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Fast Scalar Multiplication for Elliptic Curves over Prime Fields by Efficiently Computable Formulas
twisted Edwards curves Edwards curves scalar multiplication
2018/11/6
This paper addresses fast scalar multiplication for elliptic curves over finite fields. In the first part of the paper, we obtain several efficiently computable formulas for basic elliptic curves arit...
The discrete logarithm problem over prime fields: the safe prime case. The Smart attack, non-canonical lifts and logarithmic derivatives
non-canonical lifts logarithmic derivatives
2017/3/6
The discrete logarithm problem over prime fields: the safe prime case. The Smart attack, non-canonical lifts and logarithmic derivatives.
The Discrete Logarithm Problem over Prime Fields can be transformed to a Linear Multivariable Chinese Remainder Theorem
Discrete logarithm Hensel lift Multivaraible Chinese Remainder theorem
2016/12/10
The Discrete Logarithm Problem over Prime Fields can be transformed to a Linear Multivariable Chinese Remainder Theorem.
New Complexity Trade-Offs for the (Multiple) Number Field Sieve Algorithm in Non-Prime Fields
New Complexity Trade-Offs the (Multiple) Number Field Sieve Algorithm
2015/12/22
The selection of polynomials to represent number fields crucially determines the efficiency of the Number Field Sieve (NFS) algorithm for solving the discrete log problem in a finite field. An importa...
Generating Pairing-friendly Parameters for the CM Construction of Genus 2 Curves over Prime Fields
hyperelliptic curve cryprography pairing-friendly parameters
2010/10/20
We present two contributions in this paper. First, we give a quantitative analysis of the scarcity of pairing-friendly genus 2 curves, assuming the Riemann Hypothesis. This result is an improvement re...
Co-Z Divisor Addition Formulae in Jacobian of Genius 2 Hyperelliptic Curves over Prime Fields
hyperelliptic curve divisor Jacobian addition formulae exponentiation projective representation
2010/9/29
In this paper we proposed a new approach to divisor scalar multiplication in Jacobian of genus 2 hyperelliptic curves over fields with odd characteristic, without field inversion. It based on improved...
On the security of pairing-friendly abelian varieties over non-prime fields
pairing-friendly abelian non-prime fields pairing-based cryptosystem
2009/6/9
Let A be an abelian variety defined over a non-prime finite
field Fq that has embedding degree k with respect to a subgroup of prime
order r. In this paper we give explicit conditions on q, k, and r...
Accelerating the Scalar Multiplication on Elliptic Curve Cryptosystems over Prime Fields
Scalar Multiplication Elliptic Curve Cryptosystems Prime Fields
2009/6/3
Elliptic curve cryptography (ECC), independently introduced by Koblitz and Miller in the
80’s, has attracted increasing attention in recent years due to its shorter key length
requirement in compari...
Fast Algorithms for Arithmetic on Elliptic Curves Over Prime Fields
Fast Algorithms Arithmetic Elliptic Curves Prime Fields
2009/6/2
We present here a thorough discussion of the problem of fast arithmetic on elliptic
curves over prime order finite fields. Since elliptic curves were independently proposed
as a setting for cryptogr...
Software Implementation of Genus-2 Hyperelliptic Curve Cryptosystems Over Prime Fields
Hyperelliptic curve divisor addition efficient implementation
2009/6/2
This paper describes the system parameters and software implementation of a HECDSA cryptosystem based on genus-2 hyperelliptic curves over prime fields. We show how to reduce the computational complex...
New Composite Operations and Precomputation Scheme for Elliptic Curve Cryptosystems over Prime Fields（Full version）
Elliptic curve cryptosystem scalar multiplication point operation
2009/6/2
We present a new methodology to derive faster composite operations of
the form dP+Q, where d is a small integer ≥ 2, for generic ECC scalar
multiplications over prime fields. In particular, we prese...
Counting Points for Hyperelliptic Curves of type y2 = x5 + ax over Finite Prime Fields
Hyperelliptic Curves Finite Prime Fields
2009/4/15
Counting rational points on Jacobian varieties of hyperelliptic
curves over finite fields is very important for constructing hyperelliptic
curve cryptosystems (HCC), but known algorithms for general...
ASPECTS OF HYPERELLIPTIC CURVES OVER LARGE PRIME FIELDS IN SOFTWARE IMPLEMENTATIONS
HYPERELLIPTIC CURVES LARGE PRIME FIELDS SOFTWARE IMPLEMENTATIONS
2009/4/10
This paper presents an implementation of genus 2 and 3 hyperelliptic curves over prime fields,
with a comparison with elliptic curves. To achieve a fair comparison, we developed an ad-hoc
arithmetic...
A Comparison of Point Counting methods for Hyperelliptic Curves over Prime Fields and Fields of Characteristic 2
Hyperelliptic Curve Cryptography lattices point counting Jacobian
2009/4/3
Computing the order of the Jacobian of a hyperelliptic curve remains a hard problem. It is
usually essential to calculate the order of the Jacobian to prevent certain sub-exponential
attacks on the ...
Suitable Curves for Genus-4 HCC over Prime Fields: Point Counting Formulae for Hyperelliptic Curves of type y2 = x2k+1 + ax
Genus-4 HCC Hyperelliptic Curves
2009/3/27
Computing the order of the Jacobian group of a hyperelliptic
curve over a finite field is very important to construct a hyperelliptic
curve cryptosystem (HCC), because to construct secure HCC, we ne...