Dr. Najlae Falah Hameed Al Saffar

Improvement in the implementation of elliptic curves cryptography and reducing its complexity are still being actively researched. The representation of integers in non adjacent form has been the subject of various investigations in slightly different contexts. The n -digit Non Adjacent Form (NAF) representation of an integer has no two consecutive non zero digits. Fewer non zero elements mean fewer point additions and therefore more efficient when we need to compute the most popular operation: elliptic curve scalar multiplication. In the present paper we reduce the number of required operations using this representation.

Nowadays, elliptic curve based cryptosystem is an efficient public key cryptosystem, The very expensive operation in this cryptographic protocol is the elliptic curve scalar multiplication (elliptic curve point multiplication). Efforts have been mainly focused on developing efficient algorithms for representing the scalar which is involved of elliptic curve scalar multiplication. One of these is using the window- w non adjacent form method. In the present work, the accelerating elliptic curve scalar multiplication using the window- w non adjacent form method is proposed, where the number of operations in the elliptic curve scalar multiplication has been reduced. The expected gain is about 20%, 14% and 7.6% comparing with using the anther methods to compute the elliptic curve scalar multiplication. 20%

Elliptic curve scalar multiplication is the operation of successively adding a point along an elliptic curve to itself k times. It is used in elliptic curve cryptography (ECC) as a means of producing a trapdoor function. In this paper, algorithms to compute the elliptic curve scalar multiplication using a special form for integers will introduce, and then two types of signed digit representation will use. The signed digit form of the scalar is calculated by many types of algorithms such as binary, non adjacent form and direct recoding. The results indicate that the proposed methods perform better to compute the scalar multiplication on elliptic curves and it is more efficient than the existing methods.

A fast point doubling and point addition operations on an elliptic curve over prime field are proposed. This occur when we use a special coordinates system (to represent any point on elliptic curve over prime field. Using this system improved the elliptic curve point arithmetic by reducing the computation cost for point doubling and point addition operation

The group of the elliptic curve points forms an abelian group, which is considered as a suitable choice for constructing a problem similar to the Discrete Logarithm Problem. This creates and opens a new door for treatments of the special group and new operations. In 2005, Al-Saffar (2005) proposed two new methods for elliptic curve cryptosystems using the keys from the algorithm of Diffie–Hellman Key Exchange. In addition, she introduced a variant of the ElGamal scheme. Also, three propositions were introduced to develop the Menezes-Vanstone Elliptic Curves Cryptosystem (MVECC). In this paper, we will discuss all of these propositions and will compare them with the original schemes (ElGamal and MVECC) according to the complexity and the time which they took to implement each scheme.

In this paper, we introduce and study new types of separation axioms in tri-topological spaces, where we defined it using new types of closed set but not necessarily closed sets. Several properties of these concepts are proved.

The Elliptic Curve Digital Signature Algorithm (ECDSA) is the elliptic curve analogue of DSA, where it is a digital signature scheme designed to provide a digital signature based on a secret number known only to the signer and also on the actual message being signed. These digital signatures are considered the digital counterparts to handwritten signatures, and are the basis for validating the authenticity of a connection. The security of these schemes results from the infeasibility to compute the signature without the private key. In this paper we introduce a proposed to development the original ECDSA with more complexity.