A MATHEMATICAL FRAMEWORK TOWARDS EFFICIENT CLIFFORD-BASED HOMOMORPHIC CRYPTOSYSTEMS USING P-ADIC NUMBERS by DAVID WILLIAM HONORIO ARAUJO DA SILVA B.S.B.A., Universidade Potiguar (Brazil), 2012 M.S.C.S., University of Colorado Colorado Springs, 2017 A dissertation submitted to the Graduate Faculty of the University of Colorado Colorado Springs in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Computer Science 2020 © Copyright by David William Honorio Araujo da Silva 2020 All Rights Reserved This dissertation for the Doctor of Philosophy degree by David William Honorio Araujo da Silva has been approved for the Department of Computer Science by Edward C. Chow, Chair Carlos Paz de Araujo Chuan Yue Sang-Yoon Chang Philip Brown 30 November 2020 Date ii Honorio Araujo da Silva, David William (Ph.D., Engineering: Computer Science) A Mathematical Framework Towards Efficient Clifford-Based Homomorphic Cryptosystems using p-adic Numbers Dissertation directed by Professor Edward C. Chow ABSTRACT As we observe the advances in cryptography throughout history, we can see that cryptog- raphy needs to follow society’s changes in general as it gets more and more sophisticated. One critical example of this fact is that there was a time were having a message writ- ten in plain language was enough to hide information from uneducated soldiers. The next step was to apply simple replacements of letters in the message, which quickly evolved to more elaborated scramble techniques. Indeed, until the late 20th century, cryptography was considered an art. Creativity was the single most crucial strategy when producing a new encryption function. However, the advent of computers and the advances in cryptanalysis generated a demand for data security science. In the theory of cryptography, anything un- til the 1980s is considered classical cryptography. From the 1980s, modern definitions of security arose, and thus the cryptography studies and practiced from that time on is re- ferred to as modern cryptography. However, society continues to evolve. New events might change modern definitions of security even further, as is the case, to cite a few examples, the realization of large-scale quantum computers and an eventual default requirement for secure computation. If these two events became a reality today, many of the current crypto- graphic tools would be entirely or partially compromised. It seems to us that their pursuit of new ideas in cryptography must never end as we must provide proper and timely answers to such changes in society as they occur. Our motivation starts by inquiring if there are mathematical tools currently receiving none or little attention by the cryptographic com- munity that could be instrumental in producing new, efficient, and further advantageous cryptographic constructions with the ability to address the abrupt changes in the reality of cryptography such as quantum computers and secure computation. For this reason, we iii propose the use of the finite-segment p-adic arithmetic (Hensel codes) and Clifford geometric Algebra (GA) as the mathematics foundations for the construction of several homomorphic cryptographic tools such as somewhat homomorphic encryption schemes, key update, and key exchange protocols, hash algorithm, homomorphic encryption for special applications including edge computing, homomorphic image processing, and distributed computation. We discuss the security characteristics of constructions based on Hensel codes and GA by examining a leveled fully homomorphic encryption scheme based on Hensel codes whose se- curity is associated with the approximate-gcd problem and implementation of lattice-based cryptography with GA. We also introduce a mapping between arbitrary dimensional vec- tors and matrices to multivectors, which allow us to replace vector and matrix algebra by GA in constructing quantum-resistant lattice-based cryptography. We demonstrate how to use Hensel codes and GA in isolation for cryptography and different ways of combining the two mathematics foundations in a single solution. Finally, we introduce mapping between Hensel codes and multivectors in GA, which allows us to have two algebraic structures into a single one. Our work is the first exposition of a family of cryptographic functions based on Hensel codes and GA, which from concrete examples of application-specific scenarios we evolve to an application-agnostic framework, where Hensel codes and GA and explored as a mathematical framework for the production of efficient general-purpose algorithms that can satisfy modern definitions of security and also stand as candidate solutions for the era of quantum-resistant cryptography and secure computation. iv DEDICATION This dissertation is dedicated to my wife Cimaria, my son Johnathan, my daughters Samara and Sarah, and my parents Janildo and Elisabete. I could never do or be anything without you all in my life. I love you all much more than I am able to describe. v ACKNOWLEDGEMENTS I would like to thank God, first and foremost, and His Son Jesus Christ, for renewing in me every day the certainty that it is worthwhile to live a life with purpose and do not measure efforts in the search for what is good, perfect and pleasant, even in the midst of my many imperfections and limitations; Dr. Carlos Araujo for believing in me since day one, for teaching me that seeking the impossible is an honorable mission and for investing in my academic and professional growth; Dr. Edward Chow for being an encouraging ad- visor, for truly believing in the potential of my research and for keeping a supporting and positive attitude even when I faced some challenging circumstances through the course of my academic pursue; Greg Jones for being a tireless source of inspiration, a constant help- ing hand and someone I can always count on; Dr. Sang-Yoon Chang for accepting being part of my committee, for demonstrating interest in my work and for providing feedback on how to improve my research; Dr. Philip Brown for accepting being part of my committee, for being eager to contribute with my research and for the significant collaboration in one publication; Dr. Chuan Yue for accepting being part of my committee. Hanes Oliveira, Jordan Pattee, and Bhagiradh Kantheti for being trench partners, research companions, for sharing moments of tension and relaxation, for helping in my research in many ways, and for being supportive in all situations; Marcelo Xavier for countless discussions on the most varied ideas, for investing time to understand my challenges in order to help me, and for always believing that the state of impossibility of the impossible is uncertain until proven otherwise. This dissertation would not have been possible without the valuable contribution of each one of you. vi Table of Contents CHAPTER 1 Introduction 1 1.1 Motivation . 4 1.2 Contributions . 5 1.3 Two Important Related Work . 6 1.3.1 Clifford Geometric Algebra . 6 1.3.2 p-adic Numbers . 7 2 Research Questions, Metrics and Methodology 10 2.1 General-Purpose Mathematical Framework . 10 2.2 Mathematical Framework Applied to Cryptography . 12 2.3 Metrics and Methodology . 13 3 Homomorphic Encryption 16 3.1 Requirements . 18 3.2 HE Classes . 19 3.3 Key Contributions . 20 3.4 Conclusions . 26 4 p-Adic Numbers 27 4.1 A Compact Tutorial . 27 4.1.1 Basic Definitions . 28 4.1.2 Finite-Segment p-adic Arithmetic . 28 4.2 Homomorphic Data Encoding . 36 4.2.1 Performance . 37 4.3 Encrypting Rational Numbers . 38 vii 4.3.1 RSA with Rational Numbers . 40 4.4 Adding Randomness to Deterministic Algorithms . 42 4.4.1 Randomized RSA . 42 4.5 Pairing Functions . 45 4.5.1 p-adic Pairing . 46 4.6 Distributed Computation . 47 4.6.1 Description of the Scheme . 49 4.6.1.1 Security of the Scheme . 51 4.7 Conclusions . 52 5 Clifford Geometric Algebra 54 5.1 A Compact Tutorial . 55 5.1.1 Basic Definitions . 55 5.1.2 Basic Definitions in G2 ........................... 57 5.1.3 Basic Definitions in G3 ........................... 59 5.2 A First Experiment Towards FHE Based on GA . 65 5.2.1 Auxiliary Algorithms . 65 5.2.2 The Main Construction . 67 5.2.3 Performance . 68 5.2.4 General Considerations . 70 5.3 A Framework for Homomorphic Image Processing . 70 5.3.1 Auxiliary Algorithms . 71 5.3.2 The Main Construction . 72 5.3.3 Homomorphic Image Processing . 73 5.3.4 Homomorphic Results . 75 5.3.5 Performance . 75 5.3.6 General Considerations . 77 5.4 Multivector Packing Schemes . 77 5.4.1 Multivector Packing Schemes . 78 5.4.2 Clifford Eigenvalue Packing Scheme . 78 viii 5.4.3 Complex Magnitude Squared Packing Scheme . 79 5.5 Concealment Schemes . 81 5.5.1 Clifford Sylvester’s Equation Concealment (CSEC) . 82 5.5.2 Modular Concealment (MC) . 83 5.5.3 General Considerations . 84 5.6 Experimental Key Update . 85 5.6.1 HE Scheme . 85 5.6.2 Key Update Protocol . 86 5.6.3 Application . 86 5.6.4 General Considerations . 88 5.7 Further Cryptographic Experiments . 89 5.7.1 Auxiliary Algorithms . 89 5.7.2 Key Exchange Protocol . 91 5.7.3 Edge Computing Protocol . 93 5.7.4 Hash Algorithm . 94 5.7.5 Private-Key Encryption Scheme . 95 5.8 Conclusions . 97 6 Security with p-adic Numbers and GA 98 6.1 Private-Key Leveled FHE Scheme . 98 6.2 Target Definitions . 99 6.2.1 The Concrete Construction . 101 6.3 Security . 103 6.3.1 Proof by Reduction . 104 6.3.2 Weaker Version of our Scheme . 108 6.3.3 Factorization Attacks . 109 6.3.3.1 Instance With One Prime . 109 6.3.3.2 Instance With Two Primes - Option 1 . 110 6.3.3.3 Instance With Two Primes - Option 2 .