The CRC Handbook of Finite Fields is a comprehensive reference book that compiles the state-of-the-art research in finite fields and their applications. The book is organized into three main parts: an introduction to finite fields, theoretical properties of finite fields, and applications of finite fields. **Part I: Introduction** - **History of Finite Fields**: Chapter 1 provides an overview of the historical development of finite fields from the 18th to the 19th centuries, including key contributions from mathematicians like Gauss, Galois, Serret, Schönemann, and Dedekind. **Part II: Theoretical Properties** - **Basic Properties**: Chapter 2 covers fundamental properties of finite fields, including definitions, extensions, trace and norm functions, bases, linearized polynomials, and miscellaneous results. - **Polynomials**: Chapters 3 and 4 discuss irreducible and primitive polynomials, their construction, and properties. - **Bases**: Chapter 5 explores various types of bases over finite fields, such as normal bases and completely normal bases. - **Exponential and Character Sums**: Chapter 6 delves into exponential and character sums, including Gauss, Jacobi, and Kloosterman sums, and their applications. - **Equations over Finite Fields**: Chapter 7 addresses solutions of equations over finite fields, including affine and projective hypersurfaces. - **Permutation Polynomials**: Chapter 8 discusses permutation polynomials in one and multiple variables, value sets, and exceptional polynomials. - **Special Functions**: Chapter 9 covers special functions over finite fields, such as Boolean functions, APN functions, bent functions, and Dickson polynomials. - **Sequences**: Chapter 10 examines sequences over finite fields, including finite field transforms, LFSR sequences, correlation and autocorrelation, and linear complexity. - **Algorithms**: Chapter 11 provides an overview of finite field algorithms, including computational techniques, polynomial counting, factorization, and discrete logarithms. - **Curves**: Chapter 12 discusses curves over finite fields, focusing on elliptic and hyperelliptic curves, rational points, towers, and zeta functions. - **Miscellaneous Theoretical Topics**: Chapter 13 covers additional topics such as matrices, linear algebra, classical groups, and Carlitz and Drinfeld modules. **Part III: Applications** - **Combinatorial**: Chapter 14 explores combinatorial structures like Latin squares, block designs, and difference sets. - **Algebraic Coding Theory**: Chapter 15 discusses algebraic coding theory, including linear codes, cyclic codes, LDPC codes, turbo codes, and polar codes. - **Cryptography**: Chapter 16 covers cryptographic systems over finite fields, including stream and block ciphers, multivariate systems, elliptic and hyperelliptic curve systems, and systems arising from Abelian varieties. - **Miscellaneous Applications**: Chapter 17 discusses applications ofThe CRC Handbook of Finite Fields is a comprehensive reference book that compiles the state-of-the-art research in finite fields and their applications. The book is organized into three main parts: an introduction to finite fields, theoretical properties of finite fields, and applications of finite fields. **Part I: Introduction** - **History of Finite Fields**: Chapter 1 provides an overview of the historical development of finite fields from the 18th to the 19th centuries, including key contributions from mathematicians like Gauss, Galois, Serret, Schönemann, and Dedekind. **Part II: Theoretical Properties** - **Basic Properties**: Chapter 2 covers fundamental properties of finite fields, including definitions, extensions, trace and norm functions, bases, linearized polynomials, and miscellaneous results. - **Polynomials**: Chapters 3 and 4 discuss irreducible and primitive polynomials, their construction, and properties. - **Bases**: Chapter 5 explores various types of bases over finite fields, such as normal bases and completely normal bases. - **Exponential and Character Sums**: Chapter 6 delves into exponential and character sums, including Gauss, Jacobi, and Kloosterman sums, and their applications. - **Equations over Finite Fields**: Chapter 7 addresses solutions of equations over finite fields, including affine and projective hypersurfaces. - **Permutation Polynomials**: Chapter 8 discusses permutation polynomials in one and multiple variables, value sets, and exceptional polynomials. - **Special Functions**: Chapter 9 covers special functions over finite fields, such as Boolean functions, APN functions, bent functions, and Dickson polynomials. - **Sequences**: Chapter 10 examines sequences over finite fields, including finite field transforms, LFSR sequences, correlation and autocorrelation, and linear complexity. - **Algorithms**: Chapter 11 provides an overview of finite field algorithms, including computational techniques, polynomial counting, factorization, and discrete logarithms. - **Curves**: Chapter 12 discusses curves over finite fields, focusing on elliptic and hyperelliptic curves, rational points, towers, and zeta functions. - **Miscellaneous Theoretical Topics**: Chapter 13 covers additional topics such as matrices, linear algebra, classical groups, and Carlitz and Drinfeld modules. **Part III: Applications** - **Combinatorial**: Chapter 14 explores combinatorial structures like Latin squares, block designs, and difference sets. - **Algebraic Coding Theory**: Chapter 15 discusses algebraic coding theory, including linear codes, cyclic codes, LDPC codes, turbo codes, and polar codes. - **Cryptography**: Chapter 16 covers cryptographic systems over finite fields, including stream and block ciphers, multivariate systems, elliptic and hyperelliptic curve systems, and systems arising from Abelian varieties. - **Miscellaneous Applications**: Chapter 17 discusses applications of