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【内容】
冯克勤教授师从华罗庚先生，专攻数论方向，他将数论用于代数编码理论和代数图论，首发引进了“因子关联图”的概念，并对其特性进行深入研究；利用这一工具，对于有保密通信背景的M序列反馈逻辑的构作予以重大推进。另外，他利用Weil和这一深刻的数论工具得到线性移存器序列伪随机性能的很好结果：同时，还得到以表示论为背景的Ramanujan超围的谱性。近年来，他对具有密码学性质的布尔函数研究受到国内外的极大关注。《数论及其相关领域》内容定位于冯克勤教授作出重要成就及感兴趣的领域，分为两个方面：一方面是数论，包括二次丢番图方程的同余性、代数整数环的K-群、整体函数域的L-函数、自守形式以及p-adic表示等，另一方面是这些理论在设计、编码等领域的应用，反映了冯克勤教授在数论及其应用领域的发展所作的贡献和影响，不仅是数论及其应用领域专家们有价值的参考书，也是研究生开展研究时极好的人门书。
【目录】
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
References
Partial Difference Sets from Quadratic Forms and p-ary Weakly
Regular Bent Functions
1 Introduction
2 Partial difference sets from quadratic forms and uniform cyclotomy
3 Partial difference sets from weakly regular p-ary bent functions..
References
Governing 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
References
Word-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
References
Statistics of Zeros of Families of L-functions over Function Fields: A Survey
1 Introduction
2 Hyperelliptic curves
3 Cyclic/-fold covers of the projective line
4 Elliptic curves over a rational function field and generalizations
5 Concluding remarks
References
Lectures 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
References
Conjectures and Results on x2 mod p2 with 4p = x2 + dy2
1 Introduction
2 Using Apery polynomials and products of three binomial coefficients
3 Using the polynomials
4 Using the function
5 Using the function
6 Using
7 Miscellaneous things
References
Harmonic Weak Maass Forms, Automorphic Green Functions, and Period Integrals
1 Introduction
2 Shimura varieties of orthogonal type and their Kudla cycles
3 Harmonic weak Maass forms, regularized theta lifting, and automorphic Green functions
4 Eisenstein series associated to coherent and incoherent quadratic spaces
5 Period integrals of the automorphic Green function φ(z, h; f)
6 Big CM values of automorphic Green functions
References
Some 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
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
References
Partial Difference Sets from Quadratic Forms and p-ary Weakly
Regular Bent Functions
1 Introduction
2 Partial difference sets from quadratic forms and uniform cyclotomy
3 Partial difference sets from weakly regular p-ary bent functions..
References
Governing 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
References
Word-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
References
Statistics of Zeros of Families of L-functions over Function Fields: A Survey
1 Introduction
2 Hyperelliptic curves
3 Cyclic/-fold covers of the projective line
4 Elliptic curves over a rational function field and generalizations
5 Concluding remarks
References
Lectures 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
References
Conjectures and Results on x2 mod p2 with 4p = x2 + dy2
1 Introduction
2 Using Apery polynomials and products of three binomial coefficients
3 Using the polynomials
4 Using the function
5 Using the function
6 Using
7 Miscellaneous things
References
Harmonic Weak Maass Forms, Automorphic Green Functions, and Period Integrals
1 Introduction
2 Shimura varieties of orthogonal type and their Kudla cycles
3 Harmonic weak Maass forms, regularized theta lifting, and automorphic Green functions
4 Eisenstein series associated to coherent and incoherent quadratic spaces
5 Period integrals of the automorphic Green function φ(z, h; f)
6 Big CM values of automorphic Green functions
References
Some 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
