REDUCED VECTOR TECHNIQUE HOMOMORPHIC ENCRYPTION WITH VERSORS A SURVEY AND A PROPOSED APPROACH by SUNEETHA TEDLA B.Sc, Osmania University, India 1993 M.C.A, Osmania University, India 1998 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 2019 © 2019 SUNEETHA TEDLA ALL RIGHTS RESERVED ii This dissertation for the Doctor of Philosophy degree by Suneetha Tedla has been approved for the Department of Computer Science by Carlos Araujo, Co-Chair C. Edward Chow, Co-Chair T.S. Kalkur Jonathan Ventura Yanyan Zhuang Date: 3 May 2019 iii Tedla, Suneetha (Ph.D., Security) Reduced Vector Technique Homomorphic Encryption with Versors A Survey and a Proposed Approach Dissertation directed by Professors Carols Araujo and C. Edward Chow ABSTRACT In this research, a new type of homomorphic encryption technique, based on geometric algebra and versors, called Reduced Vector Technique Homomorphic Encryption (RVTHE) is designed, developed and analyzed. This new cipher method is optimized to be faster and compact in cipher length while preserving the security strength. Performance criteria are proposed to generate benchmarks to evaluate the homomorphic encryption for a fair comparison to benchmarks used for non-homomorphic encryption. The basic premise behind these performance criteria is to establish the understanding of the baseline to measure the variations of performance between different encryption methods for Cloud Storage type Solid State Drives (SSDs). Significant differences in throughput performances, up to 20-50% decreases, are observed among encryption software methods on Cloud storage SSD or encrypted SSDs. The central thesis of the research is to verify that homomorphic encryption is better accomplished with the use of versors instead of multi-vectors. Using properties of versors, it is possible to design a homomorphic cipher that has simple structure versality of key assignments while achieving a great speed that rivals existing non-homomorphic ciphers. iv In the thesis, I demonstrated that the versors based homomorphic encryption is faster than an existing non-homomorphic encryption AES Crypt which based on AES. It is shown that RVTHE is a symmetric somewhat homomorphic encryption performing addition, deletion, scalar multiplication, and scalar division. The evaluation of the implementation shows a file can be edited/appended in .001 sec. And it showed, in the case of full file encryption, RVTHE is 75% faster on encryption and 25% slower on decryption, compared with the AES-Crypt encryption software which implements the AES standard. The ciphertext sizes of RVTHE are found to be reduced on average of 25% from those of previous approaches using multi-vectors and Clifford Geometric Algebra. RVTHE has the potential for use as an encryption method on real workloads. Keywords: Encryption, Homomorphic, AES, SSD, AES-Crypt, Vectors, Versors. v DEDICATION I wish to dedicate this body of research to my husband and my best friend Shravan Tedla; with him everything is possible for me. vi ACKNOWLEDGEMENTS I am blessed with beautiful people in my life. I am very thankful to all who supported me with my journey of schooling. I really appreciate all the support, encouragement, love and understanding provided by my family, friends, colleagues and Advisory Committee. A special thank you to Dr. Carols Araujo and Dr. C. Edward Chow for their support, sharing their knowledge, and guiding me for the last several years. Dr. Xiaobo Charles Zhou advised me prior to Dr.Carols Araujo, and I am very thankful to Dr. Xiaobo providing me the skills and insight needed to pursue my Ph.D. I very much enjoyed and admired Dr. Carols Araujo’s knowledge and the way he educates his thoughts to create a new way of doing the security, and that helped me tremendously for my research. I really appreciate Dr. Chow’s support and knowledge while discussing the ideas and analyzing how to put my thoughts and ideas into actions. I am very thankful to both of you. I appreciate my Advisory Committee members: Dr. Jonathan Ventura, Dr. Yanyan Zhuang, Dr. T.S.Kalkur providing me their feedback and support. Many thanks to Ali Langfels who helps all the students with a great smile while managing all the administrative work. I am very thankful to my parents and my in-laws; one gave me the beautiful life and one provided me the beautiful life partner with their unconditional love and support. I am blessed with beautiful friend, my husband Shravan Tedla, and my kids SaiKiran and Siddhartha and my gratitude to them supporting me in all aspects of my life including my Ph.D. I am very thankful to my friend Tim Murphy spending so many hours to help me to write thesis. vii TABLE OF CONTENTS CHAPTER ONE INTRODUCTION ....................................................................................1 1.1 Security Terminology ................................................................................... 2 1.2 Security systems............................................................................................ 4 1.3 Cloud Storage Security ................................................................................. 5 1.4 Design Criteria for Cryptographic Algorithm............................................... 6 1.5 Encryption ..................................................................................................... 7 1.6 Homomorphic Encryption ............................................................................ 8 1.7 Possible fully homomorphic encryption method ........................................ 10 1.8 Vector product spaces with Clifford Geometric Algebra ........................... 13 1.9 Dissertation Contributions ......................................................................... 14 1.10 Dissertation Organization ........................................................................... 15 CHAPTER TWO BACKGROUND ..................................................................................17 2.1 Cloud Storage SSD ..................................................................................... 17 2.1.1 Data Reliability and Integrity .................................................................. 18 2.1.2 Sanitization and Secure Deletion of SSD ................................................ 18 2.2 Survey of Various Encryption Approaches ................................................ 19 2.2.1 Block Ciphers .......................................................................................... 20 viii 2.2.2 Block Cipher Modes................................................................................ 29 2.2.3 Encryption Methods for SSD .................................................................. 36 2.2.4 Comparable Encryption for Evaluations ................................................. 40 2.2.5 Homomorphic Encryption ....................................................................... 40 2.3 Mathematical Foundation ........................................................................... 41 2.3.1 Geometric Algebra Overview ................................................................. 45 2.3.2 Inner Product ........................................................................................... 47 2.3.3 Outer Product .......................................................................................... 48 2.3.4 Geometric Product................................................................................... 49 2.3.5 Inverse of Vector ..................................................................................... 52 2.3.6 Versors .................................................................................................... 52 CHAPTER THREE PROBLEMS AND LIMITATIONS .................................................55 3.1 Defining the Problem .................................................................................. 55 3.1.1 Encryption Security Limitations and Problem ........................................ 56 3.1.2 Encryption Limitations:........................................................................... 57 3.2 Other problems contributed for research motivation .................................. 57 3.2.1 Cyber Attacks .......................................................................................... 59 3.2.2 Real Randomness .................................................................................... 60 3.2.3 Storage Security Limitations ................................................................... 61 3.2.4 SSD System Level Induced Limitations ................................................. 62 ix 3.2.5 Existing research to mitigate the software limitations ............................ 70 CHAPTER FOUR STORAGE ENCRYPTION ANALYSIS ............................................78 4.1 Measurement Environment ......................................................................... 78 4.1.1 Selection of Encryption methods ............................................................ 80 4.1.2 Experimental Tools and Workloads ........................................................ 82 4.2 SSD performance without Encryption ........................................................ 83 4.2.1 Performance differences between Amazon EC2 VMs ............................ 83 4.2.2 Did various block sizes significantly affect I/O throughput? .................. 84 4.2.3 Did various levels of parallelism affect