BA, Oberlin College, 2018, Mathematics

At its core, cryptography relies on problems that are simple to construct but difficult to solve unless certain information (the “key”) is known. Many of these problems come from number theory and group theory. One method of obtaining groups from which to build cryptosystems is to define algebraic curves over finite fields and then derive a group structure from the set of points on those curves. This thesis serves as an exposition of Elliptic Curve Cryptography (ECC), preceded by a discussion of some basic cryptographic concepts and followed by a glance into one generalization of ECC: cryptosystems based on hyperelliptic curves.

Committee: Benjamin Linowitz (Advisor) Subjects: Computer Science; Mathematics

Master of Science in Mathematics, Youngstown State University, 2009, Department of Mathematics and Statistics

Wireless sensor nodes generally face serious limitations in terms of computational power, energy supply, and network bandwidth. One of the biggest challenges faced by researches today is to provide effective and secure techniques for establishing cryptographic keys between wireless sensor networks. Public-key algorithms (such as the Diffie-Hellman key-exchange protocol) generally have high energy requirements because they require computational expensive operations. So far, due to the limited computation power of the wireless sensor devices, the Diffie-Hellman protocol is considered to be beyond the capabilities of today's sensor networks. We analyzed existing methods of implementing Diffie-Hellman and proposed a new improved method of implementing the Diffie-Hellman key-exchange protocol for establishing secure keys between wireless sensor nodes. We also provide an easy-to-use implementation of the Elliptic Curve Diffie-Hellman key-exchange protocol for use in wireless sensor networks.

Committee: Graciela Perera PhD (Advisor); John Sullins PhD (Committee Member); Jamal Tartir PhD (Committee Member) Subjects: Communication; Computer Science; Information Systems; Mathematics

Master of Science in Cyber Security (M.S.C.S.), Wright State University, 2019, Computer Science

Elliptic Curve Cryptography (ECC) is a public-key cryptography system. Elliptic Curve Cryptography (ECC) can achieve the same level of security as the public-key cryptography system, RSA, with a much smaller key size. It is a promising public key cryptography system with regard to time efficiency and resource utilization. This thesis focuses on the software implementations of ECC over finite field GF(p) with two distinct implementations of the Big Integer classes using character arrays, and bit sets in C++ programming language. Our implementation works on the ECC curves of the form y^2 = x^3 + ax + b (mod p). The point addition operation and the scalar multiplication are implemented on a real SEC (Standards for Efficient Cryptography) ECC curve over a prime field with two different implementations. The Elliptic Curve Diffie-Hellman key exchange, the ElGamal encryption/decryption system, and the Elliptic Curve Digital Signature Algorithm (ECDSA) on a real SEC ECC curve with two different implementations of the big integer classes are tested, and validated. The performances of the two different implementations are compared and analyzed.

Committee: Meilin Liu Ph.D. (Advisor); Junjie Zhang Ph.D. (Committee Member); Keke Chen Ph.D. (Committee Member) Subjects: Computer Science; Information Technology

MS, University of Cincinnati, 2017, Engineering and Applied Science: Computer Engineering

This thesis focuses on elliptic curve arithmetic over the prime field GF (p) and elliptic curve cryptography (ECC). ECC over GF(p) has its own arithmetic which is done over elliptic curves of the form y2; ≡ x3;+ax+b (mod p), where p is prime. ECC is gaining importance in security because it uses smaller keys to provide the same security level as the popular RSA. It is the superior cryptographic scheme based on time efficiency and resource utilization. It is more suitable than RSA for DNSSEC and IoT systems and devices. Unlike RSA, which is easily understood, ECC is complicated because of the arithmetic involved. It is not widely understood. We provide a tutorial on elliptic curve arithmetic and also explain the working of the ElGamal cryptosystem. We also describe general hardware-efficient methods to implement ECC such as Montgomery multiplication and projective coordinates. These methods are challenging to understand. Essentially, projective coordinates help reduce the number of inversions required in doing scalar multiplication. If Montgomery multiplication is used, a time-consuming operation like reduction modulo a prime p can be simplified. In this work, we also present a user-friendly Java GUI application to provide education in elliptic curve arithmetic and its applications in cryptosystems. Lastly, we provide a module of questions and solutions to do the same and also enable senior students and graduate students to use ECC in their project work.

Committee: Carla Purdy Ph.D. (Committee Chair); Wen-Ben Jone Ph.D. (Committee Member); George Purdy Ph.D. (Committee Member) Subjects: Computer Engineering

Master of Mathematical Sciences, The Ohio State University, 2015, Mathematics

Modern cryptography relies heavily on concepts from mathematics. In this thesis we will be discussing several cryptographic ciphers and discovering the mathematical applications which can be found by exploring them. This paper is intended to be accessible to undergraduate or graduate students as a supplement to a course in number theory or modern algebra. The structure of the paper also lends itself to be accessible to a person interested in learning about mathematics in cryptography on their own, since we will always give a review of the background material which will be needed before delving into the cryptographic ciphers.

Committee: James Cogdell (Advisor); Rodica Costin (Committee Member) Subjects: Mathematics; Mathematics Education

Master of Sciences, Case Western Reserve University, 2014, Applied Mathematics

Edwards curves are a fairly new way of expressing a family of elliptic curves that contain extremely desirable cryptographic properties over other forms that have been used. The most notable is the notion of a complete and unified addition law. This property makes Edwards curves extremely strong against side-channel attacks. In the analysis and continual development of Edwards curves, it has been seen in the original Edwards form that the use of inverted coordinates creates a more efficient addition/doubling algorithm. Using inverted coordinates, the field oper- ations drop from 10M + 1S (given correctly chosen curve parameters), to 9M + 1S. The sarcrifice is the loss of completeness, but unification remains. This pa- per examines the use of the inverted coordinates system over the binary Edwards form, and shows the underlying advantages of this transformation

Committee: David Singer PhD (Advisor); Elisabeth Werner PhD (Committee Member); Johnathan Duncan PhD (Committee Member) Subjects: Computer Science; Mathematics

Master of Science in Mathematics, Youngstown State University, 2012, Department of Mathematics and Statistics

It is the intent of this thesis to study the mathematics, and applications behind the elliptic curve group over Fp. Beginning with the definition of the '+' operation,under which the points on the elliptic curves form an abelian group. Then moving to a brief introduction to both public, and private key cryptography. This will lead into an explanation of the discrete logarithm problem along with an implementation using the elliptic curve group over Fp. This thesis will conclude with an exploration Lenstra's factoring algorithm using the elliptic curve group.

Committee: Jacek Fabrykowski Ph.D. (Advisor); Neil Flowers Ph.D. (Committee Member); Thomas Smotzer Ph.D. (Committee Member) Subjects:

Master of Sciences (Engineering), Case Western Reserve University, 2012, EECS - Electrical Engineering

Elliptic curve cryptography (ECC) has emerged as a promising public-key cryptography approach for data protection. It is based on the algebraic structure of elliptic curves over finite fields. Although ECC provides high level of information security, it involves computationally intensive encryption/decryption process, which negatively affects its performance and energy-efficiency. Software implementation of ECC is often not amenable for resource-constrained embedded applications. Alternatively, hardware implementation of ECC has been investigated – in both application specific integrated circuit(ASIC) and field programmable gate array (FPGA) platforms – in order to achieve desired performance and energy efficiency. Hardware reconfigurable computing platforms such as FPGAs are particularly attractive platform for hardware acceleration of ECC for diverse applications, since they involve significantly less design cost and time than ASIC. In this work, we investigate efficient implementation of ECC in reconfigurable hardware platforms. In particular, we focus on implementing different ECC encryption algorithms in FPGA and a promising memory array based reconfigurable computing framework, referred to as MBC. MBC leverages the benefit of nanoscale memory, namely, high bandwidth, large density and small wire delay to drastically reduce the overhead of programmable interconnects. We evaluate the performance and energy efficiency of these platforms and compare those with a purely software implementation. We use the pseudo-random curve in the prime field and Koblitz curve in the binary field to do the ECC scalar multiplication operation. We perform functional validation with data that is recommended by NIST. Simulation results show that in general, MBC provides better energy efficiency than FPGA while FPGA provides better latency.

Committee: Swarup Bhunia (Advisor); Christos Papachristou (Committee Member); Frank Merat (Committee Member) Subjects: Electrical Engineering

Doctor of Philosophy, Case Western Reserve University, 2009, EECS - Computer Engineering

In order to provide security in ubiquitous, passively powered systems, especially RFID tags in the supply chain, improved asymmetric key cryptographic processors are presented, tested and compared with others from the literature. The proposed processors show a 12%-20% area and a 31%-45% time improvement. A secure protocol is also presented to minimize cryptographic effort and communication between tag and reader. A set of power management techniques is also presented to match processor performance to available power, resulting in greater range and responsiveness of RFID tags.

Committee: Christos Papachristou PhD (Committee Chair); Francis L. Merat PhD (Committee Member); Swarup Bhunia PhD (Committee Member); Xinmiao Zhang PhD (Committee Member); Francis G. Wolff PhD (Committee Member) Subjects: Computer Science; Electrical Engineering