scholarly journals Koblitz Curves and Integer Equivalents of Frobenius Expansions

2007 ◽  
pp. 126-137 ◽  
Author(s):  
Billy Bob Brumley ◽  
Kimmo Järvinen
Keyword(s):  
Koblitz Curves

Nonadjacent Radix-τ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields

2008 ◽  
Vol 60 (6) ◽  
pp. 1267-1282 ◽  
Author(s):  
Ian F. Blake ◽  
V. Kumar Murty ◽  
Guangwu Xu
Keyword(s):  
Number Fields ◽  
Quadratic Number ◽  
Efficient Point ◽  
Koblitz Curves ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number

AbstractIn his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-τ expansion of integers in the number fields and . The (window) nonadjacent form of τ -expansion of integers in was first investigated by Solinas. For integers in , the nonadjacent form and the window nonadjacent form of the τ -expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-τ expansions for integers in all Euclidean imaginary quadratic number fields.

scholarly journals Speeding up the Arithmetic on Koblitz Curves of Genus Two

2001 ◽  
pp. 106-117 ◽  
Author(s):  
Christian Günther ◽  
Tanja Lange ◽  
Andreas Stein
Keyword(s):  
Koblitz Curves ◽  
Genus Two

Design of elliptic curve cryptoprocessors over GF(2163) on Koblitz curves

2014 ◽  
Author(s):  
Paulo Realpe-Munoz ◽  
Vladimir Trujillo-Olaya ◽  
Jaime Velasco-Medina
Keyword(s):  
Elliptic Curve ◽  
Koblitz Curves

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.

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