# Dynamic projective coordinate system for elliptic curve crytography

Al-Gahtani, Theeb Ayedh (2006) Dynamic projective coordinate system for elliptic curve crytography. PhD thesis, King Fahd University of Petroleum and Minerals. PDF4Mb

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## English Abstract

Scalar multiplication is the basic operation in elliptic curve cryptography that can be performed by many algorithms. These algorithms multiply a scalar value K with an elliptic curve base point P. One of the crucial decisions when implementing an efficient elliptic curve cryptosystem is deciding which point coordinate system to use. The point coordinate system used for addition and doubling of points on the elliptic curve determines the efficiency of these routines, and hence the efficiency of the basic cryptographic operation, scalar multiplication. Although using a fixed coordinate system enhances the performance of the scalar multiplication, (by removing the intermediate inversion operations), it becomes a security weakness since it can be exploited by projective coordinates leak attacks to reveal some secure information. Therefore, finding a coordinate system that can enhance the performance of the scalar multiplication and being secure against such attacks is desired goal. This thesis introduces a new approach called Dynamic Projective Coordinate (DPC) system. DPC provides a framework that automates the selection of the projective coordinate system and uses a single mathematical formulation/software code to implement different projective coordinate systems. This framework allows the computing/encrypting device to select the projective coordinate either at random, or according to a certain rule. DPC uses dynamic transformation functions to convert coordinates of any point on the elliptic curve to any projective coordinates by using the same mathematical formula. These transformation functions are used to develop dynamic addition and doubling formulas for elliptic curve over the prime field GF(p) and over the binary field GF(2m). Also, this thesis proposes a new classification method for Side Channel Attacks (SCA). This classification is based on the type of information being leaked which can be Operation-dependent, Data-dependent, Address-dependent or any combination of them. New countermeasures for data-dependent, data-and-operation dependent and address-dependent attacks are proposed. These countermeasures are based on the fact that DPC lends itself to randomize both the data being manipulated and the number of operations being performed by randomizing the coordinate system used.

Item Type: Thesis (PhD) May 2006 Completion Computer College Of Computer Sciences and Engineering > Information and Computer Science Dept Al-Gahtani, Theeb Ayedh Al-Suwaiyel, Mohammed Amin, Alaaeldin A. M. and Ibrahim, Mohammad K. 10556 KFUPM ePrints Admin 22 Jun 2008 17:07 30 Apr 2011 15:40

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