UNLV Theses, Dissertations, Professional Papers, and Capstones May 2018 Elliptic Cryptosystem Elizabeth Dettrey Follow this and additional works at: https://digitalscholarship.unlv.edu/thesesdissertations Part of the Computer Sciences Commons Repository Citation Dettrey, Elizabeth, "Elliptic Cryptosystem" (2018). UNLV Theses, Dissertations, Professional Papers, and Capstones. 3242. http://dx.doi.org/10.34917/13568439 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 Theses, Dissertations, Professional Papers, and Capstones by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. ELLIPTIC CRYPTOSYSTEM by Elizabeth Dettrey Bachelor of Science (B.Sc.) University of Nevada, Las Vegas 2016 A thesis submitted in partial fulfillment of the requirements for the Master of Science in Computer Science Department of Computer Science Howard R. Hughes College of Engineering The Graduate College University of Nevada, Las Vegas May 2018 Thesis Approval The Graduate College The University of Nevada, Las Vegas April 10, 2018 This thesis prepared by Elizabeth Dettrey entitled Elliptic Cryptosystem is approved in partial fulfillment of the requirements for the degree of Master of Science in Computer Science Department of Computer Science Evangelos Yfantis, Ph.D. Kathryn Hausbeck Korgan, Ph.D. Examination Committee Chair Graduate College Interim Dean Hal Berghel, Ph.D. Examination Committee Member Andreas Stefik, Ph.D. Examination Committee Member Sarah Harris, Ph.D. Graduate College Faculty Representative ii Abstract The elliptic cryptographic algorithm first presented in a paper by E. F. Dettrey and E. A. Yfantis is examined and explained in this thesis. The algorithm is based on the group operations of a set of points generated from an ellipse of arbitrary radii, and arbitrary center in the case of the generalized version, modulo a large prime. The security of the algorithm depends on the difficulty of solving a discrete logarithm in the groups used by this algorithm. While the elliptic cryptographic algorithm is not the most secure among the discrete logarithm based paradigm of cryptosystems for a given prime, the algorithm can reach relatively high levels of security using a very large prime. It has similar security to RSA, which is still widely used, when the prime in elliptic cryptography is of a similar size to the modulus in RSA. It should be noted that the elliptic cryptographic algorithm is not quantum safe. iii Table of Contents Abstract iii Table of Contents iv List of Tables vi List of Figures vii List of Equations viii Chapter 1 Introduction 1 Chapter 2 Background 2 Chapter 3 A New Elliptic Cryptographic Algorithm 4 3.1 Preface . 4 3.2 Abstract . 4 3.3 Introduction . 5 3.4 The Elliptic Cryptographic Algorithm . 6 3.4.1 Ability of the Algorithm to Encrypt-Decrypt Data . 8 3.4.2 The Discrete Logarithm Problem in Elliptic Cryptography . 10 3.4.3 Key Exchange Analogous to Diffie-Hellman . 12 3.4.4 Encryption-Decryption using Elliptic Cryptography . 12 3.4.5 Digital Signing . 12 3.5 Generalization of the Algorithm to an Ellipse of Arbitrary Center . 13 3.6 Conclusion . 15 Chapter 4 Examples 16 4.1 Example Using Ep(a; b) .................................. 16 iv 4.2 Example Using Ep(a; b; c; d)................................ 19 Chapter 5 Code Demonstration 22 Chapter 6 Conclusion 28 Appendix A IEEE Permission 29 Appendix B Second Author Permission 31 Appendix C Full Code for the Demonstration in Chapter 5 32 Curriculum Vitae 43 v List of Tables 4.1 Points in the E11(5; 6) Group . 17 4.2 Order of the Points in the E11(5; 6) Group . 17 4.3 Cayley Table for the E11(5; 6) Group . 18 4.4 Points in the E11(7; 5; 7; 9) Group . 19 4.5 Order of the Points in the E11(7; 5; 7; 9) Group . 20 4.6 Cayley Table for the E11(7; 5; 7; 9) Group . 20 vi List of Figures 5.1 This is C++ code for finding the inverses of the radii. 23 5.2 This is C++ code for finding the quadratic residues in Fp. 24 5.3 This is C++ code for adding two points in E127(50; 103). 24 5.4 This is C++ code for finding a base point using E127(50; 103). 25 5.5 This is C++ code for calculating A's public key and the shared key using E127(50; 103). 26 5.6 This is C++ code for encryption using E127(50; 103). 26 5.7 This is C++ code for decryption using E127(50; 103). 27 vii List of Equations and Proofs 3.1 Defining Equation for the Points in E(a; b) ....................... 6 3.2 Definition of θ1 and θ2 in E(a; b) ............................. 6 3.3 Definition of Addition for θ in E(a; b) .......................... 6 3.4 Definition of Addition for x in E(a; b) .......................... 6 3.5 Definition of Addition for y in E(a; b) .......................... 6 3.6 Proof of Closure for E(a; b) ................................ 7 3.7 Proof of Associativity for E(a,b) . 7 3.8 Proof of the Existence of an Identity in E(a; b) ..................... 7 3.9 Proof of the Existence of Inverses in E(a; b)....................... 7 3.10 Proof of Commutativity in E(a; b) ............................ 7 3.11 Defining Equation for the Set Ep(a; b) .......................... 7 3.12 Defining Equation for Ep(a; b)............................... 7 3.13 Definition of Addition Extended to R2 .......................... 8 3.14 Inverses in Ep(a; b)..................................... 8 3.15 Special Addition for (c; 0) in R2 .............................. 8 3.16 Special Addition for (c; 0) in Ep(a; b)........................... 8 3.17 Proof (a; 0) is an Identity in R2 .............................. 8 3.18 Proof of Closure of Inverses in Ep(a; b).......................... 9 3.19 Proof of the Special Addition of (c; 0) in R2 ....................... 9 3.20 Proof of the Special Addition of (c; 0) in Ep(a; b).................... 9 3.21 Proof of the Ability to Encrypt in Ep(a; b)........................ 9 3.22 Proof of the Ability to Decrypt in Ep(a; b)........................ 9 3.23 Definition of the Order of an Element of in Ep(a; b)................... 10 3.24 General Equation of Exponentiation in Ep(a; b)..................... 10 3.25 Equation for Genus 0 Curves Used by Menezes and Vanstone . 10 viii 3.26 Proof That Ellipses Are Genus 0 Curves of the Form of Equation 3.25 . 10 3.27 Definition of Subexponential Time . 10 3.28 Definition of Addition in Ep(a; b; c; d)........................... 13 3.29 Proof of Closure in Ep(a; b; c; d).............................. 13 3.30 Proof of the Existence of an Identity in Ep(a; b; c; d) .................. 13 3.31 Proof of the Uniqueness of the Identity in Ep(a; b; c; d) . 13 3.32 Proof of the Existence of Inverses in Ep(a; b; c; d).................... 14 3.33 Proof of the Uniqueness of Inverses in Ep(a; b; c; d) ................... 14 3.34 Proof of Associativity in Ep(a; b; c; d)........................... 14 3.35 Proof of Commutativity in Ep(a; b; c; d) ......................... 14 3.36 Defining Equation for Ep(a; b; c; d) ............................ 14 4.1 Defining Equation for E11(5; 6).............................. 16 4.2 Example of Calculating A's Public Key in E11(5; 6) . 18 4.3 Example of Calculating a Shared Key in E11(5; 6) . 18 4.4 Example of Encryption Using E11(5; 6) . 18 4.5 Example of Decryption Using E11(5; 6) . 19 4.6 Defining Equation for E11(7; 5; 7; 9)............................ 19 4.7 Example of Calculating B's Public Key in E11(7; 5; 7; 9) . 20 4.8 Example of Calculating a Shared Key in E11(7; 5; 7; 9) . 20 4.9 Example of Encryption Using E11(7; 5; 7; 9) . 21 4.10 Example of Decryption Using E11(7; 5; 7; 9) . 21 5.1 Definition of an Inverse Modulo P . 22 ix Chapter 1 Introduction The mathematical difficulty of calculating discrete logarithms has long made them an easy choice for providing security in cryptographic algorithms. This thesis will present a new cryptographic algorithm using discrete logarithms for security. The elliptic cryptographic algorithm uses groups created using the points on ellipses of varying radii and center, modulo a prime. While elliptic cryptography shares some similarities with elliptic curve cryptography in the flow of the algorithms and the general premise, the use of ellipses rather than elliptic curves yields different groups. It will be shown that elliptic curve cryptography can produce significantly safer groups than this new alternative, though elliptic cryptography is arguably less complicated. The security of the proposed algorithm will be investigated in the following chapters. The proposed algorithm will be fully defined and proved to work for cryptography. The aim is to present a new cryptographic algorithm, examine its security, and thoroughly explain how it works and how to use it. 1 Chapter 2 Background Some of the most familiar cryptosystems today rely on the same mathematical problem to provide security. It is the discrete logarithm problem in finite groups. RSA, elliptic curve cryptography, El Gamal, and even the algorithm to be presented in this paper are based on different groups and discrete logarithms.