scholarly journals Scalar Multiplication on Koblitz Curves Using the Frobenius Endomorphism and Its Combination with Point Halving: Extensions and Mathematical Analysis

Algorithmica ◽  
2006 ◽  
Vol 46 (3-4) ◽  
pp. 249-270 ◽  
Author(s):  
Roberto M. Avanzi ◽  
Clemens Heuberger ◽  
Helmut Prodinger
Keyword(s):  
Mathematical Analysis ◽  
Scalar Multiplication ◽  
Koblitz Curves ◽  
Frobenius Endomorphism

The Construction of ElGamal over Koblitz Curve

2014 ◽  
Vol 931-932 ◽  
pp. 1441-1446 ◽  
Author(s):  
Krissanee Kamthawee ◽  
Bhichate Chiewthanakul
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Discrete Logarithm ◽  
Scalar Multiplication ◽  
Computer Hardware ◽  
Point Multiplication ◽  
Koblitz Curves ◽  
Elliptic Curve Cryptosystems ◽  
Elgamal Cryptosystem ◽  
Primary Advantage

Recently elliptic curve cryptosystems are widely accepted for security applications key generation, signature and verification. Cryptographic mechanisms based on elliptic curves depend on arithmetic involving the points of the curve. it is possible to use smaller primes, or smaller finite fields, with elliptic curves and achieve a level of security comparable to that for much larger integers. Koblitz curves, also known as anomalous binary curves, are elliptic curves defined over F2. The primary advantage of these curves is that point multiplication algorithms can be devised that do not use any point doublings. The ElGamal cryptosystem, which is based on the Discrete Logarithm problem can be implemented in any group. In this paper, we propose the ElGamal over Koblitz Curve Scheme by applying the arithmetic on Koblitz curve to the ElGamal cryptosystem. The advantage of this scheme is that point multiplication algorithms can be speeded up the scalar multiplication in the affine coodinate of the curves using Frobenius map. It has characteristic two, therefore it’s arithmetic can be designed in any computer hardware. Moreover, it has more efficient to employ the TNAF method for scalar multiplication on Koblitz curves to decrease the number of nonzero digits. It’s security relies on the inability of a forger, who does not know a private key, to compute elliptic curve discrete logarithm.

Short-Memory Scalar Multiplication for Koblitz Curves

2008 ◽  
Vol 57 (4) ◽  
pp. 481-489 ◽  
Author(s):  
C. Vuillaume ◽  
K. Okeya ◽  
T. Takagi
Keyword(s):  
Scalar Multiplication ◽  
Koblitz Curves ◽  
Short Memory

scholarly journals Constructing Scalar Multiplication via Elliptic Net of Rank Two

2018 ◽  
Vol 7 (4.34) ◽  
pp. 403
Author(s):  
Norliana Muslim ◽  
Mohamad Rushdan Md. Said
Keyword(s):  
Scalar Multiplication ◽  
Powerful Method ◽  
Arithmetic Properties ◽  
Initial Values ◽  
Koblitz Curves ◽  
Starting Point ◽  
Division Polynomials ◽  
Rank One ◽  
Elliptic Nets

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.  

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