Elliptic Curve Scalar Multiplication with x-Coordinate

This paper gives a comprehensive method to do Elliptic Curve Scalar Multiplication with only x-coordinate. Explicit point operation formulae for all types of defining equations of the curves are derived. For each type of curve, the performance is analyzed. The formulae are applied in Montgomery Ladder to get scalar multiplication algorithm operated with only x-coordinate. The new scalar multiplication has the same security level and computation amount with protected binary scalar multiplication （PBSM） against side channel attack, and has the advantages of higher security and little memory needed.

Efficient Scalar Multiplication for Elliptic Curves over Binary Fields

Scalar multiplication [n]P is the kernel and the most time-consuming operation in elliptic curve cryptosystems. In order to improve scalar multiplication, in this paper, we propose a tripling algorithm using Lopez and Dahab projective coordinates, in which there are 3 field multiplications and 3 field squarings less than that in the Jacobian projective tripling algorithm. Furthermore, we map P to（φε^-1（P）, and compute [n]（φε^-1（P） on elliptic curve Eε, which is faster than computing [n]P on E, where φε is an isomorphism. Finally we calculate （φε（[n]φε^-1（P）） = [n]P. Combined with our efficient point tripling formula, this method leads scalar multiplication using double bases to achieve about 23% improvement, compared with Jacobian projective coordinates.

Improved Software Implementation for Montgomery Elliptic Curve Cryptosystem

The last decade witnessed rapid increase in multimedia and other applications that require transmitting and protecting huge amount of data streams simultaneously.For such applications,a high-performance cryptosystem is compulsory to provide necessary security services.Elliptic curve cryptosystem(ECC)has been introduced as a considerable option.However,the usual sequential implementation of ECC and the standard elliptic curve(EC)form cannot achieve required performance level.Moreover,the widely used Hardware implementation of ECC is costly option and may be not affordable.This research aims to develop a high-performance parallel software implementation for ECC.To achieve this,many experiments were performed to examine several factors affecting ECC performance including the projective coordinates,the scalar multiplication algorithm,the elliptic curve(EC)form,and the parallel implementation.The ECC performance was analyzed using the different factors to tune-up them and select the best choices to increase the speed of the cryptosystem.Experimental results illustrated that parallel Montgomery ECC implementation using homogenous projection achieves the highest performance level,since it scored the shortest time delay for ECC computations.In addition,results showed thatNAF algorithm consumes less time to perform encryption and scalar multiplication operations in comparison withMontgomery ladder and binarymethods.Java multi-threading technique was adopted to implement ECC computations in parallel.The proposed multithreaded Montgomery ECC implementation significantly improves the performance level compared to previously presented parallel and sequential implementations.

An Efficient Method against Side-Channel Attacks on ECC

Side-channel attacks （SCA） may exploit leakage information to break cryptosystems. In this paper we present a new SCA resistant Elliptic Curve scalar multiplication algorithm. The proposed algorithm, builds a sequence of bit-strings representing the scalar k, characterized by the fact that all bit-strings are different from zero; this property will ensure a uniform computation behavior for the algorithm, and thus will make it secure against simple power analysis attacks （SPA）. With other randomization techniques, the proposed countermeasures do not penalize the computation time. The proposed scheme is more efficient than MOEller＇s one, its cost being about 5% to 10% smaller than MOEller＇s one.

Design of highly efficient elliptic curve crypto-processor with two multiplications over GF(2^(163))

In this article, a parallel hardware processor is presented to compute elliptic curve scalar multiplication in polynomial basis representation. The processor is applicable to the operations of scalar multiplication by using a modular arithmetic logic unit （MALU）. The MALU consists of two multiplications, one addition, and one squaring. The two multiplications and the addition or squaring can be computed in parallel. The whole computations of scalar multiplication over GF（2^163） can be performed in 3 064 cycles. The simulation results based on Xilinx Virtex2 XC2V6000 FPGAs show that the proposed design can compute random GF（2^163） elliptic curve scalar multiplication operations in 31.17 μs, and the resource occupies 3 994 registers and 15 527 LUTs, which indicates that the crypto-processor is suitable for high-performance application.

Elliptic Curves in Huff's Model

In this paper,we present the generalized Huff curves that contain Huff＇s model as a special case.First,it is proved that every elliptic curve with three points of order 2 is isomorphic to a generalized Huff curve.Then,the fast and explicit formulae are derived for generalized Huff curves in projective coordinates.This paper also enumerates the number of isomorphism classes of generalized Huff curves over finite fields.Finally,the explicit formulae are presented for the doubling step and addition step in Miller＇s algorithm to compute the Tate pairing on generalized Huff elliptic curves.