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