Elliptic Scalar Multiplication Research Articles

The elliptic scalar multiplication graph and its application in elliptic curve cryptography

In this work, a new graph has been defined as a main point to design a new version of an asymmetric encryption scheme. This graph is formed based on the scalar multiplication operation on elliptic curve defined over a prime field, which is called an elliptic scalar multiplication (ESM) graph. Some mathematical facts to show the ESM graph is a complete graph are proved theoretically. On the complete elliptic scalar multiplication (CESM) graph, an asymmetric encryption algorithm has been implemented. Fast computations on the proposed algorithm are Z obtained through the matrices representation and determination of the minimum spanning tree (MST) to compute the ciphertext. More secure computations with the CESM graphic encryption algorithm in comparison with the previous asymmetric encryption algorithms. So, the CESM graphic asymmetric encryption algorithm is considered as a new sight for elliptic curve cryptographic usages.

The soft graphic integer sub-decomposition method for elliptic scalar multiplication

A new version of the ISD method for computing a scalar multiplication on elliptic curve defined over prime field has been proposed. This version is called the soft graphic ISD (SG-ISD) method. The SG-ISD method employed the connected sub-graphs of undirected simple graph, which formed a soft graph (F, A) generated by a set valued function F, to represent the sub-scalars of ISD method. Thus, undirected simple graph can be proved as the SG-ISD version. On the SG-ISD method, the binary representations, which are all edges of the ISD sub-scalars that forms a nonempty set A, has been done directly from the connected sub-graphs. So, every ISD version of a scalar k is proven mathematically as the SG-ISD version of k in an elliptic scalar multiplication kP. New experimental results on the proposed SG-ISD method are presented. The computational complexities of the original ISD and proposed SG-ISD methods are determined mathematically on the basis of the counting operations. These operations are elliptic curve and finite field operations. The comparison results on these computational complexities of the ISD and SG-ISD methods have been discussed. The experiments on the computational complexities reveal that the SG-ISD method is faster than the ISD method. Thus, the SG-ISD method is considered as an efficient algorithm in comparison with the original ISD for cryptographic applications.

Even And Odd Nature For Pseudo Τ-Adic Non-Adjacent Form

An algorithm was developed by previous researcher for elliptic scalar multiplication (SM) on Koblitz curve where the multiplier of SM is in the form of Pseudo -adic Non-Adjacent (pseudoTNAF). PseudoTNAF of an element of the ring Z ) where is an expansion where the digits are generated by successively dividing by , allowing remainders of , 0 or 1. Such a multiplier is in the form of . In this paper, we refine some properties of the multiplier from previous researchers focusing on even and odd situation for and . We also propose two properties of when is even and is odd. As a result, the nature of and are depends on the nature of and when is even. Whereas, the nature of and are not depends on the nature of and when is odd. Keywords: Pseudo -adic Non-Adjacent Form (pseudoTNAF); scalar multiplication (SM); Koblitz curve

Constructing Scalar Multiplication via Elliptic Net of Rank Two

Elliptic nets are a powerful method for computing cryptographic pairings. The theory of rank one nets relies on the sequences of elliptic divisibility, sets of division polynomials, arithmetic upon Weierstrass curves, as well as double and double-add properties. However, the usage of rank two elliptic nets for computing scalar multiplications in Koblitz curves have yet to be reported. Hence, this study entailed investigations into the generation of point additions and duplication of elliptic net scalar multiplications from two given points on the Koblitz curve. Evidently, the new net had restricted initial values and different arithmetic properties. As such, these findings were a starting point for the generation of higher-ranked elliptic net scalar multiplications with curve transformations. Furthermore, using three distinct points on the Koblitz curves, similar methods can be applied on these curves.

FPGA High Performance Pipelined Architecture Of Elliptic Scalar Multiplication Over GF(2m) for IOT

FPGA High Performance Pipelined Architecture Of Elliptic Scalar Multiplication Over GF(2m) for IOT

Fast elliptic scalar multiplication using new double-base chain and point halving

The fast implementation of elliptic curve cryptosystems relies on the efficient computation of scalar multiplication. Based on the double-base chain representation of scalar using powers of 2 and 3, we propose a new representation with powers of 1/2 and 3 instead. Thus the efficient point halving operation can be incorporated in the new double-base chain to achieve fast scalar multiplication. Experimental results show that our approach leads to a lower complexity which contributes to the efficient implementation of elliptic curve cryptosystems.