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Cryptography in the Digital Age UNLV Retrospective Theses & Dissertations 1-1-2007 Cryptography in the digital age Thomas R. Hodge University of Nevada, Las Vegas Follow this and additional works at: https://digitalscholarship.unlv.edu/rtds Repository Citation Hodge, Thomas R., "Cryptography in the digital age" (2007). UNLV Retrospective Theses & Dissertations. 2237. http://dx.doi.org/10.25669/eb4o-rhq3 This Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Thesis has been accepted for inclusion in UNLV Retrospective Theses & Dissertations by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. CRYPTOGRAPHY IN THE DIGITAL AGE by Thomas R. Hodge, Jr. Bachelor of Science Angelo State University, San Angelo, Texas 2002 A thesis submitted in partial fulfillment of the requirements for the Master of Science Degree in Mathematical Sciences Department of Mathematical Sciences College of Sciences Graduate College University of Nevada, Las Vegas December 2007 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 1452248 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. UMI UMI Microform 1452248 Copyright 2008 by ProQuest LLC. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 E. Eisenhower Parkway PC Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Disclaimer The views expressed in this thesis are those of the author and do not reflect the official policy or position of the United States Air Force, the Department of Defense, or the U.S. government. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thesis Approval The Graduate College University of Nevada, Las Vegas November 20 _,20iLZ_ The Thesis prepared by Thomas R. Hodge, Jr. Entitled Cryptography in the Digital Age is approved in partial fulfillment of the requirements for the degree of Master of Science in Mathematical Sciences Examination Committee Chair Dean of the Graduate College Examination Committee Member Examination Cemmittee Member Ciaé-uate College Faculty Representative 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Cryptography in the Digital Age by Thomas R. Hodge, Jr. Dr. Arthur Baragar, Examination Committee Chair Professor of Mathematical Sciences University of Nevada, Las Vegas Despite it not being an armed conflict between nations, there is a war that has been waged for over 5000 years and is still being fought today. Battles have been won and lost by both sides. The battlefield is the world of cryptography. Our combatants are cryptographers, personnel who make secret codes; and cryptanalysts, personnel who try to break the secret codes. In this thesis, we examine public-key or asymmetric cryptography, the art of writing or deciphering secret codes or ciphers. We begin by taking a brief look at the overall his­ tory of cryptography. Our primary focus involves studying the mathematics behind today’s public-key cryptographic methods such as the theory of congruences by Carl Friedrich Causs, Fermat’s little theorem, Euler’s phi-function, primitive roots and indices, and elliptic curves over finite fields. Once we have explored the preliminaries we will consider some of the more popular methods of encryption and decryption, for example RSA. We not only discuss how to encrypt and decrypt plain text using these methods but explain why it is hard to break the encrypted text. We conclude our study by inspecting the shortfalls of these ciphers, techniques used to break the encryption, and what the future possibly holds. Ill Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OE CONTENTS ABSTRACT ........................................................................................................................................ iii LIST OE FICURES ......................................................................................................................... v LIST OF TABLES ............................................................................................................................ vi ACKNOWLEDCEMENTS ............................................................................................................. vii CHARTER 1 HISTORY OF CRYPTOCRAPHY ................................................................ 1 Commonly Used W ords ................................................................................................................. 1 S k y tale ................................................................................................................................................ 1 Caesar’s C ip h er ................................................................................................................................. 2 World War I I .................................................................................................................................... 3 The Dawn of a New A ge ............................................................................................................... 4 CHAPTER 2 MATHEMATICS .................................................................................................. 10 Preliminaries .................................................................................................................................... 10 Theory of Congruences ................................................. 15 Fermat’s Little Theorem ............................................................................................................... 18 CHAPTERS RSA ........................................................................................................................ 20 Euler’s Phi-function ......................................................................................................................... 20 RSA Cipher ...................................................................................................................................... 22 Security of RSA ............................................................................................................................... 27 Cryptanalysis .................................................................................................................................... 29 CHAPTER 4 EL CAMAL (ELC) ............................................................................................ 32 Primitive Roots and Indices ......................................................................................................... 32 EIC C ip h e r ........................................................................................................................................ 39 Security of E IC ................................................................................................................................. 42 CHAPTER 5 ELLIPTIC CURVES .......................................................................................... 44 Elliptic Curves over the Reals ...................................................................................................... 44 Elliptic Curves over F , ................................................................................................................... 51 Security of ECC over Fp ............................................................................................................... 55 CHAPTER 6 CONCLUSION ...................................................................................................... 57 BIBLIOCRAPHY ............................................................................................................................ 59 VITA ..................................................................................................................................................... 61 IV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES 1.1 S kytale ............................................................................................................................................. 2 1.2 The Caesar C ipher ................................................................... 3 1.3 Diffe-Hellman Public-key Cryptography ................................................................................. 6 5.1 An Elliptic Curve .........................................................................................................................
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