dortmund, FRG. From March 2006 through December 2015, Claudio Moraga was a Researcher Emeritus at the European Centre for Soft Computing, Mieres, Asturias, Spain. Claudio Moraga has served as a member of the IEEE Technical Committee on Multiple-valued Logic (USA), and a member of the editorial board of several international journals and holds several awards, including the "Long Service Award for outstanding contributions to Multiple-valued Logic since 1971" from the IEEE TC on Multiple-valued Logic, received in May 2004 and the best paper award in Multiple-valued Logic at the ISMVL 2010. In 2005, Claudio Moraga was awarded the title of the Doctor honoris causa of the University of Nis, Serbia. He is the author of few books, two of them with Radomir S. Stankovic and Jaakko Astola published by Wiley/IEEE Press and Claypool & Morgan, respectively.
Content
Basic Concepts and Notations.- Gibbs Derivatives on Finite Abelian Groups.- Gibbs Characterization of Binary Bent Functions.- Gibbs Characterisation of Ternary Bent Functions.- Gibbs Characterization of a Class of Quaternary Bent Functions.- Matrix-valued Binary Bent Functions.- Matrix-valued Ternary Bent Functions.- Construction of Bent Functions by FFT-like Permutation Matrices.- Construction of Ternary Bent Functions Trough Matrix Representations.