馮克勤教授師從華羅庚先生，專攻數(shù)論方向，他將數(shù)論用于代數(shù)編碼理論和代數(shù)圖論，首次引進了“因子關聯(lián)圖”的概念，并對其特性進行深入研究；利用這一工具，對于有保密通信背景的M序列反饋邏輯的構作予以重大推進。另外，他利用Weil和這一深刻的數(shù)論工具得到線性移存器序列偽隨機性能的最佳結果；同時，還得到以表示論為背景的Ramanujan超圖的譜性。近年來，他對具有密碼學性質的布爾函數(shù)研究受到國內外的極大關注。《數(shù)論及其相關領域》內容定位于馮克勤教授作出重要成就及感興趣的領域，分為兩個方面：一方面是數(shù)論，包括二次丟番圖方程的同余性、代數(shù)整數(shù)環(huán)的K群、整體函數(shù)域的L-函數(shù)、自守形式以及p-adic表示等，另一方面是這些理論在設計、編碼等領域的應用，反映了馮克勤教授在數(shù)論及其應用領域的發(fā)展所作的貢獻和影響，《數(shù)論及其相關領域》不僅是數(shù)論及其應用領域專家們有價值的參考書，也是研究生開展研究時極好的入門書。本書由歐陽毅等主編。
Binary Additive Counter Stream Ciphers 1 Introduction 2 Possible attacks and design criteria 3 Example 1: the Legendre cipher 4 Example 2: the Binary Additive Counter Stream Ciphers 1 Introduction 2 Possible attacks and design criteria 3 Example 1: the Legendre cipher 4 Example 2: the two-prime cipher 5 Conclusions and concluding remarks ReferencesPartial Difference Sets from Quadratic Forms and p-ary Weakly Regular Bent Functions 1 Introduction 2 Partial difference sets from quadratic forms and uniformcyclotomy 3 Partial difference sets from weakly regular p-ary bentfunctions.. ReferencesGoverning Fields of the 4-rank of K20o~./~ as p Varies 1 Introduction 2 The governing field of the 4-rank of K20F 3 The governing field of the 8-rank of K20F ReferencesWord-oriented Linear Feedback Shift Registers: a-LFSRs 1 Introduction 2 Model of a-LFSR 3 Cryptographic properties 4 a-LFSRs suitable for software implementation 5 Application of a-LFSRs 6 Conclusion ReferencesStatistics of Zeros of Families of L-functions over FunctionFields: A Survey 1 Introduction 2 Hyperelliptic curves 3 Cyclic/-fold covers of the projective line 4 Elliptic curves over a rational function field andgeneralizations 5 Concluding remarks ReferencesLectures on p-adic Zeta Functions and (ч, г)-modules 1 Introduction 2 Continuous functions, measures and distributions over Zp 3 The p-adic zeta function of Kubota-Leopoldt 4 (ч, г)-modules and Galois cohomology 5 (ч,г)-modules and Iwasawa theory ReferencesConjectures and Results on x2 mod p2 with 4p = x2 + dy2 1 Introduction 2 Using Apery polynomials and products of three binomialcoefficients 3 Using the polynomials 4 Using the function 5 Using the function 6 Using 7 Miscellaneous things ReferencesHarmonic Weak Maass Forms, Automorphic Green Functions, and PeriodIntegrals 1 Introduction 2 Shimura varieties of orthogonal type and their Kudla cycles 3 Harmonic weak Maass forms, regularized theta lifting, andautomorphic Green functions 4 Eisenstein series associated to coherent and incoherentquadratic spaces 5 Period integrals of the automorphic Green function φ(z, h; f) 6 Big CM values of automorphic Green functions ReferencesSome Recent Progress in Higher Koszulity 1 Preliminaries 2 Higher Koszulity 3 Higher Koszul complexes 4 Hilbert and Poincare series 5 Dual algebras and Ext-algebras 6 Generalized d-Koszul modules 7 Lattice distributivity and Koszulity 8 More related topics References
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