- Authors:
- Gary L. Mullen,
- Daniel Panario
Poised to become the leading reference in the field, the Handbook of Finite Fields is exclusively devoted to the theory and applications of finite fields. More than 80 international contributors compile state-of-the-art research in this definitive handbook. Edited by two renowned researchers, the book uses a uniform style and format throughout and each chapter is self contained and peer reviewed. The first part of the book traces the history of finite fields through the eighteenth and nineteenth centuries. The second part presents theoretical properties of finite fields, covering polynomials, special functions, sequences, algorithms, curves, and related computational aspects. The final part describes various mathematical and practical applications of finite fields in combinatorics, algebraic coding theory, cryptographic systems, biology, quantum information theory, engineering, and other areas. The book provides a comprehensive index and easy access to over 3,000 references, enabling you to quickly locate up-to-date facts and results regarding finite fields.
Cited By
- Zidarič N, Gong G, Aagaard M, Jurišić A and Konovalov O (2023). FFCSA - Finite Field Constructions, Search, and Algorithms, ACM Communications in Computer Algebra, 57:2, (57-64), Online publication date: 1-Jun-2023.
- Møller J (2022). Equivariant Euler Characteristics of Symplectic Buildings, Graphs and Combinatorics, 38:3, Online publication date: 1-Jun-2022.
- Barthe G, Jacomme C and Kremer S (2021). Universal Equivalence and Majority of Probabilistic Programs over Finite Fields, ACM Transactions on Computational Logic, 23:1, (1-42), Online publication date: 31-Jan-2022.
- Mesnager S, Kim K, Choe J, Lee D and Go D (2020). Solving over , Cryptography and Communications, 12:4, (809-817), Online publication date: 1-Jul-2020.
- Bojilov A, Borissov L and Borissov Y (2019). Computing the number of finite field elements with prescribed absolute trace and co-trace, Cryptography and Communications, 11:3, (497-507), Online publication date: 1-May-2019.
- Gao Z, Li C and Zhou Y (2019). Upper bounds and constructions of complete Asynchronous channel hopping systems, Cryptography and Communications, 11:2, (299-312), Online publication date: 1-Mar-2019.
- Qureshi C and Panario D (2019). The graph structure of Chebyshev polynomials over finite fields and applications, Designs, Codes and Cryptography, 87:2-3, (393-416), Online publication date: 1-Mar-2019.
- Xu X, Li C, Zeng X and Helleseth T (2018). Constructions of complete permutation polynomials, Designs, Codes and Cryptography, 86:12, (2869-2892), Online publication date: 1-Dec-2018.
- Wang Y, Zha Z and Zhang W (2018). Six new classes of permutation trinomials over $$\mathbb {F}_{3^{3k}}$$F33k, Applicable Algebra in Engineering, Communication and Computing, 29:6, (479-499), Online publication date: 1-Dec-2018.
- Liu Q, Sun Y and Zhang W (2018). Some classes of permutation polynomials over finite fields with odd characteristic, Applicable Algebra in Engineering, Communication and Computing, 29:5, (409-431), Online publication date: 1-Nov-2018.
- Bai T and Xia Y (2018). A new class of permutation trinomials constructed from Niho exponents, Cryptography and Communications, 10:6, (1023-1036), Online publication date: 1-Nov-2018.
- Faugère J and Wallet A (2018). The point decomposition problem over hyperelliptic curves, Designs, Codes and Cryptography, 86:10, (2279-2314), Online publication date: 1-Oct-2018.
- Bartoli D and Quoos L (2018). Permutation polynomials of the type $$x^rg(x^{s})$$xrg(xs) over $${\mathbb {F}}_{q^{2n}}$$Fq2n, Designs, Codes and Cryptography, 86:8, (1589-1599), Online publication date: 1-Aug-2018.
- Zha Z, Hu L and Zhang Z (2018). New results on permutation polynomials of the form (xpm ź x + ź)s + xpm + x over źź p2m, Cryptography and Communications, 10:3, (567-578), Online publication date: 1-May-2018.
- Hosseinalipour S, Sakzad A and Sadeghi M (2018). Performance of Möbius Interleavers for Turbo Codes, Wireless Personal Communications: An International Journal, 98:1, (271-291), Online publication date: 1-Jan-2018.
- Tzanakis G, Moura L, Panario D and Stevens B (2017). Covering arrays from m-sequences and character sums, Designs, Codes and Cryptography, 85:3, (437-456), Online publication date: 1-Dec-2017.
- Davis J, Huczynska S and Mullen G (2017). Near-complete external difference families, Designs, Codes and Cryptography, 84:3, (415-424), Online publication date: 1-Sep-2017.
- Castro A (2017). Quantum one-way permutation over the finite field of two elements, Quantum Information Processing, 16:6, (1-18), Online publication date: 1-Jun-2017.
- Zhou Y and Qu L (2017). Constructions of negabent functions over finite fields, Cryptography and Communications, 9:2, (165-180), Online publication date: 1-Mar-2017.
- Gyarmati K, Mauduit C and Sárközy A (2017). On finite pseudorandom binary lattices, Discrete Applied Mathematics, 216:P3, (589-597), Online publication date: 10-Jan-2017.
- Pilaram H and Eghlidos T (2017). An Efficient Lattice Based Multi-Stage Secret Sharing Scheme, IEEE Transactions on Dependable and Secure Computing, 14:1, (2-8), Online publication date: 1-Jan-2017.
- Bhattacharya S and Sarkar S (2017). On some permutation binomials and trinomials over $$\mathbb {F}_{2^n}$$F2n, Designs, Codes and Cryptography, 82:1-2, (149-160), Online publication date: 1-Jan-2017.
- Huang M and Liu L Constructing Small Generating Sets for the Multiplicative Groups of Algebras over Finite Fields Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation, (287-294)
- Hu J and Monagan M A Fast Parallel Sparse Polynomial GCD Algorithm Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation, (271-278)
- Grenet B, Hoeven J and Lecerf G (2016). Deterministic root finding over finite fields using Graeffe transforms, Applicable Algebra in Engineering, Communication and Computing, 27:3, (237-257), Online publication date: 1-Jun-2016.
- Grenet B, van der Hoeven J and Lecerf G Randomized Root Finding over Finite FFT-fields using Tangent Graeffe Transforms Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation, (197-204)
- Göloğlu F (2015). Almost perfect nonlinear trinomials and hexanomials, Finite Fields and Their Applications, 33:C, (258-282), Online publication date: 1-May-2015.
- Gao Z and Wang Q (2015). A probabilistic approach to value sets of polynomials over finite fields, Finite Fields and Their Applications, 33:C, (160-174), Online publication date: 1-May-2015.
Index Terms
- Handbook of Finite Fields
Reviews
Manish K Gupta
If you want to play football, you go to a football field. If you want to play cricket, you go to a cricket field. But what if you want to play with numbers (with all four operations of addition, subtraction, multiplication, and division)__?__ You need a field of numbers. When numbers of elements in the field are finite, we get a finite field. The smallest field (known as the binary field) has only two elements: zero and one. Our computers are based on this smallest field. We can safely say that modern information and communication technology (ICT) is based on finite fields. It provides reliability in ICT using error correction techniques from coding theory and provides security using cryptographic techniques from cryptography. The soul of coding and cryptography is finite fields. Are you ready to play with finite fields__?__ If you want to get some basic knowledge from a book and then dive into finite fields, this book is for you. This is not a textbook, but it provides comprehensive and up-to-date information in all areas of finite fields. It is divided into three parts. Part 1 gives a basic introduction to finite fields in two chapters. It begins with a historical introduction of the subject in chapter 1, continues to tables of irreducible polynomials in chapter 2, and concludes with a summary of open-source software. Part 2 provides beautiful theoretical properties for playing with finite fields in 11 chapters, covering such topics as irreducible polynomials, primitive polynomials, bases, exponential and character sums, equations over finite fields, permutation polynomials, special functions, sequences over finite fields, and curves over finite fields. Part 3 discusses applications in combinatorics, algebraic coding theory, and cryptography in the first three chapters. The last chapter deals with new fascinating and emerging areas where we find applications of finite fields, including biology, quantum computing, and some new areas in coding theory (such as space-time codes and network coding). This book is a must-read for every electrical engineer, computer scientist, mathematician, and even biologist. They will find a wealth of information and ideas that can be used to understand various systems, build beautiful things, and even enjoy the power of algebra! Online Computing Reviews Service
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