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Tutorial on Elliptic Curve Arithmetic and Introduction to Elliptic Curve Cryptography (ECC)
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

#### Keywords:

Elliptic Curve Cryptography;ECC Software;Public-Key Cryptography;RSA

EFFICIENT IMPLEMENTATION OF ELLIPTIC CURVE CRYPTOGRAPHY IN RECONFIGURABLE HARDWARE
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

#### Keywords:

Elliptic curve cryptography; ECC; MAHA; MBC; FPGA; low-power; encryption; security

A Portable and Improved Implementation of the Diffie-Hellman Protocol for Wireless Sensor Networks
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

#### Keywords:

Wireless Sensor Networks; Sun SPOTS; Diffie-Hellman Key-Exchange Protocol; Elliptic Curve Cryptography; Elliptic Curve Diffie-Hellman; Portable Diffie-Hellman

Inverted Edwards Coordinates (Maire Model of an Elliptic Curve)
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

#### Keywords:

elliptic curves; elliptic curve cryptography; edwards curves; ECDHKE; ECDSA; maire form; elliptic addition

Improved Cryptographic Processor Designs for Security in RFID and Other Ubiquitous Systems
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

#### Keywords:

Cryptography; elliptic curve cryptography; power management; RFID; embedded systems

An Exploration of Mathematical Applications in Cryptography
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

#### Keywords:

cryptography; cryptographic ciphers; number theory; elliptic curve cryptography