數(shù)論及其相關(guān)領(lǐng)域

內(nèi)容概要

《數(shù)論及其相關(guān)領(lǐng)域(英文)》內(nèi)容定位于馮克勤教授作出重要成就及感興趣的領(lǐng)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(lǐng)域的應用，反映了馮克勤教授在數(shù)論及其應用領(lǐng)域的發(fā)展所作的貢獻和影響。

書籍目錄

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 K20oas 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

章節(jié)摘錄

版權(quán)頁：   插圖：   6Conclusion We propose the study of a new family of word-oriented LFSR,s with High Efficiency: σ-LFSRs, which might be interesting in the design of modern stream ciphers.By the theory of matrix and polynomial over finite fields, we explore the cryptographic properties of σ-LFSR.s such as minimal polynomial, state graph,coordinate sequences, and several properties of primitive σ-LFSRs etc.We also discuss some principles for choosing σ-LFSR,s suitable for software implementation,and then we offer an algorithm to search for primitive cr-LFSRs and obtain many primitiveσ-LFSR.s such as HHZ-1 or HH2-2, etc, which only use few memory and basic instructions.At last, we give two applications in cryptography: constructing RNGs and stream ciphers. These studies implicate that σ-LFSRs may be used as building block in many cryptographic schemes.We believe that as the performance, security and resource consumption be increasingly important, σ-LFSRs will become an important and attractive alternative to traditional bit and byte oriented designs.However, there remain many open problems for future work.Nevertheless we hope that the results and ideas in this paper serve as an initial step in establishing a continuing research on the design of fast and secure word-oriented LFSRs for cryptographic purpose. Acknowledgement.The work is supported by a grant from the National Natural Science Foundation of China （No.61003291）, National Basic Research Program of China （No.2007CB807902）, and Foundation for the Author of National Excellent Doctoral Dissertation of China （No.FANEDD-2007874）.

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《數(shù)論及其相關(guān)領(lǐng)域(英文)》不僅是數(shù)論及其應用領(lǐng)域?qū)＜覀冇袃r值的參考書，也是研究生開展研究時極好的入門書。

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