**Thesis Type:** Doctorate

**Institution Of The Thesis:** Orta Doğu Teknik Üniversitesi, Institute of Applied Mathematics, Turkey

**Approval Date:** 2010

**Student:** BARIŞ BÜLENT KIRLAR

**Supervisor: **ERSAN AKYILDIZ

In this thesis, we explore the pairing-based cryptography on elliptic curves from the theoretical and implementation point of view. In this respect, we first study so-called pairing-friendly elliptic curves used in pairing-based cryptography. We classify these curves according to their construction methods and study them in details. Inspired of the work of Koblitz and Menezes, we study the elliptic curves in the form $y^{2}=x^{3}-c$ over the prime field $\F_{q}$ and compute explicitly the number of points $\#E(\mathbb{F}_{q})$. In particular, we show that the elliptic curve $y^{2}=x^{3}-1$ over $\mathbb{F}_{q}$ for the primes $q$ of the form $27A^{2}+1$ has an embedding degree $k=1$ and belongs to Scott-Barreto families in our classification. Finally, we give examples of those primes $q$ for which the security level of the pairing-based cryptographic protocols on the curve $y^{2}=x^{3}-1$ over $\mathbb{F}_{q}$ is equivalent to 128-, 192-, or 256-bit AES keys. From the implementation point of view, it is well-known that one of the most important part of the pairing computation is final exponentiation. In this respect, we show explicitly how the final exponentiation is related to the linear recurrence relations. In particular, this correspondence gives that finding an algoritm to compute final exponentiation is equivalent to finding an algorithm to compute the $m$-th term of the associated linear recurrence relation. Furthermore, we list all those work studied in the literature so far and point out how the associated linear recurrence computed efficiently.