A Concrete Introduction to Higher Algebra, 2nd Edition

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Lecture Notes on Cryptography Shafi Goldwasser 1 Mihir Bellare 2 August 2001 1 MIT Laboratory of Computer Science, 545 Technology Square, Cambridge, MA 02139, USA. E- mail: [email protected] ; Web page: shafi 2 Department of Computer Science and Engineering, Mail Code 0114, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA. E-mail: [email protected] ; Web page:
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Foreword This is a set of lecture notes on cryptography compiled for 6.87s, a one week long course on cryptography taught at MIT by Shafi Goldwasser and Mihir Bellare in the summers of 1996–2001. The notes were formed by merging notes written for Shafi Goldwasser’s Cryptography and Cryptanalysis course at MIT with notes written for Mihir Bellare’s Cryptography and network security course at UCSD. In addition, Rosario Gennaro (as Teaching Assistant for the course in 1996) contributed Section 9.6, Section 11.4, Section 11.5, and Appendix D to the notes, and also compiled, from various sources, some of the problems in Appendix E. Cryptography is of course a vast subject. The thread followed by these notes is to develop and explain the notion of provable security and its usage for the design of secure protocols. Much of the material in Chapters 2, 3 and 7 is a result of scribe notes, originally taken by MIT graduate students who attended Professor Goldwasser’s Cryptography and Cryptanalysis course over the years, and later edited by Frank D’Ippolito who was a teaching assistant for the course in 1991. Frank also contributed much of the advanced number theoretic material in the Appendix. Some of the material in Chapter 3 is from the chapter on Cryptography, by R. Rivest, in the Handbook of Theoretical Computer Science. Chapters 4, 5, 6, 8 and 10, and Sections 9.5 and 7.4.6, were written by Professor Bellare for his Cryptography and network security course at UCSD. All rights reserved. Shafi Goldwasser and Mihir Bellare Cambridge, Massachusetts, August 2001. 2
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Table of Contents 1 Introduction to Modern Cryptography 11 1.1 Encryption: Historical Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Modern Encryption: A Computational Complexity Based Theory . . . . . . . . . . . . . . . . 12 1.3 A Short List of Candidate One Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Security Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 The Model of Adversary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6 Road map to Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 One-way and trapdoor functions 17 2.1 One-Way Functions: Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 One-Way Functions: Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 (Strong) One Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.2 Weak One-Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.3 Non-Uniform One-Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.4 Collections Of One Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.5 Trapdoor Functions and Collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 In Search of Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.1 The Discrete Logarithm Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.2 The RSA function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.3 Connection Between The Factorization Problem And Inverting RSA . . . . . . . . . . 30 2.3.4 The Squaring Trapdoor Function Candidate by Rabin . . . . . . . . . . . . . . . . . . 30 2.3.5 A Squaring Permutation as Hard to Invert as Factoring . . . . . . . . . . . . . . . . . 34 2.4 Hard-core Predicate of a One Way Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.1 Hard Core Predicates for General One-Way Functions . . . . . . . . . . . . . . . . . . 35 2.4.2 Bit Security Of The Discrete Logarithm Function . . . . . . . . . . . . . . . . . . . . . 36 2.4.3 Bit Security of RSA and SQUARING functions . . . . . . . . . . . . . . . . . . . . . . 38 2.5
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