Electronic Thesis and Dissertation Repository


Doctor of Philosophy


Electrical and Computer Engineering


Dr. Arash Reyhani-Masoleh


The elliptic curve cryptography is an important branch in public-key cryptography. In this thesis, we consider the elliptic curve cryptography over binary extension fields from two different points of view. First, we investigate the underlying arithmetic operations in the elliptic curve cryptography. The main arithmetic operation is the scalar multiplication. This operation is based on two elliptic curve operations, known as the point addition and point doubling. Implementing these two elliptic curve operations requires finite field arithmetic, specifically, finite field addition, multiplication, squaring, and inversion. We focus on two finite field operations, namely finite field multiplication and squaring. For the finite field multiplication, we consider Montgomery multiplication algorithm and shifted polynomial basis to design bit-serial, digit-serial, bit-parallel, semi-systolic and systolic multipliers. In case of finite field squaring, we use the Montgomery multiplication algorithm for squaring using special type of irreducible pentanomials. We also investigate the finite field multiplication from the concurrent error detection point of view. This is due the fact that fault attacks have become a serious concern in cryptographic applications. In this regard, we design concurrent error detection schemes for different Montgomery multipliers. Our comparison results show that our proposed arithmetic units and concurrent error detection scheme provide improvements over their existing counterparts.