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Security Systems Based on Gaussian Integers : Analysis of Basic Operations and Time Complexity of Secret Transformations

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Security Systems Based on Gaussian Integers : Analysis of Basic Operations and Time Complexity of Secret Transformations New Jersey Institute of Technology Digital Commons @ NJIT Dissertations Electronic Theses and Dissertations Summer 8-31-2011 Security systems based on Gaussian integers : Analysis of basic operations and time complexity of secret transformations Aleksey Koval New Jersey Institute of Technology Follow this and additional works at: https://digitalcommons.njit.edu/dissertations Part of the Computer Sciences Commons Recommended Citation Koval, Aleksey, "Security systems based on Gaussian integers : Analysis of basic operations and time complexity of secret transformations" (2011). Dissertations. 277. https://digitalcommons.njit.edu/dissertations/277 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at Digital Commons @ NJIT. It has been accepted for inclusion in Dissertations by an authorized administrator of Digital Commons @ NJIT. For more information, please contact [email protected]. Copyright Warning & Restrictions The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or reproduction is not to be “used for any purpose other than private study, scholarship, or research.” If a, user makes a request for, or later uses, a photocopy or reproduction for purposes in excess of “fair use” that user may be liable for copyright infringement, This institution reserves the right to refuse to accept a copying order if, in its judgment, fulfillment of the order would involve violation of copyright law. Please Note: The author retains the copyright while the New Jersey Institute of Technology reserves the right to distribute this thesis or dissertation Printing note: If you do not wish to print this page, then select “Pages from: first page # to: last page #” on the print dialog screen The Van Houten library has removed some of the personal information and all signatures from the approval page and biographical sketches of theses and dissertations in order to protect the identity of NJIT graduates and faculty. ABSTRACT SECURITY SYSTEMS BASED ON GAUSSIAN INTEGERS: ANALYSIS OF BASIC OPERATIONS AND TIME COMPLEXITY OF SECRET TRANSFORMATIONS by Aleksey Koval Many security algorithms currently in use rely heavily on integer arithmetic modulo prime numbers. Gaussian integers can be used with most security algorithms that are formulated for real integers. The aim of this work is to study the benefits of common security protocols with Gaussian integers. Although the main contribution of this work is to analyze and improve the application of Gaussian integers for various public key (PK) algorithms, Gaussian integers were studied in the context of image watermarking as well. The significant benefits of the application of Gaussian integers become apparent when they are used with Discrete Logarithm Problem (DLP) based PK algorithms. In order to quantify the complexity of the Gaussian integer DLP, it is reduced to two other well known problems: DLP for Lucas sequences and the real integer DLP. Additionally, a novel exponentiation algorithm for Gaussian integers, called Lucas sequence Exponentiation of Gaussian integers (LSEG) is introduced and its performance assessed, both analytically and experimentally. The LSEG achieves about 35% theoretical improvement in CPU time over real integer exponentiation. Under an implementation with the GMP 5.0.1 library, it outperformed the GMP’s "mpz_powm" function (the particularly efficient modular exponentiation function that comes with the GMP library) by 40% for bit sizes 1000-4000, because of low overhead associated with LSEG. Further improvements to real execution time can be easily achieved on multiprocessor or multicore platforms. In fact, over 50% improvement is achieved with a parallelized implementation of LSEG. All the mentioned improvements do not require any special hardware or software and are easy to implement. Furthermore, an efficient way for finding generators for DLP based PK algorithms with Gaussian integers is presented. In addition to DLP based PK algorithms, applications of Gaussian integers for factoring-based PK cryptosystems are considered. Unfortunately, the advantages of Gaussian integers for these algorithms are not as clear because the extended order of Gaussian integers does not directly come into play. Nevertheless, this dissertation describes the Extended Square Root algorithm for Gaussian integers used to extend the Rabin Cryptography algorithm into the field of Gaussian integers. The extended Rabin Cryptography algorithm with Gaussian integers allows using fewer preset bits that are required by the algorithm to guard against various attacks. Additionally, the extension of RSA into the domain of Gaussian integers is analyzed. The extended RSA algorithm could add security only if breaking the original RSA is not as hard as factoring. Even in this case, it is not clear whether the extended algorithm would increase security. Finally, the randomness property of the Gaussian integer exponentiation is utilized to derive a novel algorithm to rearrange the image pixels to be used for image watermarking. The new algorithm is more efficient than the one currently used and it provides a degree of cryptoimmunity. The proposed method can be used to enhance most picture watermarking algorithms. SECURITY SYSTEMS BASED ON GAUSSIAN INTEGERS: ANALYSIS OF BASIC OPERATIONS AND TIME COMPLEXITY OF SECRET TRANSFORMATIONS by Aleksey Koval A Dissertation Submitted to the Faculty of New Jersey Institute of Technology in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Computer Science Department of Computer Science August 2011 Copyright © 2011 by Aleksey Koval ALL RIGHTS RESERVED . APPROVAL PAGE SECURITY SYSTEMS BASED ON GAUSSIAN INTEGERS: ANALYSIS OF BASIC OPERATIONS AND TIME COMPLEXITY OF SECRET TRANSFORMATIONS Aleksey Koval Dr. Boris Verkhovsky, Dissertation Advisor Date Professor of Computer Science, NJIT Dr. Frank Shih, Committee Member Date Professor of Computer Science, NJIT Dr. Cristian Borcea, Committee Member Date Associate Professor of Computer Science, NJIT Dr. James Geller, Committee Member Date Professor of Computer Science, NJIT Dr. Joon Sung, Committee Member Date Technical Manager, IBM, AT&T Labs,Middletown, NJ BIOGRAPHICAL SKETCH Author: Aleksey Koval Degree: Doctor of Philosophy Date: August 2011 Undergraduate and Graduate Education: • Doctor of Philosophy in Computer Science, New Jersey Institute of Technology, Newark, NJ, 2011 • Master of Science in Computer Science, New Jersey Institute of Technology, Newark, NJ, 2009 • Master of Science in Applied Statistics, Rutgers University, New Brunswick, NJ, 1999 • Bachelor of Science in Computer Science, Kean University, Union, NJ, 1997 • Bachelor of Science in Mathematics, Kean University, Union, NJ, 1997 Major: Computer Science Presentations and Publications: A. Koval, F. Y. Shih, and B. S. Verkhovsky, "A Pseudo-Random Pixel Rearrangement Algorithm Based on Gaussian Integers for Image Watermarking," Journal of Information Hiding and Multimedia Signal Processing, vol. 2, no. 1, pp. 60-70, 2010. A. Koval, "On Lucas Sequences Computation," Intl J. of Communications, Network and System Sciences vol. 2, no. 12, pp. 943-944 2010. iv A. Koval, and B. S. Verkhovsky, “On Discrete Logarithm Problem for Gaussian Integers,” in International Conference on Information Security and Privacy (ISP- 09), Orlando, Florida, USA, 2009, pp. 79-84. A. Koval, and B. Verkhovsky, “Analysis of RSA over Gaussian Integers Algorithm,” in Fifth International Conference on Information Technology: New Generations (ITNG 2008), Las Vegas, Nevada, USA, 2008, pp. 101-105. B. Verkhovsky, and A. Koval, “Cryptosystem Based on Extraction of Square Roots of Complex Integers,” in Fifth International Conference on Information Technology: New Generations (ITNG 2008), Las Vegas, Nevada, USA, 2008, pp. 1190-1191. v This dissertation is dedicated to the memory of my father Dr. Yevgeniy Aleksandrovich Koval. vi ACKNOWLEDGMENT I wish to thank my advisor, Dr. Boris Verkhovsky for all his guidance, dedication, patience and support. His sincere curiosity in the unexplored fields of cryptography inspired me. His dedication and professionalism helped me overcome all of the obstacles. In addition, I would like to thank the members of my committee Dr. Frank Shih, Dr. Cristian Borcea, Dr. James Geller, and Dr. Joon Sung. Dr. James Geller spent a lot of his time helping me improve this dissertation. I really appreciate the insightful observations of Dr. Christian Borcea that directed my work in the right direction. Also, I appreciate Dr. Frank Shih’s teaching and guidance, especially on image related topics. Dr. Joon Sung provided great advice, criticism and encouragement. I would like to thank Dr. Dimitri Kanevsky (IBM T.J Watson Research Center) for his support and guidance. vii TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 1.1 Problem Statement…………………………………………………………. 4 1.2 Survey of References………………………………………………………. 6 1.3 Overview of Gaussian Integers, Notation and Definitions………………… 14 1.4 Dissertation Structure……………………………………………………… 22 2 DISCRETE LOGARITHM CRYPTOGRAPHY WITH GAUSSIAN INTEGERS………………………………………………………………………... 24 2.1 Gaussian Primes P: |P| is a non-Blum Prime…………………………….… 24 2.2 Common Cryptography Algorithms Based
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