Contents Part I: 1 2

Historical introduction to finite fields Roderick Gow . . . . . . . . . . Introduction to finite fields . . . . . . . . . . . . . . . . . . . . . . . . .

2 3

Panario . . . .

2.1

2.2

Basic properties of finite fields Gary L. Mullen and Daniel 2.1.1 Basic definitions . . . . . . . . . . . . . . . . . . 2.1.2 Fundamental properties of finite fields . . . . . . 2.1.3 Extension fields . . . . . . . . . . . . . . . . . . 2.1.4 Trace and norm functions . . . . . . . . . . . . . 2.1.5 Bases . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Linearized polynomials . . . . . . . . . . . . . . 2.1.7 Miscellaneous results . . . . . . . . . . . . . . . 2.1.7.1 The finite field polynomial Φ function 2.1.7.2 Lagrange interpolation . . . . . . . . 2.1.7.3 Discriminants . . . . . . . . . . . . . 2.1.7.4 Jacobi logarithms . . . . . . . . . . . 2.1.7.5 Field-like structures . . . . . . . . . . 2.1.7.6 Galois rings . . . . . . . . . . . . . . 2.1.8 Finite field related books . . . . . . . . . . . . . 2.1.8.1 Textbooks . . . . . . . . . . . . . . . 2.1.8.2 Finite field theory . . . . . . . . . . . 2.1.8.3 Applications . . . . . . . . . . . . . . 2.1.8.4 Algorithms . . . . . . . . . . . . . . . 2.1.8.5 Conference proceedings . . . . . . . . Tables David Thomson . . . . . . . . . . . . . . . . . .

Part II: 3

Introduction

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24 24 25 26 27 28 29 29 31 31 32 35 35 35 39 39

Theoretical Properties

Irreducible polynomials 3.1

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Counting irreducible polynomials Joseph L.Yucas 3.1.1 Prescribed trace or norm . . . . . . . . . 3.1.2 Prescribed coefficients over the binary field 3.1.3 Self-reciprocal polynomials . . . . . . . . 3.1.4 Compositions of powers . . . . . . . . . . 3.1.5 Translation invariant polynomials . . . . 3.1.6 Normal replicators . . . . . . . . . . . . Construction of irreducibles Melsik Kyuregyan . . 3.2.1 Construction by composition . . . . . . . 3.2.2 Recursive Constructions . . . . . . . . . Reducible polynomials Daniel Panario . . . . . . 3.3.1 Composite polynomials . . . . . . . . . . 3.3.2 Swan-type theorems . . . . . . . . . . . . Weights of irreducible polynomials Omran Ahmadi 3.4.1 Basic definition . . . . . . . . . . . . . .

3.4.2 Existence results . . . . . . . . . . . . . . . . . . . . 3.4.3 Non-existence results . . . . . . . . . . . . . . . . . . 3.4.4 Conjectures . . . . . . . . . . . . . . . . . . . . . . . Prescribed coefficients Stephen D. Cohen . . . . . . . . . . . . 3.5.1 One prescribed coefficient . . . . . . . . . . . . . . . . 3.5.2 Prescribed trace and norm . . . . . . . . . . . . . . . 3.5.3 More prescribed coefficients . . . . . . . . . . . . . . 3.5.4 Further exact expressions . . . . . . . . . . . . . . . . Multivariate polynomials Xiang-dong Hou . . . . . . . . . . . 3.6.1 Counting formulas . . . . . . . . . . . . . . . . . . . 3.6.2 Asymptotic formulas . . . . . . . . . . . . . . . . . . 3.6.3 Results for the vector degree . . . . . . . . . . . . . . 3.6.4 Indecomposable polynomials and irreducible polynomials 3.6.5 Algorithms for gcd . . . . . . . . . . . . . . . . . . .

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. . . . . 39 . . . . . 41 . . . . . 41 3.5 . . . . . 43 . . . . . 43 . . . . . 44 . . . . . 45 . . . . . 47 3.6 . . . . . 50 . . . . . 50 . . . . . 51 . . . . . 51 . . . . . 53 . . . . . . 54 Primitive polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.1 Introduction to primitive polynomials Gary L. Mullen and Daniel Panario 56 4.2 Prescribed coefficients Stephen D. Cohen . . . . . . . . . . . . . . . . . . 60 4.2.1 Approaches to results on prescribed coefficients . . . . . . . . . . 60 4.2.2 Existence theorems for primitive polynomials . . . . . . . . . . . 61 4.2.3 Existence theorems for primitive normal polynomials . . . . . . . 63 4.3 Weights of primitive polynomials Stephen D. Cohen . . . . . . . . . . . . 66 4.4 Elements of high order Jos´e Felipe Voloch . . . . . . . . . . . . . . . . . 69 4.4.1 Elements of high order from elements of small orders . . . . . . . 69 4.4.2 Gao’s construction and generalization . . . . . . . . . . . . . . . 69 4.4.3 Iterative constructions . . . . . . . . . . . . . . . . . . . . . . . 70 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.1 Duality theory of bases Dieter Jungnickel . . . . . . . . . . . . . . . . . . 71 5.1.1 Dual bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.1.2 Self-dual bases . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.1.3 Weakly self-dual bases . . . . . . . . . . . . . . . . . . . . . . . 74 5.1.4 Binary bases with small excess . . . . . . . . . . . . . . . . . . . 76 5.1.5 Almost weakly self-dual bases . . . . . . . . . . . . . . . . . . . 77 5.2 Normal bases Shuhong Gao and Qunying Liao . . . . . . . . . . . . . . . 80 5.3 Optimal and low complexity normal bases Shuhong Gao and David Thomson 81 5.4 Completely normal bases Dirk Hachenberger . . . . . . . . . . . . . . . . 82 5.4.1 The complete normal basis theorem . . . . . . . . . . . . . . . . 82 5.4.2 A reduction to extensions of prime power degree . . . . . . . . . 83 5.4.3 The class of completely basic extensions . . . . . . . . . . . . . . 83 5.4.4 Module structures and the notion of additive orders . . . . . . . . 84 5.4.5 Cyclotomic modules and complete generators . . . . . . . . . . . 85 5.4.6 A decomposition theory for complete generators . . . . . . . . . . 86 5.4.7 The class of regular extensions . . . . . . . . . . . . . . . . . . . 88 5.4.8 Complete generators for regular cyclotomic modules . . . . . . . 89 5.4.9 Construction of complete generators . . . . . . . . . . . . . . . . 90 5.4.10 Towards a primitive complete normal basis theorem . . . . . . . . 92 5.4.11 Sequences of completely normal elements . . . . . . . . . . . . . 93 5.4.12 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Exponential and character sums . . . . . . . . . . . . . . . . . . . . . . 96 6.1 Gauss, Jacobi, and Kloosterman sums Ronald J. Evans . . . . . . . . . . 96 6.1.1 Properties of Gauss and Jacobi sums over Fq . . . . . . . . . . . 96 6.1.2 Evaluations of Jacobi and Gauss sums of small orders . . . . . . . 105

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Equations over finite fields

. . . . . . . . . . . . . . . . . . . . . . 7.2 . . . . . 7.3 . . . . . Permutation polynomials . . . . . . . . . . . . . . . . 8.1 One variable Gary L. Mullen and Qiang Wang . . . . . 8.1.1 Basics . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Criteria . . . . . . . . . . . . . . . . . . . . . 8.1.3 Enumeration and distribution of PPs . . . . . 8.1.4 Construction of PPs . . . . . . . . . . . . . . 8.1.5 PPs from permutations of multiplicative groups 8.1.6 PPs from permutations of additive groups . . . 8.1.7 Other types of PPs . . . . . . . . . . . . . . . 8.1.8 Dickson and Reversed Dickson PPs . . . . . . 8.1.9 Miscellaneous PPs . . . . . . . . . . . . . . . 8.2 Several variables Rudolf Lidl and Gary L. Mullen . . . 7.1

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6.1.3 Prime ideal divisors of Gauss and Jacobi sums . . . . . . . . 6.1.4 Kloosterman sums over Fq . . . . . . . . . . . . . . . . . . . 6.1.5 Gauss and Kloosterman sums over finite rings . . . . . . . . . More general exponential and character sums Antonio Rojas Le´ on . . 6.2.1 One variable character sums . . . . . . . . . . . . . . . . . . 6.2.2 Additive character sums . . . . . . . . . . . . . . . . . . . . 6.2.3 Multiplicative character sums . . . . . . . . . . . . . . . . . 6.2.4 Generic estimates . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 More general types of character sums . . . . . . . . . . . . . Some-products theorems and applications Moubariz Z. Garaev . . . . 6.3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 The sum-product estimate and its variants . . . . . . . . . . 6.3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . Some applications of character sums Alina Ostafe and Arne Winterhof 6.4.1 Applications of a simple character sum identity . . . . . . . . 6.4.2 Applications of Gauss and Jacobi sums . . . . . . . . . . . . 6.4.3 Applications of the Weil bound . . . . . . . . . . . . . . . . . 6.4.4 Applications of Kloosterman sums . . . . . . . . . . . . . . . 6.4.5 Incomplete character sums . . . . . . . . . . . . . . . . . . . 6.4.6 Other character sums . . . . . . . . . . . . . . . . . . . . . . 6.4.7 Other applications and links to other chapters . . . . . . . . . General forms Daqing Wan . . . . . . . . . . . . . . 7.1.1 Affine hypersurfaces . . . . . . . . . . . . . . 7.1.2 Projective hypersurfaces . . . . . . . . . . . 7.1.3 Toric hypersurfaces . . . . . . . . . . . . . . 7.1.4 Artin-Schreier hypersurfaces . . . . . . . . . 7.1.5 Kummer hypersurfaces . . . . . . . . . . . . 7.1.6 p-Adic estimates . . . . . . . . . . . . . . . Quadratic forms Robert Fitzgerald . . . . . . . . . . 7.2.1 Basic definitions . . . . . . . . . . . . . . . . 7.2.2 Quadratic forms over finite fields . . . . . . . 7.2.3 Trace forms . . . . . . . . . . . . . . . . . . 7.2.4 Applications . . . . . . . . . . . . . . . . . . Diagonal equations Francis Castro and Ivelisse Rubio 7.3.1 Preliminaries . . . . . . . . . . . . . . . . . 7.3.2 Solutions of diagonal equations . . . . . . . . 7.3.3 Generalizations of diagonal equations . . . . 7.3.4 Waring’s problem in finite fields . . . . . . .

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8.3

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Value sets of polynomials Gary L. Mullen and Michael 8.3.1 Large value sets . . . . . . . . . . . . . . . . 8.3.2 Small value sets . . . . . . . . . . . . . . . . 8.3.3 General polynomials . . . . . . . . . . . . . 8.3.4 Lower bounds . . . . . . . . . . . . . . . . . 8.3.5 Examples . . . . . . . . . . . . . . . . . . . 8.3.6 Further value set papers . . . . . . . . . . . Exceptional polynomials Michael E. Zieve . . . . . . 8.4.1 Fundamental properties . . . . . . . . . . . . 8.4.2 Classification results . . . . . . . . . . . . . 8.4.3 Low-degree exceptional polynomials . . . . . 8.4.4 Potpourri . . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special functions over finite fields . . . . . . . . . . . . . . . . . . . 9.1 Boolean functions Claude Carlet . . . . . . . . . . . . . . . . . . . . 9.1.1 Representation of Boolean functions . . . . . . . . . . . . . . 9.1.2 The Walsh transform . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Parameters of Boolean functions . . . . . . . . . . . . . . . . 9.1.4 Boolean functions and cryptography . . . . . . . . . . . . . . 9.1.5 Constructions of cryptographic Boolean functions . . . . . . . . . 9.1.6 Boolean functions and error correcting codes . . . . . . . . . . . 9.1.7 Boolean functions and sequences . . . . . . . . . . . . . . . . . . 9.2 PN and APN functions Pascale Charpin . . . . . . . . . . . . . . . . . . 9.2.1 Functions from F2n into F2m . . . . . . . . . . . . . . . . . . . . 9.2.2 Perfect Nonlinear (PN) functions . . . . . . . . . . . . . . . . . . 9.2.3 Almost Perfect Nonlinear (APN) and Almost Bent (AB) functions 9.2.4 APN permutations . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.5 Properties of stability . . . . . . . . . . . . . . . . . . . . . . . 9.2.6 Coding theory point of view . . . . . . . . . . . . . . . . . . . . 9.2.7 Quadratic APN functions . . . . . . . . . . . . . . . . . . . . . 9.2.8 APN monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Bent and related functions Alexander Kholosha and Alexander Pott . . . 9.3.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Basic properties of bent functions . . . . . . . . . . . . . . . . . 9.3.3 Constructions of bent functions . . . . . . . . . . . . . . . . . . 9.3.4 Bent functions and other combinatorial objects . . . . . . . . . . 9.3.5 Special classes of bent functions . . . . . . . . . . . . . . . . . . 9.3.6 Hyper bent, normal and self-dual bent functions . . . . . . . . . 9.3.7 Constructions using PN and s-plateaued functions . . . . . . . . 9.3.8 p-ary bent functions in univariate form . . . . . . . . . . . . . . 9.4 κ-polynomials and related algebraic objects Robert Coulter . . . . . . . . 9.4.1 Definitions and preliminaries . . . . . . . . . . . . . . . . . . . . 9.4.2 Pre-semifields, semifields and isotopy . . . . . . . . . . . . . . . . 9.4.3 Semifield constructions . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Semifields and nuclei . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Planar functions and commutative semifields Robert Coulter . . . . . . . 9.5.1 Definitions and preliminaries . . . . . . . . . . . . . . . . . . . . 9.5.2 Constructing affine planes using planar functions . . . . . . . . . 9.5.3 Examples, constructions and equivalence . . . . . . . . . . . . . . 9.5.4 Classification results, necessary conditions and the Dembowski-Ostrom Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.5 Planar DO polynomials and commutative semifields of odd order .

189 189 189 190 190 191 191 193 193 193 194 194 196 196 197 198 199 201 202 204 204 205 205 206 207 208 209 210 210 212 214 214 216 217 219 219 221 221 222 225 225 226 227 228 230 230 230 231 232 233

. . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 . . . . . . . . . . . . . . . . . . . . . . Sequences over finite fields . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Finite field transforms Gary McGuire . . . . . . . . . . . . . . . . . . . . 10.1.1 Basic definitions and Important Examples . . . . . . . . . . . . . 10.1.2 Functions between two groups . . . . . . . . . . . . . . . . . . . 10.1.3 Sequence and Matrix Formulation . . . . . . . . . . . . . . . . . 10.1.4 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . 10.1.5 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.5.1 Fourier Spectrum . . . . . . . . . . . . . . . . . . . . 10.1.5.2 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . 10.1.5.3 Characteristic Functions . . . . . . . . . . . . . . . . 10.1.5.4 Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . 10.1.5.5 Uncertainty Principle . . . . . . . . . . . . . . . . . . 10.2 LFSRs and maximum length sequences Solomon Golomb . . . . . . . . . 10.3 Correlation and autocorrelation of sequences Tor Helleseth . . . . . . . . 10.3.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Autocorrelation of sequences . . . . . . . . . . . . . . . . . . . . 10.3.3 Sequence families with low crosscorrelation . . . . . . . . . . . . 10.3.4 Quaternary sequences . . . . . . . . . . . . . . . . . . . . . . . . 10.3.5 Aperiodic correlation . . . . . . . . . . . . . . . . . . . . . . . . 10.3.6 The merit factor . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.7 Partial period correlation . . . . . . . . . . . . . . . . . . . . . . 10.3.8 The Hamming correlation . . . . . . . . . . . . . . . . . . . . . 10.4 Linear complexity of sequences and multisequences Wilfried Meidl and Arne Winterhof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Linear complexity measures . . . . . . . . . . . . . . . . . . . . 10.4.2 Analysis of the linear complexity . . . . . . . . . . . . . . . . . . 10.4.3 Average behaviour of the linear complexity . . . . . . . . . . . . 10.4.4 Some sequences with large nth linear complexity . . . . . . . . . 10.4.4.1 Explicit sequences . . . . . . . . . . . . . . . . . . . . 10.4.4.2 Recursive nonlinear sequences . . . . . . . . . . . . . . 10.4.4.3 Legendre sequence and related bit sequences . . . . . . 10.4.4.4 Elliptic curve sequences . . . . . . . . . . . . . . . . . 10.4.5 Related measures . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.5.1 Kolmogorov complexity . . . . . . . . . . . . . . . . . 10.4.5.2 Lattice test . . . . . . . . . . . . . . . . . . . . . . . 9.6

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Dickson polynomials Qiang Wang and Joseph L. Yucas . . . . . . 9.6.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2.1 a-reciprocals of polynomials . . . . . . . . . . . 9.6.2.2 Φa and Ψa . . . . . . . . . . . . . . . . . . . . 9.6.2.3 Factors of Dickson polynomials . . . . . . . . . 9.6.2.4 a-cyclotomic polynomials . . . . . . . . . . . . 9.6.3 Dickson polynomial of the (k + 1)-th kind . . . . . . . . . 9.6.4 Multivariate Dickson polynomials . . . . . . . . . . . . . Schur’s conjecture and exceptional covers Michael D. Fried . . . . 9.7.1 Rational function definitions . . . . . . . . . . . . . . . . 9.7.2 MacCluer’s Theorem and Schur’s Conjecture . . . . . . . 9.7.3 Fiber product of covers . . . . . . . . . . . . . . . . . . . 9.7.4 Combining exceptional covers; the (Fq , Z) exceptional tower 9.7.5 Exceptional rational functions; Serre’s Open Image Theorem 9.7.6 Davenport pairs and Poincar´e series . . . . . . . . . . . .

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235 235 236 237 237 238 239 239 241 243 243 244 247 249 250 253 256 256 256 258 259 260 261 261 261 261 262 262 263 264 264 264 266 268 268 269 269 269 270 270 273 275 277 277 278 278 279 280 280 280

10.4.5.3 Correlation measure of order k . . . . . . . . . 10.4.5.4 Discrepancy . . . . . . . . . . . . . . . . . . . 10.5 Algebraic dynamical systems over finite fields Igor Shparlinski . . . 10.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Background and Main Definitions . . . . . . . . . . . . . 10.5.3 Degree growth . . . . . . . . . . . . . . . . . . . . . . . . 10.5.4 Linear independence and other algebraic properties of iterates 10.5.5 Multiplicative independence of iterates . . . . . . . . . . . 10.5.6 Trajectory length . . . . . . . . . . . . . . . . . . . . . . 10.5.7 Irreducibility of iterates . . . . . . . . . . . . . . . . . . . 10.5.8 Diameter of partial trajectories . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Computational techniques Christophe Doche . . . . . . . . . . . . . . . . 11.2 Basic polynomial counting Daniel Panario . . . . . . . . . . . . . . . . . 11.2.1 Classical counting results . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Flajolet’s analytic combinatorics approach . . . . . . . . . . . . . 11.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Relations to algorithms . . . . . . . . . . . . . . . . . . . . . . . 11.3 Algorithms for irreducibility testing and constructing irreducible polynomials Mark Giesbrecht . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Testing irreducibility of univariate polynomials . . . . . . . . . . Early Irreducibility Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rabin’s irreducibility test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Constructing irreducible polynomials: randomized algorithms . . . 11.3.3 Constructing Irreducible Polynomials: Deterministic Algorithms . 11.4 Factorization of univariate polynomials Joachim von zur Gathen . . . . . 11.5 Factorization of multivariate polynomials Erich Kaltofen and Gr´egoire Lecerf 11.5.1 Factoring dense multivariate polynomials . . . . . . . . . . . . . 11.5.1.1 Separable factorization . . . . . . . . . . . . . . . . . 11.5.1.2 Squarefree factorization . . . . . . . . . . . . . . . . . 11.5.1.3 Bivariate irreducible factorization . . . . . . . . . . . . 11.5.1.4 Reduction from any number to two variables . . . . . . 11.5.2 Factoring sparse multivariate polynomials . . . . . . . . . . . . . 11.5.2.1 Ostrowski’s theorem . . . . . . . . . . . . . . . . . . . 11.5.2.2 Irreducibility tests based on indecomposability of polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2.3 Sparse bivariate Hensel lifting driven by polytopes . . . 11.5.2.4 Convex-dense bivariate factorization . . . . . . . . . . 11.5.3 Factoring straight-line programs and black boxes . . . . . . . . . 11.6 Primary decomposition of ideals over finite fields Shuhong Gao . . . . . . 11.7 Grobner bases and solving polynomial systems over finite fields Shuhong Gao 11.8 Discrete logarithms over finite fields Andrew Odlyzko . . . . . . . . . . . 11.9 Standard models for finite fields Hendrik Lenstra and Bart de Smit . . . . Curves over finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction to function fields and curves Arnaldo Garcia and Henning Stichtenoth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Valuations and places . . . . . . . . . . . . . . . . . . . . . . . . 12.1.2 Divisors and Riemann–Roch theorem . . . . . . . . . . . . . . . 12.1.3 Extensions of function fields . . . . . . . . . . . . . . . . . . . . 12.1.4 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.5 Function fields and curves . . . . . . . . . . . . . . . . . . . . .

280 281 282 282 282 283 285 286 286 287 288 290 290 291 291 291 292 293 294 294 294 295 295 297 300 301 301 301 302 303 305 306 306 307 307 308 308 312 313 314 315 316 316 317 319 323 330 332

12.2 Elliptic curves Joseph Silverman . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Weierstrass equations . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 The group law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Isogenies and endomorphisms . . . . . . . . . . . . . . . . . . . . 12.2.4 The number of points in E(Fq ) . . . . . . . . . . . . . . . . . . 12.2.5 Twists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.6 The torsion subgroup and the Tate module . . . . . . . . . . . . 12.2.7 The Weil pairing and the Tate pairing . . . . . . . . . . . . . . . 12.2.8 The endomorphism ring and automorphism group . . . . . . . . . 12.2.9 Ordinary and supersingular elliptic curves . . . . . . . . . . . . . 12.2.10 The zeta function of an elliptic curve . . . . . . . . . . . . . . . 12.2.11 The elliptic curve discrete logarithm problem . . . . . . . . . . . 12.3 Hyperelliptic curves Michael John Jacobson, Jr. and Renate Scheidler . . 12.3.1 Hyperelliptic equations . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 The degree zero divisor class group . . . . . . . . . . . . . . . . . 12.3.3 Divisor class arithmetic over finite fields . . . . . . . . . . . . . . 12.3.4 Endomorphisms and supersingularity . . . . . . . . . . . . . . . 12.3.5 Class number computation . . . . . . . . . . . . . . . . . . . . . 12.3.6 The Tate-Lichtenbaum pairing . . . . . . . . . . . . . . . . . . . 12.3.7 The hyperelliptic curve discrete logarithm problem . . . . . . . . 12.4 Rational points on curves Arnaldo Garcia and Henning Stichtenoth . . . . 12.4.1 Rational places . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 The Zeta function of a function field . . . . . . . . . . . . . . . . 12.4.3 Bounds for the number of rational places . . . . . . . . . . . . . 12.4.4 Maximal function fields . . . . . . . . . . . . . . . . . . . . . . . 12.4.5 Asymptotic bounds . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Towers Arnaldo Garcia and Henning Stichtenoth . . . . . . . . . . . . . . 12.5.1 Introduction to towers . . . . . . . . . . . . . . . . . . . . . . . 12.5.2 Examples of towers . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 (t, m, s)-nets and (t, s)-sequences Harald Niederreiter . . . . . . . . . . . 12.6.1 (t, m, s)-nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.2 Digital (t, m, s)-nets . . . . . . . . . . . . . . . . . . . . . . . . 12.6.3 Constructions of (t, m, s)-nets . . . . . . . . . . . . . . . . . . . 12.6.4 (t, s)-sequences and (T, s)-sequences . . . . . . . . . . . . . . . . 12.6.5 Digital (t, s)-sequences and digital (T, s)-sequences . . . . . . . . 12.6.6 Constructions of (t, s)-sequences and (T, s)-sequences . . . . . . 12.7 Zeta functions and L-functions Lei Fu . . . . . . . . . . . . . . . . . . . 12.7.1 Zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.2 L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.3 The case of curves . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 P-adic estimates of zeta functions and L-functions R´egis Blache . . . . . . 12.8.1 Lower bounds for the first slope . . . . . . . . . . . . . . . . . . 12.8.2 Uniform lower bounds for Newton polygons . . . . . . . . . . . . 12.8.3 Variation of Newton polygons in a family . . . . . . . . . . . . . 12.8.4 The case of curves, and abelian varieties . . . . . . . . . . . . . . 12.9 Computing the number of rational points and zeta functions Daqing Wan 12.9.1 Point counting: sparse input . . . . . . . . . . . . . . . . . . . . 12.9.2 Point counting: dense input . . . . . . . . . . . . . . . . . . . . . 12.9.3 Computing zeta functions: general case . . . . . . . . . . . . . . 12.9.4 Computing zeta functions: curve case . . . . . . . . . . . . . . .

13

Miscellaneous theoretical topics

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334 334 336 338 341 342 343 344 347 348 350 350 352 352 353 355 357 358 359 360 361 361 362 363 365 366 368 368 370 373 373 374 376 378 380 381 384 384 388 391 394 395 396 398 400 403 403 404 405 406 407

13.1 Relations between integers and polynomials over finite fields Gove Effinger 13.1.1 The density of primes . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2 Primes in arithmetic progression . . . . . . . . . . . . . . . . . . 13.1.3 Twin primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.4 The generalized Riemann hypothesis . . . . . . . . . . . . . . . . 13.1.5 The Goldbach problem . . . . . . . . . . . . . . . . . . . . . . . 13.1.6 The Waring problem . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Matrices over finite fields Dieter Jungnickel . . . . . . . . . . . . . . . . . 13.2.1 Matrices of specified rank . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Matrices of specified order . . . . . . . . . . . . . . . . . . . . . 13.2.3 Matrix representations of finite fields . . . . . . . . . . . . . . . . 13.2.4 Circulant and orthogonal matrices . . . . . . . . . . . . . . . . . 13.2.5 Symmetric and skew-symmetric matrices . . . . . . . . . . . . . 13.2.6 Hankel and Toeplitz matrices . . . . . . . . . . . . . . . . . . . . 13.2.7 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Linear algebra over finite fields Jean-Guillaume Dumas and Cl´ement Pernet 13.3.1 Dense matrix multiplication . . . . . . . . . . . . . . . . . . . . 13.3.1.1 Tiny finite fields . . . . . . . . . . . . . . . . . . . . . 13.3.1.2 Word size prime fields . . . . . . . . . . . . . . . . . . 13.3.1.3 Large finite fields . . . . . . . . . . . . . . . . . . . . 13.3.1.4 Large matrices: subcubic time complexity . . . . . . . 13.3.2 Dense Gaussian elimination and echelon forms . . . . . . . . . . 13.3.2.1 Building blocks . . . . . . . . . . . . . . . . . . . . . 13.3.2.2 PLE decomposition . . . . . . . . . . . . . . . . . . . 13.3.2.3 Echelon forms . . . . . . . . . . . . . . . . . . . . . . 13.3.3 Minimal and characteristic polynomial of a dense matrix . . . . . 13.3.4 Blackbox iterative methods . . . . . . . . . . . . . . . . . . . . . 13.3.4.1 Minimal Polynomial and the Wiedemann algorithm . . 13.3.4.2 Rank, Determinant and Characteristic Polynomial . . . 13.3.4.3 System solving and the Lanczos algorithm . . . . . . . 13.3.5 Sparse and structured methods . . . . . . . . . . . . . . . . . . . 13.3.5.1 Reordering . . . . . . . . . . . . . . . . . . . . . . . . 13.3.5.2 Structured matrices and displacement rank . . . . . . 13.3.6 Hybrid methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.6.1 Hybrid sparse-dense methods . . . . . . . . . . . . . . 13.3.6.2 Block-iterative methods . . . . . . . . . . . . . . . . . 13.4 Classical groups over finite fields Zhe-Xian Wan . . . . . . . . . . . . . . 13.4.1 Linear groups over finite fields . . . . . . . . . . . . . . . . . . . 13.4.2 Symplectic groups over finite fields . . . . . . . . . . . . . . . . . 13.4.3 Unitary groups over finite fields . . . . . . . . . . . . . . . . . . 13.4.4 Orthogonal groups over finite fields of characteristic not two . . . 13.4.5 Orthogonal groups over finite fields of characteristic two . . . . . 13.5 Carlitz and Drinfeld modules David Goss . . . . . . . . . . . . . . . . . . 13.5.1 Quick review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 Drinfeld modules: definition and analytic Theory . . . . . . . . . 13.5.3 Drinfeld modules over finite fields . . . . . . . . . . . . . . . . . 13.5.4 The reduction theory of Drinfeld modules . . . . . . . . . . . . . 13.5.5 The A-module of rational points . . . . . . . . . . . . . . . . . . 13.5.6 The invariants of a Drinfeld module . . . . . . . . . . . . . . . . 13.5.7 The L-series of a Drinfeld module . . . . . . . . . . . . . . . . . 13.5.8 Special values . . . . . . . . . . . . . . . . . . . . . . . . . . . .

407 408 409 409 410 411 412 415 415 416 417 418 421 422 423 425 425 425 427 427 428 428 428 429 430 431 432 432 432 433 433 433 434 435 435 435 437 437 439 441 443 446 448 448 449 451 451 452 452 453 454

13.5.9 13.5.10 13.5.11 13.5.12

Part III: 14

Measures and symmetries . . Multizeta . . . . . . . . . . Modular theory . . . . . . . Transcendency results . . . .

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461 462 463 463 464 465 466 467 468 468 468 469 470 471 473 474 476 476 477 477 479 480 482 483 484 485 487 487 489 491 494 496 496 498 500 502 502 502 505 507 508 509 510

Applications

Combinatorial

14.1 Latin squares Gary L. Mullen . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Prime powers . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.2 Non-prime powers . . . . . . . . . . . . . . . . . . . . . . . . 14.1.3 Frequency squares . . . . . . . . . . . . . . . . . . . . . . . . 14.1.4 Hypercubes . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.5 Connections to affine and projective planes . . . . . . . . . . 14.1.6 Other finite field constructions for MOLS . . . . . . . . . . . 14.2 Lacunary polynomials over finite fields Simeon Ball and Aart Blokhuis 14.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Lacunary polynomials . . . . . . . . . . . . . . . . . . . . . . 14.2.3 Directions and the R´edei polynomial . . . . . . . . . . . . . . 14.2.4 Sets of points determining few directions . . . . . . . . . . . . 14.2.5 Lacunary polynomials and blocking sets . . . . . . . . . . . . 14.2.6 Lacunary polynomials and blocking sets in planes of prime order 14.2.7 Lacunary polynomials and multiple blocking sets . . . . . . . 14.3 Affine and projective planes Gary Ebert and Leo Storme . . . . . . . 14.3.1 Projective planes . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Affine planes . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 Translation planes and spreads . . . . . . . . . . . . . . . . . 14.3.4 Nest planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.5 Flag-transitive affine planes . . . . . . . . . . . . . . . . . . . 14.3.6 Subplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.7 Embedded unitals . . . . . . . . . . . . . . . . . . . . . . . . 14.3.8 Maximal arcs . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.9 Other results . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Projective spaces James W.P. Hirschfeld and Joseph A. Thas . . . . . 14.4.1 Projective and affine spaces . . . . . . . . . . . . . . . . . . . 14.4.2 Collineations, correlations and coordinate frames . . . . . . . 14.4.3 Polarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.4 Partitions and cyclic projectivities . . . . . . . . . . . . . . . 14.4.5 k -Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The three problems of Segre . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.6 k -Arcs and linear MDS codes . . . . . . . . . . . . . . . . . . 14.4.7 k -Caps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Block designs Charles J. Colbourn and Jeffrey H. Dinitz . . . . . . . . 14.5.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.2 Triple systems . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.3 Difference families and balanced incomplete block designs . . 14.5.4 Nested designs . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.5 Pairwise balanced designs . . . . . . . . . . . . . . . . . . . . 14.5.6 Group divisible designs . . . . . . . . . . . . . . . . . . . . . 14.5.7 t-designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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14.6

14.7

14.8

14.9

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14.5.8 Packing and covering . . . . . . . . . . . . . . . . . . . . . . . . 14.5.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Difference sets Alexander Pott . . . . . . . . . . . . . . . . . . . . . . . . 14.6.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.2 Difference sets in cyclic groups . . . . . . . . . . . . . . . . . . . 14.6.3 Difference sets in the additive groups of finite fields . . . . . . . . 14.6.4 Difference sets and Hadamard matrices . . . . . . . . . . . . . . 14.6.5 Further families . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.6 Difference sets and character sums . . . . . . . . . . . . . . . . . 14.6.7 Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications and divisibility of polynomials Brett Stevens . . . . . . . . . 14.7.1 Weights of multiples of polynomials . . . . . . . . . . . . . . . . 14.7.1.1 Applications . . . . . . . . . . . . . . . . . . . . . . . 14.7.1.2 Weights of multiples of polynomials . . . . . . . . . . 14.7.2 Card Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ramanujan and Expander Graphs M. Ram Murty and Sebastian M. Cioab˘ a 14.8.1 Graphs, Adjacency Matrices and Eigenvalues . . . . . . . . . . . 14.8.2 Cayley Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8.3 Ramanujan Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 14.8.4 Expander Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8.5 Explicit Constructions of Ramanujan Graphs . . . . . . . . . . . 14.8.6 Combinatorial Constructions of Expanders . . . . . . . . . . . . 14.8.7 The Ihara Zeta Function . . . . . . . . . . . . . . . . . . . . . . Other combinatorial structures Jeffrey H. Dinitz and Charles J. Colbourn 14.9.1 Association Schemes . . . . . . . . . . . . . . . . . . . . . . . . 14.9.2 Costas Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9.3 Conference Matrices . . . . . . . . . . . . . . . . . . . . . . . . 14.9.4 Covering Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9.5 Hall Triple Systems . . . . . . . . . . . . . . . . . . . . . . . . . 14.9.6 Ordered Designs and Perpendicular Arrays . . . . . . . . . . . . 14.9.7 Perfect Hash Families . . . . . . . . . . . . . . . . . . . . . . . . 14.9.8 Room Squares and Starters . . . . . . . . . . . . . . . . . . . . . 14.9.9 Strongly Regular Graphs . . . . . . . . . . . . . . . . . . . . . . 14.9.10 Whist Tournaments . . . . . . . . . . . . . . . . . . . . . . . . . 14.9.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Algebraic coding theory

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15.1 Basic coding properties and bounds Ian Blake and W. Cary 15.1.1 Channel models and error correction . . . . . . . . 15.1.2 Linear codes . . . . . . . . . . . . . . . . . . . . . 15.1.2.1 Standard array decoding of linear codes 15.1.2.2 Hamming codes . . . . . . . . . . . . . 15.1.2.3 Reed-Muller codes . . . . . . . . . . . . 15.1.2.4 Subfield and trace codes . . . . . . . . . 15.1.2.5 Modifying linear codes . . . . . . . . . 15.1.2.6 Bounds on codes . . . . . . . . . . . . . 15.1.2.7 Asymptotic bounds . . . . . . . . . . . 15.1.3 Cyclic codes . . . . . . . . . . . . . . . . . . . . . 15.1.3.1 Algebraic prerequisites . . . . . . . . . 15.1.3.2 Properties of cyclic codes . . . . . . . . 15.1.3.3 Classes of cyclic codes . . . . . . . . . . 15.1.4 A spectral approach to coding . . . . . . . . . . .

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510 510 512 512 514 516 517 517 518 519 520 520 520 521 530 532 532 534 537 539 540 542 545 546 546 546 547 548 549 551 552 553 556 556 558 559 559 559 561 565 566 567 568 569 570 573 574 575 576 577 589

15.2

15.3 15.4

15.5 15.6

16

15.1.5 Codes and combinatorics . . . . . . . . . . . . . . . . . . . . . . 15.1.6 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.6.1 Decoding BCH codes . . . . . . . . . . . . . . . . . . 15.1.6.2 The Peterson-Gorenstein-Zierler decoder . . . . . . . . 15.1.6.3 Berlekamp-Massey decoding . . . . . . . . . . . . . . 15.1.6.4 Extended Euclidean algorithm decoding . . . . . . . . 15.1.6.5 Welch-Berlekamp decoding of GRS codes . . . . . . . 15.1.6.6 Majority logic decoding . . . . . . . . . . . . . . . . . 15.1.6.7 Generalized minimum distance decoding . . . . . . . . 15.1.6.8 List decoding - decoding beyond the minimum distance bound . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.7 Codes over Z4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic-geometry codes Harald Niederreiter . . . . . . . . . . . . . . . 15.2.1 Classical algebraic-geometry codes . . . . . . . . . . . . . . . . . 15.2.2 Generalized algebraic-geometry codes . . . . . . . . . . . . . . . 15.2.3 Function-field codes . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.4 Asymptotic bounds . . . . . . . . . . . . . . . . . . . . . . . . . LDPC codes over finite fields Oscar Takeshita . . . . . . . . . . . . . . . Turbo codes over finite fields Oscar Takeshita . . . . . . . . . . . . . . . 15.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1.1 Historical background . . . . . . . . . . . . . . . . . . 15.4.1.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . 15.4.2 Convolutional codes . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.2.1 Non-recursive convolutional codes . . . . . . . . . . . 15.4.2.2 Distance properties of non-recursive convolutional codes 15.4.2.3 Recursive convolutional codes . . . . . . . . . . . . . . 15.4.2.4 Distance properties of recursive convolutional codes . . 15.4.3 Permutations and interleavers . . . . . . . . . . . . . . . . . . . 15.4.4 Encoding and decoding . . . . . . . . . . . . . . . . . . . . . . . 15.4.5 Design of turbo codes . . . . . . . . . . . . . . . . . . . . . . . . 15.4.5.1 Design of the recursive convolutional code . . . . . . . 15.4.5.2 Design of the interleaver . . . . . . . . . . . . . . . . Polar codes Simon Litsyn . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum codes Harriet Pollatsek . . . . . . . . . . . . . . . . . . . . . .

Cryptography

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

16.1 Introduction Alfred Menezes . . . . . . . . . . . . 16.1.1 Goals of cryptography . . . . . . . . . . 16.1.2 Symmetric-key cryptography . . . . . . . 16.1.2.1 Stream ciphers . . . . . . . . . 16.1.2.2 Block ciphers . . . . . . . . . 16.1.3 Public-key cryptography . . . . . . . . . 16.1.3.1 RSA . . . . . . . . . . . . . . 16.1.3.2 Discrete logarithm cryptosystems 16.1.3.3 DSA . . . . . . . . . . . . . . 16.1.4 Pairing-based cryptography . . . . . . . . 16.1.5 Post-quantum cryptography . . . . . . . 16.2 Stream and block ciphers Guang Gong and Kishan 16.2.1 Basic Concepts of Stream Ciphers . . . . 16.2.2 (Alleged) RC4 Algorithm . . . . . . . . . 16.2.3 WG Stream Cipher . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chand . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gupta . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

590 591 591 592 593 594 594 595 596 597 599 601 603 603 605 608 610 613 614 614 614 614 615 616 618 618 619 620 620 621 622 622 623 624 625 625 625 626 626 627 628 628 629 630 631 633 635 635 637 638

16.2.4 Basic Structures of Block Ciphers . . . . . . . . . . . . . . . . . 16.2.5 RC6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.6 AES (Advanced Encryption Standard) RIJNDAEL . . . . . . . . 16.3 Multivariate cryptographic systems Jintai Ding . . . . . . . . . . . . . . . 16.3.1 The Basics of Multivariate PKCs . . . . . . . . . . . . . . . . . . 16.3.1.1 The Standard (Bipolar) Construction of MPKCS . . . 16.3.1.2 Other Constructions . . . . . . . . . . . . . . . . . . . 16.3.1.3 Implicit Form MPKCs . . . . . . . . . . . . . . . . . . 16.3.1.4 Isomorphism of Polynomials . . . . . . . . . . . . . . 16.3.2 Main Constructions and Variations . . . . . . . . . . . . . . . . . 16.3.2.1 Historical Constructions . . . . . . . . . . . . . . . . . 16.3.2.2 Triangular Constructions . . . . . . . . . . . . . . . . 16.3.2.3 Big-Field Families: Matsumoto-Imai (C ∗ ) and HFE . . 16.3.2.4 Oil and Vinegar (Unbalanced and Balanced) and Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.2.5 UOV as a Booster Stage . . . . . . . . . . . . . . . . 16.3.2.6 Plus-Minus Variations . . . . . . . . . . . . . . . . . . 16.3.2.7 Internally Perturbation . . . . . . . . . . . . . . . . . 16.3.2.8 Vinegar as an external perturbation and Projection . . 16.3.2.9 TTM and Related Schemes: “Lock” or Repeated Triangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.2.10 Intermediate Fields: MFE and `IC . . . . . . . . . . . 16.3.2.11 Odd Characteristics . . . . . . . . . . . . . . . . . . . 16.3.2.12 Other constructions . . . . . . . . . . . . . . . . . . . 16.3.3 Standard Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.3.1 Linearization Equations . . . . . . . . . . . . . . . . . 16.3.3.2 Critical Bilinear Relations . . . . . . . . . . . . . . . . 16.3.3.3 HOLEs (Higher-Order Linearization Equations) . . . . 16.3.3.4 Differential Attacks . . . . . . . . . . . . . . . . . . . 16.3.3.5 Attacking Internal Perturbations . . . . . . . . . . . . 16.3.3.6 The Skew Symmetric Transformation . . . . . . . . . . 16.3.3.7 The Multiplicative Symmetry . . . . . . . . . . . . . . 16.3.3.8 Rank Attacks . . . . . . . . . . . . . . . . . . . . . . 16.3.3.9 MinRank Attacks on Big-Field Schemes . . . . . . . . 16.3.3.10 Distilling Oil from Vinegar and Other Attacks on UOV 16.3.3.11 Reconciliation . . . . . . . . . . . . . . . . . . . . . . 16.3.3.12 Direct attack using polynomial Solvers . . . . . . . . . 16.3.4 The Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Elliptic curve cryptographic systems Andreas Enge . . . . . . . . . . . . . 16.4.1 Cryptosystems based on elliptic curve discrete logarithms . . . . . 16.4.1.1 Key sizes . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1.2 Cryptographic primitives . . . . . . . . . . . . . . . . 16.4.1.3 Special curves . . . . . . . . . . . . . . . . . . . . . . 16.4.1.4 Random curves: point counting . . . . . . . . . . . . . 16.4.2 Pairing based cryptosystems . . . . . . . . . . . . . . . . . . . . 16.4.2.1 Cryptographic pairings . . . . . . . . . . . . . . . . . 16.4.2.2 Pairings and twists . . . . . . . . . . . . . . . . . . . 16.4.2.3 Explicit isomorphisms . . . . . . . . . . . . . . . . . . 16.4.2.4 Curve constructions . . . . . . . . . . . . . . . . . . . 16.4.2.5 Hashing into elliptic curves . . . . . . . . . . . . . . . 16.5 Hyperelliptic curve cryptographic systems Tanja Lange . . . . . . . . . .

642 643 644 648 649 649 650 650 651 651 651 652 653 654 655 656 656 657 657 658 658 658 659 659 659 659 660 660 661 661 662 662 662 663 663 664 666 666 666 666 667 669 670 670 673 674 674 678 680

16.6 Cryptosystems arising from abelian varieties Kumar Murty . . . . . . . . 16.7 Finite field arithmetic in hardware Anwar Hasan . . . . . . . . . . . . . .

17

Miscellaneous applications

. . . . . . . . . . . . . . . . . . . . . . . . . .

17.1 Finite Fields in Biology Franziska Hinkelmann and Reinhard Laubenbacher 17.1.1 Polynomial dynamical systems as framework for discrete models in systems biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2 Polynomial dynamical systems . . . . . . . . . . . . . . . . . . . 17.1.3 Discrete model types and their translation into PDS . . . . . . . 17.1.3.1 Boolean network models . . . . . . . . . . . . . . . . 17.1.3.2 Logical models . . . . . . . . . . . . . . . . . . . . . . 17.1.3.3 Petri nets and agent-based models . . . . . . . . . . . 17.1.4 Reverse engineering and parameter estimation . . . . . . . . . . . 17.1.4.1 The minimal-sets algorithm . . . . . . . . . . . . . . . 17.1.4.2 Parameter estimation using the Gr¨ obner fan of an ideal 17.1.5 Software for biologists and computer algebra software . . . . . . . 17.1.6 Specific polynomial dynamical systems . . . . . . . . . . . . . . . 17.1.6.1 Nested canalyzing functions . . . . . . . . . . . . . . . 17.1.6.2 Parameter estimation resulting in nested canalyzing functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.6.3 Linear polynomial dynamical systems . . . . . . . . . 17.1.6.4 Conjunctive/disjunctive networks . . . . . . . . . . . . 17.2 Finite fields in quantum information theory Arne Winterhof . . . . . . . . 17.3 Finite fields in engineering Jonathan Jedwab and Kai-Uwe Schmidt . . . . 17.3.1 Binary sequences with small aperiodic autocorrelation . . . . . . 17.3.2 Sequence sets with small aperiodic auto- and crosscorrelation . . . 17.3.3 Binary Golay sequence pairs . . . . . . . . . . . . . . . . . . . . 17.3.4 Optical orthogonal codes . . . . . . . . . . . . . . . . . . . . . . 17.3.5 Sequences with small Hamming correlation . . . . . . . . . . . . 17.3.6 Rank distance codes . . . . . . . . . . . . . . . . . . . . . . . . 17.3.7 Space-time coding . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.8 Coding over networks . . . . . . . . . . . . . . . . . . . . . . . .

Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

681 682 683 683 683 684 685 685 687 688 689 689 689 689 690 690 692 692 692 693 694 694 695 695 696 698 699 699 701 823

Actual appearance, pages 290–302

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9.7

243

Schur’s conjecture and exceptional covers

.

Michael D. Fried,

9.7.1

University of California Irvine

Rational function definitions

9.7.1 Remark (Extend values) The historical functions of this section are polynomials and rational

functions: f (x) = Nf (x)/Df (x) with Nf and Df relatively prime (nonzero) polynomials, denoted f ∈ F (x), F a field (almost always Fq or a number field). The subject takes off by including functions f – covers – where the domain and range are varieties of the same dimension. Still, we emphasize functions between projective algebraic curves (nonsingular), often where the target and domain are projective 1-space. 9.7.2 Definition The degree of f ∈ F (x), deg(f ), is the maximum of deg(Nf ) and deg(Df ).

Add a point at ∞ to F , F ∪ {∞} = P1x (F ), to get the F points of projective 1-space. 9.7.3 Remark (Plug in ∞) Using Definition 9.7.2 requires plugging in and getting out ∞. We

sometimes use the notion of value sets Vf and their cardinality #Vf (Section 8.3). 1. The value of f (x0 ) for x0 ∈ F is ∞ if x0 is a zero of Df (x). 2. The value of f (∞) is respectively ∞, 0, or the ratio of the Nf and Df leading coefficients, if the degree of Nf is greater, less than, or equal to the degree of Df . If z is a variable indicating the range, this gives f as a function from P1x (F ) to P1z (F ). We abbreviate this as f : P1x → P1z . 9.7.4 Definition (M¨ obius equivalence) Denote the group – under composition – of

M¨ obius transformations x 7→ ax+b with ad − bc = 6 0, a, b, c, d ∈ F by PGL(F ). Refer to cx+d f1 , f2 ∈ F (x) as M¨ obius equivalent if f2 = α ◦ f1 ◦ β for α, β ∈ PGL(F ).

9.7.5 Example If f (x) = xn , with gcd(n, q − 1) = 1, then #Vf = q k + 1 on P1x (Fqk ) exactly for

those infinitely many k with gcd(n, q k − 1) = 1.

9.7.6 Remark Initial motivation came from Schur’s Conjecture Thm. 9.7.32, which starts over a

number field K – a finite extension of Q, the rational numbers – with its ring of integers OK . That asks about Vf over residue class fields, OK /pp of prime ideals p , denoting this Vf (O/pp) (Vf (Fp ) if O = Z). Assume Nf and Df have coefficients in OK . Avoid p – it is a bad prime – if it contains the leading coefficient of either Nf or Df . 9.7.7 Definition For f ∈ F (x), if f = f1 ◦ f2 with f1 , f2 ∈ F (x), deg(fi ) > 1, i = 1, 2, we say f

decomposes over F . Then, the fi s are composition factors of f . 9.7.8 Definition (Cofinite) For B a subset of A, we say B is cofinite in A if A \ B is finite. 9.7.9 Proposition [1638, p. 390] Consider Xh0 = {(x, y) | h(x, y) = 0}, an algebraic curve, defined

by h ∈ K[x, y]. Then, there is a unique nonsingular curve Xh – the normalization of Xh0 – and a morphism µh : Xh0 → Xh that is an isomorphism on the complement of a finite subset of points in Xh . Indeed, every variety Xh0 has such a unique normalization, but in higher dimensions it may be singular, and µh is an isomorphism off a codimension 1 set.

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9.7.10 Definition (Components) A definition field for an algebraic set W is a field containing all

coefficients of all polynomials defining W . Components of W over F are algebraic subsets which are not the union of two closed non-empty proper algebraic subsets over F [1054, p. 3]. We say W is a variety if it has just one component. It is absolutely irreducible if it has just one component over F¯ , an algebraic closure of F. 9.7.11 Remark (Points on varieties) [1054, Chapters 1 and 2] and [1638, §2] introduce affine and

projective algebraic sets, and their components (Definition 9.7.10), except they are over an algebraically closed field. For perfect fields F (including finite fields and number fields) this extends for normal varieties. Since their components do not meet, taking any disjoint union of distinct varieties under the action of the absolute Galois group of F defines components in general. Points on an algebraic set X over F refers here to geometric points: points with coordinates in F¯ . It is an F point if its coordinates are in F . 9.7.12 Definition A general f : X → Z is a cover means it is a finite, flat morphism (see Definition

9.7.25) of quasi-projective varieties [1638, p. 432, Proposition 2]. 9.7.13 Lemma Definition 9.7.12 simplifies for curves, because all our varieties will be normal, and

so for curves, nonsingular. Then, any nonconstant morphism is a cover: That includes any nonconstant rational function f : P1x → P1z . 9.7.14 Example If f : X → Z is finite and X and Z are nonsingular, generalizing what happens

for curves, and no matter their dimension, then f is automatically flat [1054, p. 266, 9.3a)]. This doesn’t extend to weakening nonsingular to normal varieties. [1638, p. 434] has a finite morphism, where X is nonsingular (it is affine 2-space), and Z is normal. But, the fiber degree is 2 over each z ∈ Z, excluding one point where it is 3. 9.7.15 Remark (Assuming normality) Starting with §9.7.2 all results assume that the algebraic

sets are normal. Some constructions (especially Def. 9.7.45) momentarily produce nonnormal sets, that we immediately replace with their normalizations.

9.7.2

MacCluer’s Theorem and Schur’s Conjecture

9.7.16 Definition An f ∈ Fq (x) is exceptional if it maps one-one on P1 (Fqk ) for infinitely many

k. Similarly, with K a number field, f ∈ K(x) is exceptional if it is exceptional mod p for infinitely many primes p . 9.7.17 Remark We use K, allowing decoration, for a number field. §8.3 refers to the splitting field,

Ωf (resp. Ωf F¯ ), of f (x) − z over F (z) (resp. over F¯ (z)). The automorphism group of the extension Ωf /F (z) (resp. Ωf,F¯ /F¯ (z)) is the arithmetic (resp. geometric) monodromy group A (resp. G) of a separable function (Definition 9.7.25) f ∈ F (x). When there are several functions, we denote these Af and Gf . They act on the zeros, {x1 , . . . , xn } (often denoted {1, . . . , n}), of f (x) − z, giving a natural permutation representation on n symbols. 9.7.18 Definition Every cover f : X → Z over a field F with X irreducible has an associated

extension of function fields that determines the cover up to birational morphisms (see Lemma 9.7.43). 9.7.19 Remark Essentially all the Galois theory of fields translates to useful statements about a

cover f : X → Z (over F ) of an irreducible variety Z. It does this by corresponding to f the

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composite of the function field extensions F (X 0 )/F (Z) where X 0 runs over the components of X [1638, p. 396]. Several papers in our references (say, [842, §0.C]) give oft-used examples, with Lemma 9.7.20 a simple archetype. 9.7.20 Lemma (See Remark 9.7.21) Any separable cover f : X → Y over F has a Galois closure

ˆ → Z over F . Then, Af is the group of fˆ with its natural permutation reprecover fˆ : X sentation TAf (of degree the degree of f ). Do this over F¯ to get the geometric monodromy Gf . Then, X is irreducible (resp. absolutely irreducible) if and only if TAf (resp. TGf ) is transitive. For f a rational function it is automatic that TGf (and so TAf ) is transitive. 9.7.21 Remark [846, §2.1] explains how to form the Galois closure cover of a cover using fiber

products (see Remark 9.7.54). This shows how to form the Galois closure cover of any collection of covers as in Lemma 9.7.50. 9.7.22 Remark Normalization gives a nearly invertible process to Remark 9.7.19: going from field

extensions of F (Z) to covers of Z. While this doesn’t translate all arithmetic cover problems to Galois theory, we apply the phrase “monodromy precision” (Remark 9.7.26) to when it does. Example: It does in the topic of exceptional covers, as in Proposition 9.7.28. 9.7.23 Definition Denote the elements of a group G, under a representation TG , that fix 1 by G(1).

When TG is transitive, refer to TG as primitive (resp. doubly transitive) if there is no group properly between G(1) and G (resp. G(1) is transitive on {2, . . . , n}). 9.7.24 Theorem [840, Theorem 1]: An f ∈ Fq (x) is exceptional if and only if the following holds

for each orbit O of Af (1) on {2, . . . , n}: O breaks into strictly smaller orbits under Gf (1).

(9.23)

(y) Denote the projective normalization of {(x, y) | f (x)−f = 0} by Xf,f \ ∆. Also equivalent x−y to (9.23): Each Fq component of Xf,f \ ∆ has at least 2 components over F¯q . Similarly, an f ∈ K(x), K a number field, is exceptional if and only if (9.23) holds for f mod p for infinitely many primes p .

9.7.25 Definition (Covers) Let f ∈ Fq (x) be nonconstant and separable: not g(xp ) for some

g ∈ Fq (x). Then, f : P1x (F¯q ) → P1z (F¯q ) by x 7→ f (x) has these cover properties.

1. Excluding a finite set {z1 , . . . , zr } ⊂ P1z (F¯q ), branch points of f , there are exactly n = deg(f ) points over z 0 . 2. For z 0 a branch point, counting zeros, x0 of f (x) − z 0 with multiplicity, the sum at all x0 s over z 0 is still n. An x0 ∈ P1x with multiplicity > 1 is a ramified point. For K a number field, the same properties hold, without any separable condition. 9.7.26 Remark (MacCluer’s Theorem) Theorem 9.7.24 has a surprise: (9.23) implies exceptionality

over Fq . An error term in applying Chebotarev’s density theorem with branch points (as in §8.3, in §8.3.3) vanishes. A ramified point with p not dividing its multiplicity is tame. Macluer’s thesis [1472] responded to a Davenport-Lewis conjecture [577] by showing Theorem 9.7.24 for a polynomial tame at every point. We say: MacCluer’s Theorem shows tame polynomial exceptional covers exhibit monodromy precision [847, §3.2.1]. Proposition 9.7.28 shows monodromy precision holds for general exceptional covers. 9.7.27 Example A polynomial f over Fq for which p| deg(f ) is not tame at ∞. 9.7.28 Proposition [840] combined with [846, Principle 3.1]: Let f : X → Z be any cover (Defi-

nition 9.7.12) over Fq with X absolutely irreducible. Then [846, Corollary 2.5]:

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1. the extended meaning of (9.23) is that the 2-fold fiber product (§9.7.3) of f minus the diagonal has no absolutely irreducible Fq components; and 2. (9.23) is equivalent to f being exceptional: X(Fqk ) → Y (Fqk ) is one-one (and onto) for infinitely many k. 9.7.29 Remark As noted in [846, Comments on Principle 3.1], the proof of [840] applies without

change to give Proposition 9.7.28 Part 2 when X and Z are non-singular; indeed, it applies to pr-exceptionality (Definition 9.7.93). Without, however, this nonsingularity assumption, there are complications considered in [847, §A.4.1] (see Example 9.7.14). 9.7.30 Definition Let f in Proposition 9.7.28 over Fq be an exceptional cover. Denote values k

where (9.23) holds with Fqk replacing Fq , by Ef,q : the exceptionality set of f . Similarly, for f satisfying the hypotheses of Proposition 9.7.28 over a number field K, denote those primes p where f mod p has Ef,O/pp infinite, by Ef,K . 9.7.31 Definition The equation Tu (cos(θ)) = cos(uθ) defines the u-th Chebychev polynomial, Tu .

From it define a Chebychev conjugate: α ◦ Tu ◦ α−1 with α(x) = αz0 (x) = z 0 x and either z 0 = 1, or z 0 and −z 0 are conjugate in a quadratic extension of K. 9.7.32 Theorem (Schur’s Conjecture) [837, Theorem 2]: With K a number field, the f ∈ O[x]

for which Ef,K is infinite are compositions with maps a 7→ ax + b (affine) over K with polynomials of the following form for some odd prime u: xu (cyclic) or, α ◦ Tu ◦ α−1 , u > 3, a Chebychev conjugate.

(9.24)

9.7.33 Remark Many still refer to Theorem 9.7.32 as Schur’s Conjecture, though Schur conjectured

it only over Q. [837] refers to all Chebychev conjugates as Chebychev polynomials, rather than Dickson as in Remark 9.7.34. [1441] assiduously distinguishes Dickson polynomials. Here is a simple branch point Chebychev Conjugate characterization: f has two finite ¯ which identify with the unique unramified points (in (6= ∞) branch points, ±z 0 ∈ P1z (Q), 1 ¯ Px (Q)) over the branch points, as in [837, Proof of Lemma 9]. 1. A corollary of [853, Theorem 3.5] is that any cover with a unique totally and tamely ramified point decomposes over F if and only if it decomposes over F¯ . This applies if f ∈ F [x] has deg(f ) prime to the characteristic of F . 2. If f from Part 1 is indecomposable, then Gf is primitive (see Definition 9.7.23) and it contains an n-cycle. 3. If f ∈ K[x] is exceptional, since (9.23) says Gf cannot be doubly transitive, up to composing with K affine maps, f from Part 2 is in (9.24). 9.7.34 Remark (Dickson doppelgangers, see §9.6) Each Chebychev conjugate is a constant times

a Dickson polynomial [846, Proposition 5.3]. The Remark 9.7.33 characterization – by locating their branch points – avoids using equations. That is the distinction at the last step between the proof of Theorem 9.7.32 and [1441, Chapter 6]. 9.7.35 Remark Use the notation in Theorem 9.7.32. Suppose f ∈ OK [x] is an exceptional poly-

nomial. Define nf,c (resp. nf,C ) to be the product of distinct primes s for which f has a degree s cyclic (resp. Chebychev conjugate) composition factor. The referee of [1507] noted Corollary 9.7.36 follows from 9.7.28 combined with 9.7.32. 9.7.36 Corollary For f ∈ OK [x] an exceptional polynomial, one can determine Ef,K (excluding

bad primes, Remark 9.7.6) from nc,f and nf,C by congruences. When OK = Z, then p ∈ Ef,Q if and only if gcd(p − 1, s) = 1 for each s|nf,c and gcd(p2 − 1, s) = 1 for each s|nf,C .

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9.7.37 Example (Infinite Ef,Q ) It is necessary that gcd(2, nc ) = 1 and gcd(6, nC ) = 1 for there

to be infinitely many p that satisfy the conclusion of Corollary 9.7.36. But it is sufficient, too. Without loss, assume gcd(nc , nC ) = 1. If 36 | nc , then Dirichlet’s Theorem on primes in arithmetic progressions gives an infinite set of p ≡ 3 mod nc nC . They are in Ef,Q . If 3|nc , the Chinese remainder theorem gives an arithmetic progression of p satisfying p ≡ 3 mod nC and p ≡ −1 mod nc . So, Ef,Q is infinite whenever it has a chance to be. 9.7.38 Remark Combine [841, Lemma 1] with monodromy precision in Proposition 9.7.39. This

shows, the Proposition 9.7.28 fiber product statement is equivalent to f ∈ K(x) being exceptional, and therefore permutation, mod p . If OK /pp is sufficiently large, the fiber product statement is also necessary for f to be permutation (well-known, for example [837, proof of Theorem 2, last paragraph]). 9.7.39 Proposition (Permutation functions) From Remark 9.7.38, for f ∈ Fq (x), those k where f

permutes P1 (Fqk ) contains Ef,Fq as a cofinite subset. Similarly, for K a number field, those p where f functionally permutes P1 (O/pp) contains Ef,K as a cofinite subset. 9.7.40 Remark §8.1 shows permutation polynomials are abundant. Exceptional polynomials sat-

isfy a much stronger property, but Corollary 9.7.36 shows they are abundant, too. One difference: §9.7.3 combines them in ways with no analog for permutation polynomials. 9.7.41 Corollary An analog of Theorem 9.7.32 holds over Fq to characterize exceptional polyno-

mials of degree prime to p ([848, Introduction to §5] or [846, Proposition 5.1]). There, z 0 in αz0 is either 1 or in the unique quadratic extension of Fq . Consider a Chebychev conjugate αz0 ◦ Tn ◦ αz−1 as a permutation polynomial on Fqk with gcd(q 2k − 1, n) = 1. Then, when 0 n · m ≡ 1 ( mod q 2k − 1), αz0 ◦ Tm ◦ αz−1 is its functional inverse. 0

9.7.3

Fiber product of covers

9.7.42 Definition For any field extension F1 /F2 containing Fp , there is the notion of being sep-

arable [849, p. 111]. For f ∈ Fq (x), the extension F¯q (x)/F¯q (f (x)) being separable is equivalent to f is separable (Definition 9.7.25). Many of our examples inherit separableness from this special case.

9.7.43 Lemma (Curve covering maps [1054, Chapter I, §6]) Any nonsingular projective algebraic

curve X over a perfect field F has a field of functions F (X) that uniquely determines X up to isomorphism over F . Each non-constant element f ∈ F (X) determines a finite map X → P1z over F [1054, Chapter I, Exercise 6.4]. If F (X)/F (f ) is separable, then f has the covering properties of (9.7.25): finite number of branch points, and uniform count of points in a fiber over F¯ (including multiplicity in the fiber) [1054, Chapter IV, Proposition 2.2]. 9.7.44 Definition Refer to any f in the conclusion of Lemma 9.7.43 as a nonsingular cover of P1z . 9.7.45 Definition (Fiber product) Let fi : Xi → P1z , i = 1, 2, be two nonsingular covers of P1z . The

set theoretic fiber product consists of the algebraic curve {(x1 , x2 ) ∈ X1 × X2 |f1 (x1 ) = f2 (x2 )}. Denote this X1 ×set P1z X2 . Its normalization (Proposition 9.7.9), X1 ×P1z X2 , is the fiber product of f1 and f2 .

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9.7.46 Remark Definition 9.7.45 works equally for any covers Xi → Z, i = 1, 2, with Z a normal

projective variety. Then, X1 ×Z X2 is normal and projective (possibly with several components) with natural maps pri : X1 ×Z X2 → Xi , i = 1, 2, given by its projection on each factor. The functions fi ◦ pri , i = 1, 2 are identical, giving a well-defined map: (f1 , f2 ) : X1 ×Z X2 → Z.

(9.25)

9.7.47 Remark (Fiber equations) Consider x0 ∈ X1 ×Z X2 that is simultaneously over x0i ∈ Xi ,

i = 1, 2, where both x0i s ramify over pr1 (x01 ) = pr2 (x02 ). Then, f1 (x1 ) = f2 (x2 ), with xi in a neighborhood of x0i , is not a correct local description around x0 . There is another complication when Z is not a curve (dimension 1). The fiber product might be singular even when the Xi s are not. So (f1 , f2 ) in (9.25) may not be a cover because it is not flat (Remark 9.7.67).

9.7.48 Example Consider two polynomials, f1 , f2 ∈ K[x], of the same degree n. They define

fj : P1xj → P1z , j = 1, 2. Then, there are n points over z = ∞ on P1y1 ×P1z P1y2 , but only one point on the set theoretic fiber product over ∞. [839, Proposition 1] gives the generalization of this, showing – when the covers are tame – how to compute the genus of the fiber product components from the covers fj , j = 1, 2. 9.7.49 Definition The fiber product Xf,f = X ×Z X for a cover f : X → Z of degree exceeding 1

has at least two components. One is the diagonal : the set ∆(X) = {(x, x) | x ∈ X}. The normal variety X ×Z X \ ∆(X) generalizes the set in Theorem 9.7.24. 9.7.50 Lemma (Fiber product monodromy [846, §2.1.3]) Consider the covers in Definition (9.7.45).

To each fj there is an arithmetic (resp. geometric) monodromy group Afj (resp. Gfj ), j = 1, 2. Similarly, for (f1 , f2 ) in (9.25). Then, A(f1 ,f2 ) maps naturally, surjectively, to Afj by homomorphisms pr∗j , j = 1, 2. There is a largest simultaneous quotient, H, of both Afj s given by homomorphisms mi : Afj → H, j = 1, 2, so that A(f1 ,f2 ) = {(σ1 , σ2 ) ∈ Af1 × Af2 | m1 (σ1 ) = m2 (σ2 )}. Similarly with geometric replacing arithmetic monodromy. 9.7.51 Corollary (Components) With the hypotheses of Lem. 9.7.50, let {1j , . . . , nj }, be integers

on which Aj acts, j = 1, 2. Then, A(f1 ,f2 ) acts on the pairs (i1 , i2 ) and on each of the sets {1j , . . . , nj } separately. If X1 is absolutely irreducible, then the components of X1 ×Z X2 over F (resp. F¯ ) correspond to the orbits of A(f1 ,f2 ) (11 ) (resp. G(f1 ,f2 ) (11 ); see Definition 9.7.23) on {12 , . . . , n2 }. Note: The degrees n1 and n2 may be different. 9.7.52 Definition (Absolute components) Given X1 ×Z X2 in Corollary 9.7.51, denote the union of

its absolutely irreducible F components by X1 ×abs Z X2 . Denote the complementary cp set, X1 ×Z X2 \ X1 ×abs Z X2 , of components by X1 ×Z X2 . 9.7.53 Theorem (Explicit Ef,q – see Remark 9.7.54) Let f : X → Z (as in Proposition 9.7.28) be

an exceptional cover over Fq . For Xi0 , an Fq component of X ×cp Z X, denote the number of components in its breakup over F¯q by si , i = 1, . . . , u. With sexc = lcm(s1 , . . . , su ), Ef,q = {k mod sexc | gcd(k, si ) < si , i = 1, . . . , u}.

The group G(Fqsexc /Fq ) is naturally a quotient of Af /Gf . We can interpret all quantities using Af and Gf .

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9.7.54 Remark All but the last sentence of Theorem 9.7.53 is [846, Corollary 2.8]. The last sentence

is from [846, Lemma 2.6], using that the Galois closure cover of f is a(ny) component (over Fq ) of the deg(f ) = n-fold fiber product of f with itself. Project that fiber product onto the 2-fold fiber product of f over Fq to finish. Corollary 9.7.51 shows the orbit lengths of Af (1) on {2, . . . , n} divided by the corresponding orbit lengths of Gf (1), give the si s. 9.7.55 Theorem (Explicit Ef,K – see Remark 9.7.56): Now change Fq to K (number field) in the

first sentence of Theorem 9.7.53. For each cyclic subgroup C ≤ Af /Gf denote those σ ∈ Af that map to C by AC . As previously, denote the stabilizers of 1 in the representation by AC (1) and GC (1). Consider this set, Cf,K , of cyclic C (as in (9.23)): {C | each orbit of AC (1) on {2, . . . , n} breaks into strictly smaller orbits under GC (1)}. Then, f is exceptional over K if and only if Cf,K is nonempty. Further, Ef,K consists of those primes p for which the Frobenius attached to p is a generator of some C ∈ Cf,K . 9.7.56 Remark Theorem 9.7.55 comes from applying [841, §2] exactly as in Remark 9.7.28. If Ef,K

is infinite, then X ×Z X \ ∆(X) has no absolutely irreducible component. The converse, however, does not hold.

9.7.4

Combining exceptional covers; the (Fq , Z) exceptional tower

9.7.57 Definition (Category of exceptional covers) For Z absolutely irreducible over Fq , denote

the collection of exceptional covers of Z over Fq by TZ,Fq . 9.7.58 Theorem [846, §4.1]: Given (fi , Xi ) ∈ TZ,Fq , i = 1, 2, X1 ×abs Z X2 (Definition 9.7.52) has

one component. We conclude that: (f1 ◦ pr1 , X1 ×abs Z X2 ) ∈ TZ,Fq . Also, there is at most one morphism between any two objects in TZ,Fq . 9.7.59 Remark (When f1 = f2 in Theorem 9.7.58) We definitely include the fiber product of

a cover in TZ,Fq with itself. Then, the only absolutely irreducible component of the fiber product is the diagonal (Definition 9.7.49), which is equivalent to the original cover. 9.7.60 Definition We call X1 ×abs Z X2 the fiber product of f1 and f2 in TZ,Fq , and continue to

denote its morphism to Z by (f1 , f2 ). This defines TZ,Fq as a category with fiber products. Theorem 9.7.53 shows Ef1 ,q ∩ Ef2 ,q = E(f1 ,f2 ),Fq is infinite. 9.7.61 Remark Consider (fi , Xi ) ∈ TZ,Fq , i = 1, 2, for which there exists ψ : X1 → X2 over Fq

that factors through f2 : f2 ◦ ψ = f1 . Then, Theorem 9.7.58 says ψ is unique. 9.7.62 Corollary For (f, X) ∈ TZ,Fq , denote the group of the Galois closure cover of f over X by

Af (1). Then, Af has the representation Tf by acting on cosets of Af (1). If (fi , Xi ) ∈ TZ,Fq , i = 1, 2, we write (f1 , X1 ) > (f2 , X2 ) if f1 factors through X2 . [846, Prop. 4.3] produces from these pairs a canonical group AZ,Fq with a profinite permutation representation TZ,Fq . 9.7.63 Remark (A projective limit) Given (fi , Xi ) ∈ TZ,Fq , i = 1, 2, there is a 3rd (f, X) ∈ TZ,Fq ,

given by the fiber product, that factors through both. This is the condition defining a projective sequence. So, AZ,Fq in Corollary 9.7.62 is a projective limit.

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9.7.64 Definition (AZ,Fq , TZ,Fq ) is the (arithmetic) monodromy group, in its natural permutation

representation, of the Exceptional Tower TZ,Fq . 9.7.65 Theorem Let fi : Xi → Z, i = 1, 2, be exceptional covers over K: Efi ,K is infinite, i = 1, 2.

Then, X1 ×Z X2 is exceptional in the sense that X1 mod p ×abs Z mod p X2 mod p is exceptional for infinitely p if and only if Ef1 ,K ∩ Ef2 ,K is infinite. 9.7.66 Remark Theorem 9.7.65 forces considering if there is an infinite intersection of two excep-

tionality sets over K. As Theorem 9.7.53 shows, this is automatic over Fq . Example 9.7.37 shows it is not automatic over a number field. §9.7.5 and §9.7.6 have examples along these lines: If both fi s, i = 1, 2, are exceptional rational functions, then their composition is again exceptional over K if and only if Ef1 ,K ∩ Ef2 ,K is infinite. Beyond cyclic and Chebychev situations, it is very difficult to decide when this intersection is infinite. 9.7.67 Remark The same definition for exceptional works for any finite, surjective, map of normal

varieties over Fq . Such maps may not be flat (say, when Remark 9.7.14 doesn’t apply), so they may not be covers. Normalization of any projective variety is projective: Segre’s Embedding [1638, Thm. 4, p. 400]. For irreducible X, flatness says the multiplicity sum of points in the fiber over z is constant in z: the function field extension degree, [K(X) : K(Z)] [1638, Proposition 2, p. 432]. That is, Definition 9.7.25, Part 2, holds. For finite morphisms that characterizes flatness [1638, Corollary p. 432]. With normality, but not flatness, this may hold only outside a codimension 2 set in the target. [847, Appendix A.4] has a liesurely discussion. See Example 9.7.14.

9.7.5

Exceptional rational functions; Serre’s Open Image Theorem

9.7.68 Definition Definition 9.7.31 explains Chebychev conjugates. Consider lz 0 : x 7→

x−z 0 x+z 0 ,

mapping ±z 0 to 0, ∞, with a = (z 0 )2 ∈ K, z 0 ∈ 6 K. Then, for n odd, characterize Rn,a = (lz0 )−1 ◦ (lz0 (x))n , a cyclic conjugate, by these conditions: ±z 0 are its sole ramified points, Rn,a (±z 0 ) = ±z 0 and it maps ∞ 7→ ∞.

(9.26)

9.7.69 Remark According to [1441, Chapter 2, §5]), Rn,a in Definition 9.7.68 is a Redei function.

From [1441, Theorem 3.11]: Under the hypotheses on z 0 , the exceptionality set ERn,a ,K is {pp | (|OK /pp| − 1, n) = 1} {pp | (|OK /pp| + 1, n) = 1}

if z 0 is a quadratic residue if not.

mod p , and

9.7.70 Remark (Addendum Remark 9.7.69) Quadratic reciprocity determines nonempty arithmetic

progressions for which z 0 is a quadratic residue and those for which it is not. If z 0 in Definition 9.7.68 were in K, then – of course – the exceptional set is the same as for xn . Whether or not z 0 ∈ K, we refer to Rn,a as a cyclic conjugate. 9.7.71 Definition [846, §4.2] Suppose a collection C of covers from an exceptional tower TY,Fq is

closed under the categorical fiber product. We say C is a subtower. We also speak of the (minimal) subtower any collection generates under fiber product.

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9.7.72 Remark [846, §4.3] uses that the fiber product of two unramified covers is unramified to cre-

ate cryptographic exceptional subtowers. [846, §5.2.3] computes the arithmetic monodromy attached to the Dickson subtower generated by all the exceptional Chebychev conjugates over Fq . The analog of Remark 9.7.69 over Fq gives a similar – Redei – subtower of TP1z ,Fq generated by exceptional cyclic conjugates. 9.7.73 Remark Theorem 9.7.65 requires common exceptional intersection (Remark 9.7.66) to form

fiber products in TZ,K , Z absolutely irreducible over a number field K. For fiber products (or composites) of Chebychev and cyclic conjugates, we easily decide if exceptional sets have infinite intersection. Exceptional rational functions from Serre’s O(pen) I(mage) T(heorem) give much harder versions of such problems. 9.7.74 Definition (j-line P1j ) A special copy of projective 1-space, the j-line, occurs in the study

¯ = A1 (Q) ¯ has an of modular curves (see Theorem 9.7.76). Each j ∈ P1j \ {∞}(Q) j attached isomorphism class of elliptic curves Ej . For each integer n > 0, consider a special case of a modular curve, µ0 (n) : X0 (n) → P1j , with its cover of P1j . Denote the points of X0 (n) not lying over j = ∞ by Y0 (n).

9.7.75 Definition For E an elliptic curve, denote by E → E/C an isogeny from quotienting E

by a (finite) torsion subgroup C of E. When C is a cyclic, generated by e0 ∈ E (resp. all torsion points killed by multiplication by n), write C = he0 i (resp. Cn ). 9.7.76 Theorem There are two approaches to giving “meaning” to each algebraic point y ∈ Y0 (n),

whose image in P1j is jy . 1. [1718, p. 108] or [843, p. 158]: y 7→ [Ejy → Ejy /he0y i] with e0y ∈ Ejy of order n where brackets, [ ] , indicate an isomorphism class of isogenies. ¯ 2. [843, Lemma 2.1]: y 7→ fy ∈ Q(x) (up to M¨obius equivalence) of degree n. 9.7.77 Theorem [843, Theorem 2.1]: Suppose f ∈ K(x) is exceptional and of prime degree u.

Then, f is M¨ obius equivalent over K to either: 1. a cyclic (Remark 9.7.70) or a Chebychev (Remark 9.7.34) conjugate; 2. or to some fy (u = n) in Theorem 9.7.76, Part 2. 9.7.78 Definition For a dense set of j 0 ∈ A1j , we say the corresponding Ej 0 is of CM -type if its ring

of isogenies, tensored by Q, has dimension 2 over Q. Such isogenies form a complex quadratic extension of Q (containing j 0 , which is an algebraic integer; [1904, II-28] or [1928, Chapter 2, §5.2]). Otherwise, j 0 is of GL2 -type. 9.7.79 Theorem [843, (2.10)]: Continue the notation of Theorem 9.7.77. Except for the two cases

where jy is one of the two finite branch points of µ0 (u), the geometric monodromy Gfy is the order 2u dihedral group Du , and fy has four branch points (Definition 9.7.25). For u in Theorem 9.7.77, Part 2, for which Ej 0 has good reduction, the coordinates of e0y generate a constant extension of K with group Afy /Gfy (explained in Theorem 9.7.85). 9.7.80 Theorem [843, §2.B]: For j 0 of CM-type, complex multiplication theory gives (an infinite)

Efy ,K . Computing this would use [1010, §6.3.1-§6.3.2]. 9.7.81 Remark (Addendum to Theorem 9.7.80) Using adelic (modular) arithmetic gives analogs

of Corollary 9.7.36; and Corollary 9.7.41 for explicitly finding the functional inverse of a CM-type reduced modulo a prime in the exceptional set Efy ,K . If K = Q(j 0 ), then Efy ,K depends on the congruence defining the Frobenius in the (cyclic of degree u−1 over K)

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constant field. Only finitely many j 0 in Q have CM-type, corresponding to class number 1 for complex quadratic extensions. 9.7.82 Problem Take one of the CM-type j s in Q. Then, consider two allowed values of u, ui ,

i = 1, 2, denoting the corresponding fy s by fi , i = 1, 2. Test for explicitness in Remark 9.7.81 as to whether Ef1 ,Q ∩ Ef2 ,Q is infinite. 9.7.83 Definition (Composition factor definition field ) For f ∈ F (x) consider a minimal field

Ff (ind) over which f decomposes into composition factors indecomposable over F¯ . Similarly, denote the minimal field over which Xf,f \ ∆ in Theorem 9.7.24 breaks into absolutely irreducible components by Fˆf (2). 9.7.84 Proposition [846, Proposition 6.5]: If f : X → Z is a cover over F , then Ff (ind) ⊂ Fˆf (2). 9.7.85 Theorem See Remark 9.7.87: Assume j 0 ∈ A1j is of GL2 -type. For K = Q(j 0 ), consider

C = Cu in Definition 9.7.75 with u a prime. The corresponding fu ∈ K(x) has degree u2 . Use the monodromy groups of Definition 9.7.17. There is a constant M1,j 0 so that if u > M1,j 0 , then the arithmetic/geometric monodromy quotient Afu /Gfu is GL2 (Z/u)/{±1}. Further, fu decomposes into two degree u rational functions over Kf (ind), but it is indecomposable over K. 9.7.86 Theorem [846, Proposition 6.6]: Continue Theorem 9.7.85 hypotheses. For a second constant

M2,j 0 , and for any prime p of OK with |OK /pp| > M2,j 0 assume Ap ∈ GL2 (Z/u)/h±1i represents the conjugacy class of the Frobenius for p . Then, fu mod p is an exceptional indecomposable rational function, and it decomposes over the algebraic closure of OK /pp, precisely when hAp i acts irreducibly on (Z/p)2 = Vp . This holds for infinitely many primes p . In particular, fu is exceptional over K (Definition 9.7.30). 9.7.87 Remark (Using Serre’s OIT ) [1904] lays the groundwork for [1905]. The latter has the

¯ existence of the constant M1,j 0 . [1904, App. A.1, §3.2] proves it exists when j 0 ∈ A1j (Q) is not an algebraic integer. Then, the computation of Mi,j 0 , i = 1, 2, in Theorems 9.7.85 and 9.7.86 is effective. Even after all these years, there is no effective computation of these constants when j is not CM-type, but is an algebraic integer. [843, §2] gets Theorem 9.7.85 from the OIT using the relation between Parts 1 and 2 in Theorem 9.7.76. 9.7.88 Remark (More elementary, but less precise, Theorem 9.7.86) [843, Theorem 2.2] shows,

for every K and any prime u > 3, the j 0 ∈ K, with fu satisfying the exceptionality and decomposability conclusions of Theorem 9.7.86, are dense. Applying the [841, Theorem 3] (or [849, Theorem 12.7]) version of Hilbert’s Irreducibility Theorem to X0 (u) gives the corresponding M2,j 0 explicitly. 9.7.89 Example (M1,j 0 effectiveness?) [1904, App. A.1, §3.3] gives Ogg’s example [1717] with

j 0 ∈ Q. [846, §6.2.2] reviews this case, where M2,j 0 = 6, to show how to pick an Ap acting irreducibly as in Theorem 9.7.86 (for infinitely many p ), assuring that Efu ,Q is infinite for u > M2,j 0 . [846, §6.3.2] – still Ogg’s case – aims at finding an automorphic function, a la Langland’s Program, that would characterize the primes in Efu ,Q . This is akin to the unrelated examples of [1913], but uses results on automorphic functions in [1908, Theorem 22]. Primes of Efu ,Q do not lie in arithmetic progressions. So, Problem 9.7.90 is much harder than Problem 9.7.82. 9.7.90 Problem (Analog of Problem 9.7.82) For the Ogg curve in Example 9.7.89, consider two

allowed values of u, ui , i = 1, 2, denoting the corresponding fy s by fi , i = 1, 2. Test for explicitness in Remark 9.7.81 as to whether Ef1 ,Q ∩ Ef2 ,Q is infinite.

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253

9.7.91 Remark [852] connects “variables separated factors” of Xf,f and composition factors of f .

[61] et. al. used this to effectively test for composition factors (and primitivity) of covers. 9.7.92 Theorem [1010, Chapter 3]: Excluding finitely many degrees, all indecomposable excep-

tional f ∈ K(x) (K a number field) are M¨obius equivalent to a cyclic or Chebychev conjugate, or to a CM function from Theorem 9.7.77 of prime degree; or they are from Theorem 9.7.86 and of prime degree squared.

9.7.6

Davenport pairs and Poincar´ e series

9.7.93 Definition [846, Definition 2.2] Consider f : X → Z, a cover of normal varieties over Fq ,

with Z absolutely irreducible, but X possibly reducible. Then f is pr-exceptional if it is surjective on Fqk points for infinitely many k. There is a similar definition extending Definition 9.7.30 over a number field, and for both a notation for exceptional sets. 9.7.94 Definition Use the value set notation of Remark 9.7.3. We say fi ∈ Fq (x), i = 1, 2, is a

Davenport pair over Fq if Vf1 (P1 (Fqk )) = Vf2 (P1 (Fqk )) for infinitely many k. So, take f2 (x) = x to see Davenport pairs generalize exceptional functions. The notion applies to any pair of covers fi : Xi → Z, i = 1, 2. For K a number field, this similarly generalizes Definition 9.7.30: f1 , f2 ∈ K(x) are a Davenport pair if they are a Davenport pair for infinitely many residue class fields. 9.7.95 Theorem [846, Corollary 3.6]: Monodromy precision (Definition 9.7.26) applies to pr-

exceptional covers and so to Davenport pairs. That is, generalizing Theorems 9.7.53 and 9.7.55, a precise monodromy statement generalizes MacCluer’s Theorem (Proposition 9.7.28) to pr-exceptional covers and to Davenport pairs. 9.7.96 Theorem [846, §3.1.2]: With the notation of Definition 9.7.93, a pr-exceptional cover over

Fq is exceptional if and only if X is absolutely irreducible. 9.7.97 Remark The proof of Schur’s Conjecture began the solution of Davenport’s problem for

polynomial pairs (f1 , f2 ) over a number field, the main result of [838]. [846, §3.2] shows the exceptional set characterization for Davenport pairs in general is given by the intersection of exceptionality sets for pr-exceptionality correspondences. A full description of many authors’ results that came from the solution of Davenport’s problem – especially the study of general zeta functions attached to diophantine problems – is in [847, §7.3]. 9.7.98 Remark (The Genus 0 Problem) Geometric monodromy groups of rational functions are

¯ severely limited. The mildest statement for f ∈ Q(x) is that excluding cyclic and alternating groups the composition factors of Gf fall among a finite set of simple groups. That is the original genus 0 problem. ¯ There is a large literature distinguishing between geometric monodromy of f ∈ Q(x) and those in F¯q (x), because of wild (not tame; Remark 9.7.26) ramification. The contrast starts from the [1631, §8.1.2, Guralnick’s Optimistic Conjecture] list of all primitive monodromy ¯ groups of indecomposable f ∈ Q[x]. 9.7.99 Example (Davenport pairs) A significant part of the exceptional primitive monodromy

groups (Remark 9.7.98), without cyclic or alternating group composition factors, came from the finitely many possible degrees of Davenport pairs f1 , f2 ∈ K[x] (polynomials) over number fields, with f1 indecomposable and Vf1 (OK /pp) = Vf2 (OK /pp). Important hints about what to expect for primitive monodromy groups of f ∈ F¯q (x) came also from Davenport pairs. [846, §3.3.3] (explicitly in [247]): Over every Fq , there are

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infinitely many degrees of Davenport pairs, where (deg(f1 ), p) = 1, f1 is indecomposable, and Vf1 (Fqk ) = Vf2 (Fqk ) for all k. 9.7.100 Example [512, Theorem 14.1] described the geometric monodromy (PSL2 (pa ), p = 2, 3, a

odd) of the only possible exceptional polynomials over Fp whose degrees were neither prime to p or a power of p. Then, [848] produced these: the first exceptional polynomials over finite fields with nonsolvable monodromy. 9.7.101 Remark (Zeta functions attached to problems) [849, Chapter 25 and 26] details how Dav-

enport pairs led to attaching Poincar´e series – based on the Galois stratification procedure of [845] – to counting the values of parameters for any diophantine problem interpretable over all extensions of Fq , or for infinitely many primes p of K. w , x), g(w w , y) ∈ Fq [w w , x, y]. Denote the car9.7.102 Example Denote w1 , . . . , wu by w . Suppose f (w dinality of w 0 ∈ Au (Fqk ) with

w 0 , x))(P1 (Fqk )) = V (g(w w 0 , x))(P1 (Fqk )) V (f (w by Nf,g,k . Define Pf,g,Fq (t) to be the Poincar´e series

P∞

i=1

(9.27)

Nf,g,k tk .

w , x), g(w w , y) ∈ Z[w w , x, y]. 9.7.103 Example With notation over Z, as in Example 9.7.102, suppose f (w

Denote the cardinality of w 0 ∈ Au (Fpk ) with (9.27) holding over Fpk by Nf,g,Z/p,k . Define P∞ Pf,g,Z/p (t) to be i=1 Nf,g,Z/p,k tk .

9.7.104 Theorem [849, Chapter 25], [847, §7.3.3]: For any diophantine problem over Fq expressed

in a first order language, the attached Poincar´e series is a rational function. Further, there is an effective computation of the coefficients of its numerator and denominator based on expressing those coefficients in p-adic Dwork cohomology. 9.7.105 Theorem [Theorem 9.7.104 continued] Given a diophantine problem D over Z (or OK )

expressed in a first order language, there is an effective split of the primes of Q (or over K) into two sets: LD,1 and LD,2 , with LD,2 finite. Further, there is a set of varieties V1 , . . . , Vs over Z, from which we produce linear equations in variables Y1 , . . . , Ys0 that serve as the coefficients of the numerator and denominator of a rational function PD (t). To each (p, Yi ), p ∈ LD,1 there is a universal attachment of a p-adic Dwork cohomology group, H(p, Yi ), computed in the category of such Dwork cohomology attached to V1 , . . . , Vs . The corresponding Poincar´e series PD,p at p ∈ LD,1 comes by substituting H(p, Yi ) for each Y1 , . . . , Ys0 in PD (t). Then apply the Frobenius operator at p to these coefficients. 9.7.106 Remark [608]: In Theorem 9.7.105 it is possible to take V1 , . . . , Vs to be nonsingular projec-

tive varieties with Yi representing a Chow motive (over Q). Applying the Frobenius operator at p is meaningful as Chow motives are formed from ´etale cohomology groups of V1 , . . . , Vs . 9.7.107 Remark The effectiveness of Theorem 9.7.104 is based on Dwork cohomology [713], and the

explicit calculations of [256]. Theorem 9.7.105 and Remark 9.7.106 both rest on the Galois stratification procedure of [845] or [849, Chapter 24]. On the plus side, the uniform use of ´etale cohomology from characteristic 0 produces wonderful invariants – like, Euler characteristics – attached to diophantine problems. On the negative, all the effectiveness disappears. In particular, the relation between the sets denoted LD,1 in the two results is a mystery. 9.7.108 Remark Relating exceptional covers (and Davenport pairs) and other problems about al-

gebraic equations is a running theme in [846] and [847]. Detecting these relations comes from pr-exceptional correspondences [846, §3.2]. We catch the possible appearance of such correspondences when two Poincar´e series have infinitely many identical coefficients.

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255

9.7.109 Example Example: An exceptional cover, X → P1z , over Q, will be a curve whose Poincar´e

series is the same as that of P1z at infinitely many primes. The systematic use of such characterizations combines monodromy precision (where it applies) and Theorem 9.7.110.

9.7.110 Theorem [847, Proposition 7.17], based on [775]: The zero support of the difference of two

Poincar´e series consists of the union of arithmetic progressions. See Also

§8.1 Discusses the large literature on permutation polynomials (as in Proposition 9.7.39). This contrasts with the use of a cover given by an exceptional polynomial, where one fixed polynomial works for infinitely many finite fields. §8.3 Section §8.3.3 mentions several explicit Chebotarev density theorem error terms. Such error terms have improved over time, but, like Proposition 9.7.28, this sections’ results exhibit monodromy precision: the error term vanishes. §9.6 Discusses Dickson polynomials in detail, including their various combinatorial formulas. This contrasts with the Remark 9.7.34 formula free characterization. References Cited: [61, 247, 256, 512, 577, 608, 713, 775, 837, 838, 839, 840, 841, 845, 842, 843, 846, 847, 848, 849, 853, 852, 845, 1010, 1054, 1441, 1472, 1507, 1631, 1638, 1717, 1718, 1904, 1905, 1908, 1913, 1928]

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Index (n, m)-function, 205 (t, m, s)-net, 373 (t, s)-sequence, 379 (T, s)-sequence, 379 L-function, 389, 392, 454 k-arc, 496 complete, 496 secant, 497 tangent, 497 k-cap, 500 complete, 500 k-normal, 29 m-sequence, 212 p-adic gamma function, 110 p-rank, 401 q-clan normalized, 485 t-polynomial, 28 t-reciprocal polynomial, 29 A(q), 367 AB function, 207 abelian variety has Fq point, 336 addition algorithm, 337 additive white Gaussian noise channel, 560 adjacent, 532 affine plane, 477 affine plane, 466 classical, 477 Desarguesian, 477 affine space, 489 agent-based model, 688 algebraic curve X0 (n), 251 Y0 (n), 251 modular, 251 normalization, 243 algebraic dynamical system, 282–287 algebraic entropy, 283 algebraic set absolutely irreducible, 244 component, 244 components, 244 definition field, 244 variety, 244

algebraic-geometry code, 604 almost perfect nonlinear, 182 alternate, 439 anomalous elliptic curve, 351 aperiodic correlation, 268 APN function, 207 approximation theorem, 319 Araki, Kiyomichi, 351 arc, 484 maximal, 484 trivial, 484 Artin, Emil, 409 asymptotic bounds, 573 asymptotic Gilbert-Varshamov bound, 610 asymptotic normalized rate, 573 Aut(E), 338 autocorrelation, 136, 264 automorphic collineation, 489 automorphism, 477 group, 477 of elliptic curve, 338 automorphism group of a BIBD, 503, 505 of elliptic curve, 340, 347 Baer subplane, 482 Baer subplane partition, 482 classical, 482 perfect, 483 balanced incomplete block design automorphism group, 503, 505 complete, 502 cyclic, 505 decomposable, 502 derived, 502 generated by a difference family, 505 isomorphic, 503 m-multiple, 502 nontrivial, 502 quasi-symmetric, 502 simple, 502 starter blocks, 503 symmetric, see symmetric design Barker sequence, 268 bases almost self-dual, 74 823

824 almost weakly self-dual, 78 almost weakly self-dual polynomial, 78 characterization, 12 complementary, 72 dual, 12, 72 normal, 12, 72 number of, 11 polynomial, 12, 73 primitive normal, 13 self dual, 12 self-dual, 72, 74 trace-orthogonal, 72 weakly self-dual, 74 weakly self-dual polynomial, 75 weakly self-dual polynomial over F2 , 75 bent function, 517 Maiorana-McFarland, 518 bent functions, 111 Berlekamp algorithm, 653 Bertini theorem, 305 BIBD, see balanced incomplete block design big-field, 653 binary erasure channel, 560 binary symmetric channel, 560 binomial coefficients congruences, 99 biquadratic reciprocity law, 138 birational, 652 Birch, Bryan, 341 bit-packing, 425, 426 bit-slicing, 425 black box, 308, 310 Blahut’s Theorem, 273 BLAS, 427 block cipher, 205 block weight, 609 blocking set, 486 small linear, 486 Boolean function, 197, 205 affine, 199 algebraic degree, 199 algebraic immunity, 201 algebraic normal form, 197 annihilator, 201 balanced, 199 bent, 200 derivative, 200 Hamming distance, 200 Hamming weight, 199 inverse Walsh transform, 199

Handbook of Finite Fields nonlinearity, 200 Parseval’s relation, 199 propagation criterion, 202 quadratic, 199 resilient, 201 semi-bent, 205 sign function, 198 strict avalanche criterion, 202 trace representation, 198 Walsh support, 198 Walsh transform, 198 Boolean network, 687 Boolean network model, 685, 687 bound, 570–574 asymptotic, 573 asymptotic Gilbert-Varshamov, 610 Elias, 573 Griesmer, 572 Hamming, 572 linear programming, 573 MRRW, 574 ¨ Niederreiter-Ozbudak, 611 Plotkin, 572 Singleton, 572, 604 sphere covering bound, 571 sphere packing, 572 TVZ, 611 Varshamov-Gilbert, 571 Weil, 212 Brewer sums, 99, 104 Cameron-Liebler line class, 486 canalyzing function, 690 nested, 690 parametrization of nested, 691 Car, Mireille, 413 Cartesian group, 225 Cauchy matrix, 435 Cayley graph, 112, 535 channel capacity, 561 character Hecke, 103 lifted, 101 power residue, 100, 109 restriction, 99 multiplicative, 96 order, 96 quadratic, 96, 694 trivial, 96 character sum, 518 Characteristic polynomial, 431

Miscellaneous applications characteristic polynomial of sequence, 270 Characteristic polynomials, 431–433, 436 Chebyshev polynomial, 115 check digit system, 136 chromatic number, 534 circle, 479 circle geometry, 510 circle method, 412 class group, 109 class number, 100 class number (of a function field), 362 Clifford’s theorem, 323 code, 560–601 Z4 , 599 Gray image, 599 Lee distance, 600 Lee weight, 600 residue, 599 torsion, 599 type, 599 algebraic-geometry, 604 alternant, 583 asymptotically good, 586 BCH, 210, 578 designed distance, 578 narrow sense, 578 primitive, 578 concatenated, 570 constant-dimension, 701 cyclic, 210, 575–589 defining set, 578 generator matrix, 576 generator polynomial, 575 parity check matrix, 576 cyclic with two zeros, 212 direct sum, 570 distance distribution, 564 distance invariant, 565 doubly even, 591 duadic, 581 splitting, 581 dual, 562 encoding, 565 Euclidean geometry, 588 even, 591 even-like, 581 external distance, 565 finite geometry, 587 formally self-dual, 591 four fundamental parameters, 565

825 function-field, 608 generalized Reed-Muller (GRM), 586 generalized RS (GRS), 580 generator matrix, 562 Golay binary, 583, 590 Golay ternary, 583, 590 Goppa, 584 Hamming, 566 Hermitian, 604 information set, 580 Justesen, 588 Kerdock, 601 linear, 210, 561–565 maximum distance separable (MDS), 572 MDS, 499 Melas, 212 minimum distance, 560 minimum distance decoding, 565 modifying, 569 nonlinear, 560 NXL, 606 octacode, 600 odd-like, 581 optical orthogonal, 696 parity check matrix, 562 perfect, 572 polynomial, 588 Preparata, 601 product, 570 projective geometry, 588 quadratic residue (QR), 582 rank distance, 699 rate, 560 Reed-Muller, 211, 567, 696 Reed-Solomon, 579, 604 self-dual, 562, 591 self-orthogonal, 562 simplex, 567 space-time, 699 subfield, 568 trace, 568 XNL, 607 coding theory, 100, 109, 111 coefficient ith, 43 first, 25 last, 25 coefficients first, 43 last, 43

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cogredient, 440, 441 hermitian, 483 collineation, 477, 489 Newton polygon, 400 automorphic, 489 non-singular curve, 333 group, 477 ordinary, 400 complete mapping, 136 projective curve, 333 completely inseparable, 339 supersingular, 400 complex multiplication, 358 cyclic codes, 114 computational Diffie–Hellman problem, 351, cyclic digital net, 378 360 cyclic projectivity, 495 conductor, 347 cyclotomic coset, 135, 577 conic, 492 cyclotomic number, 104 conjecture cyclotomic numbers, 516 Barker sequence, 517 cyclotomy, 516 circulant Hadamard, 517 Davenport pair Lander, 515 over Fq , 253 Ryser, 515 over a number field, 253 conjugates, 10 de Jonqui`eres map, 652 conorm (of a divisor), 325 decisional Diffie–Hellman problem, 351 convex-dense decoder factorization, 308 maximum a posteriori, 560 coordinate frame, 490 maximum likelihood, 560 coordinate vector minimum distance, 561 of a hyperplane, 488 decoding, 591–599 of a point, 487 BCH code, 592 coordinates Berlekamp-Massey, 593 dual, 72 error evaluator polynomial, 593 primal, 72 error locator polynomial, 592 correlation, 490 extended Euclidean, 594 aperiodic, 694, 695 generalized minimum distance, 596 Hamming, 698 key equation, 593 periodic, 695, 696 list, 597 correlation measure, 280 majority logic, 595 cover Peterson-Gorenstein-Zierler, 592 branch points, 245 standard array, 565, 566 elliptic curve isogeny, 251 syndrome, 592 exceptional, 245 Welch-Berlekamp, 594 pr-exceptional, 253 Dedekind eta function, 114 properties, 245 Dedekind’s different theorem, 327 ramified point, 245 degree covering radius, 140 of an isogeny, 339 critical orbit, 142 degree zero divisor class group, 354 crosscorrelation, 264 degree zero divisor class number, 355 cryptanalysis Deligne’s theorem, 387, 390 differential, 205 dense polynomial representation, 302 linear, 205 density of primes, 408 cryptosystem dependency graph, 684 multivariate public key, 648 derivation, 330 symmetric, 205 derivative, 206 cubic reciprocity law, 138 Desarguesian, 467 curve algebraic curve, 332 design, 590

Miscellaneous applications Assmus-Mattson theorem, 590 symmetric, 513 determinant, 424 Moore, 424 Deuring, Max, 342, 349 diagonal equations, 100 Dickson polynomial, 113 difference families multiplier, 506 radical, 506 relative, 510 difference set, 512 cyclotomy, 516 Gordon-Mills-Welch, 515 Hadamard, 514 multiplier, 519 Paley, 514 planar, 514 Singer, 514 difference sets, 109 different (of a field extension), 327 different exponent, 327 differential (of a function field), 330 divisor of a differential, 331 differential module, 330 differential uniformity, 207 Diffie-Hellman triple, 148 digital (t, m, s)-net, 374 digital (t, m, s)-net over R, 374 digital (t, s)-sequence, 380 digital (t, s)-sequence over R, 380 digital (T, s)-sequence, 380 digital (T, s)-sequence over R, 380 digital method, 374, 380 digital strict (t, m, s)-net, 374 digital strict (t, m, s)-net over R, 374 digital strict (t, s)-sequence, 380 digital strict (t, s)-sequence over R, 380 digital strict (T, s)-sequence, 380 digital strict (T, s)-sequence over R, 380 dimension translation plane, 478 Dirichlet character, 410 discrepancy, 139, 281 discrete Fourier transform, 148, 274 discrete log cryptosystems, 100 discrete memoryless channel, 559 discrete model, 683 discriminant, 334 Displacement rank, 434 distinct degree factorization

827 multivariate, 306 division polynomial, 343 divisor balanced, 356 Cantor’s algorithm, 356 defined over L, 354 finitely effective, 355 Mumford representation, 356 NUCOMP, 356 reduced, 356 semi-reduced, 355 divisor (of a function field), 320 canonical class, 322 canonical divisor, 322, 331 class group Cl0 (F ), 362 degree of a divisor, 320 dimension of a divisor, `(A), 322 divisor class [D], 321 divisor class group Cl(F ), 321 divisor group Div(F ), 320 divisor of a differential, 331 divisor of poles (x)∞ , 321 equivalent divisors, 321 positive divisor, 320 prime divisor, 320 principal divisor, 321 principal divisor div(x), 321 zero divisor (x)0 , 321 divisor group of elliptic curve, 341 Drinfeld–Vlˇadut¸ bound, 367 dual basis multiplier, 75 dual isogeny, 339, 340 Frobenius map, 348 dual space chain, 381 duality theory, 381 duplication formula, 337 dynamical system polynomial, 684 ECDHP, 351 ECDLP, 351 Effinger, Gove, 410, 412 Eichler, Martin, 349 Eisenstein sum, 99 elation projective plane, 477 elements Bernoulli-Carlitz, 454 Elkies, Noam, 349 elliptic curve, 334

828 GL2 -type, 251 addition algorithm, 337 anomalous, 351 automorphism, 338 automorphism group, 340, 347 CM-type, 251 Diffie–Hellman problem, 351 discrete logarithm problem, 351 division polynomial, 343 divisor group, 341 dual isogeny, 339 duplication formula, 337 ECDHP, 351 ECDLP, 351 endomorphism, 338 endomorphism ring, 340, 347 formal group, 348 Frobenius map, 339 group law, 336 Hasse–Weil estimate, 341, 350 isogeny, 338 isogeny of degree 2, 339 isogeny theorem, 342, 344 isomorphic, 335, 336 kernel of multiplication-by-m, 338, 343 mass formula, 349 multiplication-by-m map, 338 nonsingular projective genus one, 335 number of points, 341, 350 ordinary, 348 over F2 , 336 Picard group, 341 point at infinity, 334 points defined over a field, 334 principal divisor iff deg 0 and sums to O, 341 supersingular, 348 Tate module, 344 Tate pairing, 346 torsion subgroup, 338, 343 transformation of coordinates, 335 Weil pairing, 344 zeta function, 350 elliptic curves, 103 complex multiplication, 251 embedding degree, 351 End(E), 338 endomorphism, 358 of elliptic curve, 338 endomorphism ring, 358 of elliptic curve, 340, 347

Handbook of Finite Fields entropy function, 573, 610 equation Artin-Schreier, 142 diagonal, 140, 164 hyperelliptic, 142 Kloosterman, 144 Equidistribution Kloosterman angles, 113 Sali´e angles, 113 equivalent, 466 ErdHos-Turan inequality, 139 error evaluator polynomial, 593 error locator polynomial, 592 error-rate exponent, 561 Euclidean geometry, 587 exceptional cover MacCluer’s Theorem, 245 Serre’s OIT Theorem, 252 Exceptional tower, 249 arithmetic monodromy, 249 cryptographic subtower, 251 Dickson subtower, 251 Redei subtower, 251 subtower, 250 exceptionality set number field, 246 over Fq , 246 excess, 76 exponential Carlitz, 450 Drinfeld, 450 extended Euclidean algorithm, 594 extension algebraic, 8 finite, 8 simple, 8 extension (of function fields), 323 Artin–Schreier extension, 329 constant field extension, 329 Kummer extension, 328 external distance, 565 factor of a symmetric matrix, 421 factorization convex-dense, 308 distinct degree multivariate, 306 irreducible bivariate, 303 multivariate, 305 separable, 301

Miscellaneous applications sparse, 306 squarefree multivariate, 302 Faltings, Gerd, 344 Family A, 268 Faure sequence, 381 FFLAS, 427 fiber product, 246 absolute components, 248 complementary components, 248 in TZ,Fq , 249 normalized, 247 set theoretic, 247 field cardinality, 4 cyclotomic, 450 definition, 3 existence and uniqueness, 6 number, 700 prime, 4 skew, 3 splitting, 6 subfield criterion, 6 field extension separable, 247 field-like structures division semiring, 16 nearfield, 16 neofield, 16 prequasifield, 15 presemifield, 16 quasifield, 16 semifield, 16 figure of merit, 377 Fill-in, 433 finite field embedding degree, 351 fixed point, 684 flock quadratic cone, 485 form algebraic normal, 696 modular, 457 quadratic, 159 trace, 161 formal group, 348 four fundamental parameters, 565 Four Russians (Method of), 425 Fourier coefficients, 145 frequency square, 464

829 orthogonal, 464 Frobenius automorphism, 10 Frobenius eigenvalues, 112 Frobenius endomorphism, 358 Frobenius endomorphism (acting on the Tate module), 363 eigenvalues of Frobenius, 363 Frobenius map, 339 dual of, 348 is purely inseparable, 339 isogeny factors through, 340 function almost bent, 207 almost perfect nonlinear, 207 balanced, 205 bent, 206 CCZ-equivalent, 209 component, 205 crooked, 211 Dobbertin, 213 EA-equivalent, 209 Euler’s Φ, 14, 432 Euler’s φ , 7 exponential, 449 Gold, 211 inverse, 208 M¨obius, 5 perfect nonlinear, 206 planar, 206, 230 plateaued, 205 R´eidi, 236 trace, 205 Welch, 213 function field, 317 constant field, 317 elliptic function field, 317, 331 Fermat function field, 365 Giulietti–Korchm´aros function field, 366 Hermitian, 604 Hermitian function field, 365 hyperelliptic function field, 317, 331 maximal function field, 365 rational function field, 317, 319, 321, 323 function-field code, 608 functional equation, 385, 387, 393 Fundamental Theorem of Projective Geometry, 489 fundamental unit, 355 Galois ´ Evariste, 6

830 field, 6 group, 10 ring, 17 theory, 6 Gauss multiplication formula for gamma functions, 103 Gauss sum estimates, 98 generalized quadratic, 116 Hecke, 117 in multi-quadratic field, 106 lifted, 101 primitive, 117 quadratic over Z/kZ, 116 quintic, 106 reciprocity, 116 reduction formula, 117 absolute value, 97 cubic, 107 equidistribution, 97, 107 of first kind, 137 of second kind, 137 prime ideal factorization, 109 pure, 102 quadratic, 106 quartic, 107 uniform distribution, 97 with character over Fq , 97 Gaussian elimination, 433 Gaussian period, 98 cubic, 107 duodecic, 108 quartic, 108 sextic, 107 generalized dual coordinates, 76 generalized quadrangle, 485 generalized Riemann hypothesis, 410 generating matrices, 375, 380 generator cyclotomic, 137 inversive, 144 linear congruential, 139 generator matrix, 562 genus (of a function field), 322 genus (of a plane curve), 323 genus one curve, 335 has Fq point, 336 geometric Frobenius, 389, 390 geometric Frobenius correspondence, 385 geometry affine, 697

Handbook of Finite Fields projective, 697 GMW sequences, 265 Gold exponents, 211 Gold sequences, 266 Goldbach problem, 411 Gr¨obner basis, 55 Gr¨obner fan, 689 graph, 532 adjacency matrix, 532 bipartite, 533 complete, 532 complete bipartite, 533 connected, 533 cycle, 533 degree, 532 diameter, 533 distance, 533 edge set, 532 eigenvalue, 532 loop, 532 Ramanujan, 532 regular, 532 simple, 532 spectrum, 532 strongly regular, 502 vertex set, 532 Gray map, 599 greatest common divisor (gcd), 6 Gross-Koblitz formula for Gauss sums, 110 for Jacobi sums, 110 Grothendieck trace formula, 390 Grothendieck’s formula L-function, 390 Zeta function, 386 Grothendieck-Ogg-Shafarevich formula, 393 group abelian, 3 doubly transitive, 245 general linear, 415, 437 multiplicative is cyclic, 6 orthogonal, 443, 446 primitive, 245 projective general linear, 437 projective orthogonal, 445 projective proper orthogonal, 445 projective special linear, 437 projective special unitary, 442 projective symplectic, 440 projective unitary, 442 proper orthogonal, 444

Miscellaneous applications regular automorphism, 513 special linear, 415, 437 special unitary, 441 symplectic, 439 unitary, 441 group law on elliptic curve, 336

831

HOLE, 659 Hom(E1 , E2 ), 338 homogeneous coordinates, 476 Hurwitz genus formula, 327 hyper-Kloosterman sum, 111 hyperbolic fibration, 485 agrees on a line, 485 regular, 485 Hadamard design, 502 hypercube, 465 Hadamard matrix, 106, 517 orthogonal, 465 Hall, Chris, 410 Hyperelliptic curve, 352 Hammersley net, 373, 375 finite points, 353 Hamming correlation, 269 imaginary, 352 Hamming distance, 499, 560 infinite places, 352 Hamming space, 375 points, 353 Hamming weight, 560 points at infinity, 353 Handshaking Lemma, 532 real, 352 Hankel matrix, 435 unusual, 352 hard, 30 hyperelliptic curve Hardy, G.H., 411, 412 baby step, 357 Hasse–Weil estimate, 341, 350 balanced divisor, 356 Hasse–Weil bound, 363 Cantor’s algorithm, 356 Hasse–Weil theorem, 363 complex multiplication, 358 Hasse-Davenport product formula for Gauss degree zero divisor class group, 354 sums, 103, 112 degree zero divisor class number, 355 Hasse-Davenport theorem on lifted Gauss Diffie–Hellman problem, 360 sums, 102 discrete logarithm problem, 360 Hasse-Davenport theorem on lifted Jacobi distance, 357 sums, 102 divisor defined over L, 354 Hasse-Weil estimante, 358 endomorphism, 358 Hayes, David, 409, 412 endomorphism ring, 358 HCDHP, 360 finitely effective divisor, 355 HCDLP, 360 Frobenius endomorphism, 358 Hecke L-function, 117 fundamental unit, 355 Hecke characters, 103 giant step, 357 Hensel lifting Hasse-Weil interval, 358 sparse, 307 HCDHP, 360 Hermite/Dickson criterion, 191 HCDLP, 360 hermitian IDLP, 360 curve, 483 index-calculus, 360 Hermitian code, 604 infrastructure, 357 Hermitian curve, 492 infrastructure discrete logarithm probhermitian curve, 483 lem, 360 Hermitian function field, 604 Jacobian, 355 hermitian matrix, 441 Miller’s algorithm, 359 Hermitian surface, 492 modified Tate-Lichtenbaum pairing, 359, Hermitian variety, 492 360 HFE, 653 Mumford representation, 356 Hilbert NUCOMP, 356 theorem, 305 reduced divisor, 356 Hilbert, David, 412 regulator, 355

832 semi-reduced divisor, 355 supersingular, 358, 360 Tate-Lichtenbaum pairing, 359 zeta function, 358 hyperelliptic curves, 106 Hyperelliptic equation, 352 Hyperelliptic involution, 353 hypergeometric character sums, 103, 104 hypergraph, 538 hyperoval, 484, 497 hyperplane at infinity, 489 hyperplane coordinates, 488 hyperplane net, 378 ideal, 575 principal, 575 IDLP, 360 Ihara’s bound, 365 Ihara’s quantity A(q), 367 Implicit Form, 650 independence number, 534 independent set, 533 infinity line, 477 point, 477 infrastructure, 357 baby step, 357 discrete logarithm problem, 360 distance, 357 giant step, 357 inseparability degree, 339 inseparable isogeny, 339 integer Weil, 120 integral basis, 326 integral closure, 325 integral domain, 3 integral equation, 325 intrinsic rank, 654 inversive plane, 479, 501, 510 circle, 501 classical, 501 egglike, 501 Miquelian, 480, 501 involution, 441 IP, 651 irreducibility test multivariate, 307 irreducible factorization

Handbook of Finite Fields bivariate, 303 multivariate, 305 isogenous, 338 isogeny, 338 defined over K, 338 degree of, 339 Drinfeld, 450 dual, 340 factors through Frobenius, 340 inseparable, 339 is a homomorphism, 340 is constant or surjective, 339 is unramified, 340 of degree 2, 339 product of, 340 separable, 339 sum of, 340 zero, 339 isogeny theorem, 342, 344 isomorphism, 477 of elliptic curves, 335, 336 isomorphism of polynomials, 651 isotopism, 226 Jacobi sum, 99, 137 congruences, 104 lifted, 102 quintic, 106 reduction formula, 100 cubic, 105 duodecic, 106 equidistribution, 101 multiple, 100 octic, 106 prime ideal factorization, 110 pure, 102 quadratic, 106 quartic, 105 sextic, 105 uniform distribution, 101 Jacobian, 355 Jacobian (of a curve), 362 Jacobsthal sum, 104 j-invariant, 334 equal iff isomorphic, 335 joint linear complexity, 271 k-error, 272 nth, 272 profile, 272 Kasami exponent, 210

Miscellaneous applications Kasami sequences, 267 kernel, 478 kernel of multiplication-by-m, 338, 343 key equation, 593 Kloosterman angle, 112 code, 144 sum, 144 Kloosterman sum, 111 congruences, 111, 114 degree, 111 equidistribution, 113 estimates, 112, 115 over Z/kZ, 117 reduction formula, 117 symmetric powers, 115 zeros, 111 lifted, 113 multiple, 111 power moments, 114 Kolmogorov complexity, 280 Krawtchouk polynomials, 564 Kronecker substitution, 425–427 Kronecker product, 464 Kronecker-product construction, 378 Kubota, R.M., 413 Kummer’s theorem, 326, 361 L-polynomial (of a function field), 363 functional equation, 363 lacunary polynomial, 308 Lagrange Interpolation Formula, 462 Lagrange interpolation formula, 14 lambda phage, 687 Lanczos, 433 Lang, Serge, 336, 349 largest prime survives, 45 lps pair, 45 latin orthogonal, 462 latin square, 374, 462 infinite, 463 mateless, 463 mutually orthoognal, 463 reduced, 462 lattice profile, 280 lattice test, 280 Laumon’s product formula, 393 Laurent polynomial non-degenerate, 122

833 law of quadratic reciprocity, 138 Lefschetz fixed point theorem, 386 Legendre sequence, 147, 266 lifted character, 101 lifted Gauss sum, 101 lifted Jacobi sum, 102 lifted Kloosterman sums, 113 limit cycle, 684 linear complexity, 147, 270 Fq -, 272 k-error, 271 nth, 270 profile, 271 linear feedback shift register, 270 linear recurring sequence, 270 linear translator, 181 linearity conjecture, 486 Littlewood, J.E., 411, 412 Logical Model, 687 logical model, 687 m-sequence, 265 M¨obius equivalence, 243 transformations, 243 MacWilliams identities, 563 MacWilliams transform, 564 mass formula, 349 matrix circulant, 419 circulant Hadamard, 517 companion, 418 generator, 562, 576 Hadamard, 135, 517 Hankel, 423 Hasse Witt, 401 involutory, 417 nilpotent, 417 orthogonal, 420 orthogonal , 444 orthogonal circulant, 420 parity check, 562, 576 proper orthogonal, 444 skew-symmetric, 421 symmetric, 421 systematic, 563 Toeplitz, 422 Vandermonde, 578 matrix-product construction, 378 maximal order, 347 maximal partial spread, 486

834 MDS code, 499 measure R-valued, 455 correlation, 146 well distribution, 146 Menezes, Alfred, 351 merit factor, 269 Miller’s algorithm, 359 Miller, Victor, 345 Miller’s algorithm, 345 minihyper, 486 Minimal polynomial, 431–433, 435, 436 minimal polynomial joint, 271 of sequence, 270 minimal sampling algorithm, 689 minimal sets algorithms, 689 minimum block weight, 609 minimum distance, 376, 499 Minkowski sum, 306 minus, 656 model selection, 689 model space, 689 modified Tate-Lichtenbaum pairing, 359, 360 module Carlitz, 450 class, 453 Drinfeld, 450 Hayes, 450 Tate-Drinfeld, 451 Tate-Shafarevich, 453 M¨ obius plane, 510 monodromy group arithmetic, 244 geometric, 244 monodromy precision Davenport pairs, 253 exceptional polynomial, 245 general exceptional covers, 245 pr-exceptional covers, 253 monomial, 212 Morita’s p-adic gamma function, 110 morphism cover, 244 Drinfeld, 450 flat, 250 multigraph, 532 multinomial coefficients, 109 multiplication-by-m map, 338 multiplier, 506, 519 multiply nested BIBD, 507

Handbook of Finite Fields multisequence, 271 Nq (g), 364 nebentypus, 115 nest, 479 plane, 479 replaceable, 479 nested canalyzing function, 690, 692 nested design multiply, 507 net (t, m, s)-, 373 cyclic digital, 378 digital (t, m, s)-, 374 digital strict (t, m, s)-, 374 Hammersley, 373, 375 hyperplane, 378 strict (t, m, s)-, 373 Netto triple system, 503, 504 newform, 114 Newton polyhedron, 122 Newton polytope, 306 Niederreiter sequence, 381 ¨ Niederreiter-Ozbudak bound, 611 Niederreiter-Xing sequence, 382 No sequences, 267 nonlinearity, 205 norm definitions, 11 properties, 11 normal rational curve, 496 NP-hard, 648 NP-hardness, 309 NRT space, 375 NRT weight, 375 NXL code, 606 Okamoto, Tatsuaki, 351 operator hyperdifferential, 455 optical orthogonal code(OOC), 269 orbit, 439 orbit length, 142 order, 347 affine plane, 477 conductor of, 347 in quadratic imaginary field, 347 of a finite field, 4 of an element, 6 projective plane, 476 ordered orthogonal array, 374

Miscellaneous applications

835

ordinary, 396 ordinary elliptic curve, 348 orthogonal array, 520 orthogonal system, 186 orthomorphism, 467 Ostrowski theorem, 306 oval, 497 complete, 497 ovoid, 500 Tits, 500

place at infinity, 319 pole of x, 319 prime element at a place, 318 ramification index, 324 ramified extension, 325 rational place, 319, 361 number of rational places N (F ), 361 relative degree, 324 residue class field of a place, 319 residue class map, 319 unramified extension, 325 zero of x, 319 planar equivalence, 231 planar function, 185 plane affine, 477 flag-transitive, 480 Andr´e, 479 Hall, 479 inversive, 479 nest, 479 projective, 476 PLE decomposition, 429 plus, 656 PN function, 206 Poincar´e duality, 386, 392 Poincar´e series, 112 point at infinity, 334 point set, 373 points special, 453 polar, 492 polarity, 490 Hermitian, 492 null, 492 ordinary, 492 orthogonal, 492 pseudo-, 492 symplectic, 492 unitary, 492 pole, 492 Pollack, Paul, 410 Polya-Vinogradov-Weil bound, 145 polygon generic, 398 Hodge, 396, 397 Newton, 394 polynomial κ-, 225 completely normal, 64 Dirichlet L-function, 411

p-density, 395 packed matrix multiplication, 426 pairing Miller’s algorithm, 359 modified Tate-Lichtenbaum, 359, 360 Tate-Lichtenbaum, 359 Paley construction, 135 parallelism, 477 parameter estimation, 689 parity check matrix, 562 partial-period correlation, 269 partition Baer subplane, 482 classical, 482 perfect, 483 path, 533 period Fourier expansion, 98 Carlitz, 450 period polynomial, 98 periodic correlation, 264 periodic point, 684 permanent, 424 permutation apn, 208 permutation polynomial, 376, 467 perspectivity projective plane, 477 elation, 477 Petri net, 688 phase space, 684 Picard–Fuchs differential operator, 349 Picard group of elliptic curve, 341 place, 318 completely splitting place, 325 degree of a place, 319 extension of a place, 324

836 normal, 60 primitive, 43 strong primitive normal, 64 absolute value of, 408 affine, 13 all one, 40 characteristic, 178 Chebychev conjugate, 246 complete mapping, 181 Dembowski-Ostrom, 232 Dickson, 191, 246 Dickson polynomial of the first kind, 235 Dickson polynomial of the second kind, 235 discriminant, 15, 36 even, 412 exceptional, 173, 212 existence of irreducible, 8 feedback, 273 Hasse, 398 indecomposable, 53 irreducible, 5 linearized, 13 Mattson-Solomon, 589 minimal, 8, 577 minimal value set, 189 monic original, 53 multivariate quadratic, 648 norm, 25, 43, 60 number of irreducible, 5 odd, 412 permutation, 171, 187, 376 permutation in several variables, 186 planar, 230 primitive, 7, 56 primitive normal, 57 reciprocal, 7, 56 ring of, 5 special, 454 stable, 142 syndrome, 584 trace, 25, 43, 60 Polynomial 3-Primes Theorem, 412 Polynomial Dynamical System, 684 Polynomial Generalized Riemann Hypothesis, 411 polynomial interpolation problem, 598 polynomial lattice, 376 Polynomial Prime Number Theorem, 408 Polynomial Twin Primes Theorem, 410 Polynomial Waring Theorem, 413

Handbook of Finite Fields polynomials with prescribed trace and norm, 100 power residue character, 100 power residue symbol, 109 primality testing, 100 prime ideal factorization of p, 108 primes in arithmetic progression, 409 primitive element, 6, 56 primitive part, 54 principal divisor on elliptic curve, 341 principal ideal domain (PID), 575 Principle of Duality, 488 problem hidden number, 141 sparse polynomial noisy interpolation, 141 Waring, 140 projective completion, 477 plane, 476 projective 1-space F points, 243 j-line, 251 projective geometry, 587 projective plane, 466 classical, 476 Desarguesian, 476 projective space, 487 hyperplane, 487 line, 487 plane, 487 point, 487 solid, 487 subspace, 487 projective spaces, 487–501 projectivity, 489 cyclic, 495 propagation rule, 378 pseudorandom graph, 534 pseudorandom number generator, 282, 283 pure number of weight w, 390 quadratic imaginary field order in, 347 quadratic nonresidue, 582 quadratic residue, 582 quadratic space, 159 Arf invariant, 160 non-degenerate, 160 radical, 159

Miscellaneous applications rank, 160 quadric, 492 elliptic, 494 hyperbolic, 494 parabolic, 494 quadric surface, 492 quality parameter, 373, 379 quantum computer, 648 quasifield, 225, 467, 478 quaternion algebra, 347, 348 radical, 28, 61 rainbow structure sequence, 655 Ramanujan sum, 116 ramification tame ramification, 327 wild, 253 wild ramification, 327 ramification locus (of a tower), 369 Rank, 429, 432–435 rank, 652 rational, 652 rational function composition factor definition field, 252 composition factors, 243 cyclic conjugate, 250 decomposable, 243 exceptional over Fq , 244 exceptional over a number field, 244 permutation over Fq , 247 Redei, 250 separable, 247 tame, 245 rational functions Davenport pair, 253 genus 0 problem, 253 rational point (rational place), 361 reciprocity, 490 REDQ, 426 Compression, 426, 427 Correction, 426, 427 reduction good-Drinfeld, 452 potentially good, 452 stable-Drinfeld, 452 Reed-Solomon code, 604 regulator, 355 regulus, 478 opposite, 478 Reordering, 433 replicator, 29

837 representation matrix, 439 residuacity, 104 resolvable BIBD Bose’s condition, 502 reverse engineering, 689 Riemann hypothesis, 385, 387, 390, 393 Riemann hypothesis (for function fields), 363 Riemann’s inequality, 323 Riemann’s theorem, 323 Riemann–Roch space L(A), 322 Riemann–Roch theorem, 322, 331 Riemann-Roch space, 603 Riemann–Roch theorem, 335 ring, 3 characteristic, 4 commutative, 3 division, 3 R¨ uck, Hans-Georg, 342 Sali´e angle, 113 Sali´e sum over Z/kZ, 117 over Fq , 112 Samaev, I., 351 Sato-Tate measure, 113, 114 Satoh, Takakazu,, 351 Schur’s Conjecture, 243 semifield definition, 225 nuclei, 228 separable factorization, 301 separable isogeny, 339 sequence, 694 (t, s)-, 379 (T, s)-, 379 Barker, 694 digital (t, s)-, 380 digital (T, s)-, 380 digital strict (t, s)-, 380 digital strict (T, s)-, 380 elliptic curve congruential, 279 explicit inversive congruential, 277 Faure, 381 frequency hopping, 698 generalized Lucas, 178 Golay, 695 inversive, 278 Legendre, 278, 694, 695 maximum length, 694, 695

838 Niederreiter, 381 Niederreiter-Xing, 382 nonlinear congruential, 278 power, 278 quadratic exponential, 277 recursive nonlinear, 278 Sidelnikov, 279 Sobol’, 382 strict (t, s)-, 379 strict (T, s)-, 379 van der Corput, 379, 380 Serre bound, 364 Serre’s explicit formulas, 364 Serre, J.P., 413 Serre, Jean-Pierre, 349 set difference, 212 simplest cubic, 100 simplex of reference, 490 Singer cycle, 495 Singer group, 495 Singleton bound, 604 singular point, 334 small-field, 653 Smart, Nigel, 351 Sobol’ sequence, 382 space affine, 489 Hamming, 375 NRT, 375 projective, 487 Riemann-Roch, 603 sparse factorization, 306 Sparse matrix, 433 sparse polynomial representation, 306 spectrum Walsh, 205 sphere, 571 spherical geometry, 510 spin, 30 splitting, 581 splitting locus (of a tower), 369 spread, 478, 494, 701 automorphism group, 478 partial maximal, 486 regular, 478 subregular, 479 square-free divisor, 61 W (r), 61

Handbook of Finite Fields radical, 61 squares, 512 St¨ohr–Voloch theory, 366 standard array, 566 starter block, 504 state space, 684 steady state, 684 Stein generator, 434 Steiner triple system 2-homogeneous, 504 Stickelberger’s congruence for Gauss sums, 109 straight-line program, 308, 309 straight-line programs without divisions, 309 strict (t, m, s)-net, 373 strict (t, s)-sequence, 379 strict (T, s)-sequence, 379 strict sum of polynomials, 413 strongly regular graph constructed from a quasi-symmetric design, 502 Structured matrix, 433 subgeometry, 495 subplane, 482 Baer, 482 subregular spread, 479 translation plane, 479 sum Kloosterman, 212 supersingular, 358, 360 supersingular elliptic curve, 348 mass formula, 349 supersingularity, 103 supersparse polynomial, 308 Swan theorem, 36 Sylvester generator, 434 symmetric, 665 symmetric design, 502 symmetric differential, 663 symmetry, 444 syndrome, 584, 592 syndrome polynomial, 584 Tame Transformation Method (TTM), 664 tangential coordinates, 488 Tate module, 344 Weil pairing on, 345 Tate pairing, 346 modified, 346 Tate, John, 342, 344, 346

Miscellaneous applications Tate-Lichtenbaum pairing, 359 tight set, 486 Tits ovoid, 500 Toeplitz matrix, 435 torsion subgroup, 338, 343 total degree, 388 tower (of function fields), 368 asymptotically good tower, 369 limit of a tower, 369, 370 recursive tower, 370 tame tower, 370 wild tower, 371 trace, 568 definitions, 10 properties, 11 trace of Frobenius, 341, 344, 350 trajectory, 282, 286, 287, 289 transform n-th order, 29 translation affine plane, 478 group, 478 line, 478 projective plane, 478 translation invariant, 29 transvection orthogonal, 446 symplectic , 440 unitary , 442 transversal, 478 regulus, 478 triangle inequality, 318 triangular map, 651 trinomial Mersenne, 66 triple system Netto, see Netto triple system Trotter, Hale, 349 TRSM, 429 Tsfasman–Vlˇ adut¸–Zink theorem, 367 TVZ bound, 611 twin primes, 409 twisted cubic, 496 Uniform distribution Kloosterman angles, 113 Sali´e angles, 113 unital, 483 Buekenhout nonsingular, 483 orthogonal, 483

839 embedded, 483 update schedule, 684 valuation, 318 valuation corresponding to a place, 318 valuation ring, 318 value set, 189 van der Corput sequence, 379, 380 Vandermonde matrix, 435 Vanstone, Scott, 351 variety Drinfeld modular, 457 function field, 244, 245 geometric point, 244 vector degree, 51 Vinogradov’s formula, 147 Vinogradov, I.M., 411 walk, 533 closed, 533 Waring problem, 412 Waring’s formula, 235 Waring’s number, 168 existence, 168 Waterhouse, William, 342 Webb, W.A., 413 Wedderburn, 4 Weierstrass ℘-function, 107 Weierstrass equation, 334 discriminant, 334 j-invariant, 334 nonsingular, 334 singular, 334 transformation of coordinates, 335 weight, 76 NRT, 375 Weil bound, 141 Weil conjecture, 385 Weil pairing, 344 computation of, 345 formulas for, 345 Weil, Andr´e, 344 Weil, Andr´e, 411, 412 Wiedemann, 432 wiring diagram, 684 XNL code, 607 zero isogeny, 339 Zeta function, 384 zeta function of a hyperelliptic curve, 358

840 of elliptic curve, 350 Poincar´e duality, 350 zeta function (of a function field), 362 Zsigmondy prime, 45 largest, 45

Handbook of Finite Fields