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E-raamat: Algebraic Aspects of Cryptography

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This book is intended as a text for a course on cryptography with emphasis on algebraic methods. It is written so as to be accessible to graduate or advanced undergraduate students, as well as to scientists in other fields. The first three chapters form a self-contained introduction to basic concepts and techniques. Here my approach is intuitive and informal. For example, the treatment of computational complexity in Chapter 2, while lacking formalistic rigor, emphasizes the aspects of the subject that are most important in cryptography. Chapters 4-6 and the Appendix contain material that for the most part has not previously appeared in textbook form. A novel feature is the inclusion of three types of cryptography - "hidden monomial" systems, combinatorial-algebraic sys­ tems, and hyperelliptic systems - that are at an early stage of development. It is too soon to know which, if any, of these cryptosystems will ultimately be of practical use. But in the rapidly growing field of cryptography it is worthwhile to continually explore new one-way constructions coming from different areas of mathematics. Perhaps some of the readers will contribute to the research that still needs to be done. This book is designed not as a comprehensive reference work, but rather as a selective textbook. The many exercises (with answers at the back of the book) make it suitable for use in a math or computer science course or in a program of independent study.

Arvustused

"... This is a textbook in cryptography with emphasis on algebraic methods. It is supported by many exercises (with answers) making it appropriate for a course in mathematics or computer science. ... Overall, this is an excellent expository text, and will be very useful to both the student and researcher."



M.V.D.Burmester, Mathematical Reviews 2002



"... I think this book is a very inspiring book on cryptography. It goes beyond the traditional topics (most of the cryptosystems presented here are first time in a textbook, some of Paturi's work is not published yet). This way the reader has the feeling how easy to suggest a cryptosystem, how easy to break a safe looking system and hence how hard to trust one. The interested readers are forced to think together with ther researchers and feel the joy of discovering new ideas. At the same time the importance of "hardcore" mathematics is emphasized and hopefully some application driven students will be motivated to study theory."



Péter Hajnal, Acta Scientiarum Mathematicarum, 64.1998, p. 750



"... Overall, the book is highly recommended to everyone who has the requisite mathematical sophistication."



E.Leiss, Computing Reviews 1998, p. 506



"... Der Autor, der ... vielen Lesern dieses Rundbriefs gut bekannt sein wird, hat hier ein kleines Werk vorgelegt, das man wohl am Besten als "Lesebuch zu algebraischen Aspekten der Kryptographie mit öffentlichem Schlüssel" charakterisieren kann. ...



Mit zunehmender Schwierigkeit des Material werden die Ausführungen dabei skizzenhafter und beschränken sich immer stärker auf den Hinweis auf entsprechende Quellen, was den Charakter eines guten "Lesebuchs", wie ich es oben bezeichnet habe, ausmachen sollte. Das Buch eignet sich damit selbst für "advanced undergraduates", wie es im Klappentext heißt, als Einstieg und erster Überblick über ein Gebiet, in dem sich in den letzten Jahren auf überraschendeWeise praktische Anwendungsmöglichkeiten für tief innermathematische Themen ergeben haben."



Hans-Gert Gräbe, Computeralgebra Rundbrief 1999, Issue 25



"... Es gelingt Koblitz, anschaulich und mit elementaren Mitteln auch Dinge zu erläutern, die in vergleichbaren Texten kaum zu finden sind: z.B. den Hilbertschen Basis- und Nullstellensatz, sowie Gröbnerbasen. ..."



Franz Lemmermeyer, Mathematische Semesterberichte 1999, 46/1 



 



 


Chapter
1. Cryptography 1(17)
1. Early History 1(1)
2. The Idea of Public Key Cryptography 2(3)
3. The RSA Cryptosystem 5(3)
4. Diffie-Hellman and the Digital Signature Algorithm 8(2)
5. Secret Sharing, Coin Flipping, and Time Spent on Homework 10(2)
6. Passwords, Signatures, and Ciphers 12(1)
7. Practical Cryptosystems and Useful Impractical Ones 13(4) Exercises 17(1)
Chapter
2. Complexity of Computations 18(35)
1. The Big-O Notation 18(4) Exercises 21(1)
2. Length of Numbers 22(2) Exercises 23(1)
3. Time Estimates 24(10) Exercises 31(3)
4. P, NP, and NP-Completeness 34(10) Exercises 41(3)
5. Promise Problems 44(1)
6. Randomized Algorithms and Complexity Classes 45(3) Exercises 48(1)
7. Some Other Complexity Classes 48(5) Exercises 52(1)
Chapter
3. Algebra 53(27)
1. Fields 53(2) Exercises 55(1)
2. Finite Fields 55(8) Exercises 61(2)
3. The Euclidean Algorithm for Polynomials 63(2) Exercises 64(1)
4. Polynomial Rings 65(5) Exercises 70(1)
5. Grobner Bases 70(10) Exercises 78(2)
Chapter
4. Hidden Monomial Cryptosystems 80(23)
1. The Imai-Matsumoto System 80(7) Exercises 86(1)
2. Patarins Little Dragon 87(9) Exercises 95(1)
3. Systems That Might Be More Secure 96(7) Exercises 102(1)
Chapter
5. Combinatorial-Algebraic Cryptosystems 103(14)
1. History 103(1)
2. Irrelevance of Brassards Theorem 104(1) Exercises 105(1)
3. Concrete Combinatorial-Algebraic Systems 105(6) Exercises 109(2)
4. The Basic Computational Algebra Problem 111(1) Exercises 112(1)
5. Cryptographic Version of Ideal Membership 112(1)
6. Linear Algebra Attacks 113(1)
7. Designing a Secure System 114(3)
Chapter
6. Elliptic and Hyperelliptic Cryptosystems 117(38)
1. Elliptic Curves 117(14) Exercises 129(2)
2. Elliptic Curve Cryptosystems 131(6) Exercises 136(1)
3. Elliptic Curve Analogues of Classical Number Theory Problems 137(2) Exercises 139(1)
4. Cultural Background: Conjectures on Elliptic Curves and Surprising Relations with Other Problems 139(5)
5. Hyperelliptic Curves 144(4) Exercises 148(1)
6. Hyperelliptic Cryptosystems 148(7) Exercises 154(1) Appendix. An Elementary Introduction to Hyperelliptic Curves 155(24) Alfred J. Menezes Yi-Hong Wu Robert J. Zuccherato
1. Basic Definitions and Properties 156(3)
2. Polynomial and Rational Functions 159(2)
3. Zeros and Poles 161(6)
4. Divisors 167(2)
5. Representing Semi-Reduced Divisors 169(2)
6. Reduced Divisors 171(1)
7. Adding Reduced Divisors 172(6) Exercises 178(1) Answers to Exercises 179(14) Bibliography 193(8) Subject Index 201
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