818 research outputs found
Refinements of Miller's Algorithm over Weierstrass Curves Revisited
In 1986 Victor Miller described an algorithm for computing the Weil pairing
in his unpublished manuscript. This algorithm has then become the core of all
pairing-based cryptosystems. Many improvements of the algorithm have been
presented. Most of them involve a choice of elliptic curves of a \emph{special}
forms to exploit a possible twist during Tate pairing computation. Other
improvements involve a reduction of the number of iterations in the Miller's
algorithm. For the generic case, Blake, Murty and Xu proposed three refinements
to Miller's algorithm over Weierstrass curves. Though their refinements which
only reduce the total number of vertical lines in Miller's algorithm, did not
give an efficient computation as other optimizations, but they can be applied
for computing \emph{both} of Weil and Tate pairings on \emph{all}
pairing-friendly elliptic curves. In this paper we extend the Blake-Murty-Xu's
method and show how to perform an elimination of all vertical lines in Miller's
algorithm during Weil/Tate pairings computation on \emph{general} elliptic
curves. Experimental results show that our algorithm is faster about 25% in
comparison with the original Miller's algorithm.Comment: 17 page
Families of fast elliptic curves from Q-curves
We construct new families of elliptic curves over \FF_{p^2} with
efficiently computable endomorphisms, which can be used to accelerate elliptic
curve-based cryptosystems in the same way as Gallant-Lambert-Vanstone (GLV) and
Galbraith-Lin-Scott (GLS) endomorphisms. Our construction is based on reducing
\QQ-curves-curves over quadratic number fields without complex
multiplication, but with isogenies to their Galois conjugates-modulo inert
primes. As a first application of the general theory we construct, for every
, two one-parameter families of elliptic curves over \FF_{p^2}
equipped with endomorphisms that are faster than doubling. Like GLS (which
appears as a degenerate case of our construction), we offer the advantage over
GLV of selecting from a much wider range of curves, and thus finding secure
group orders when is fixed. Unlike GLS, we also offer the possibility of
constructing twist-secure curves. Among our examples are prime-order curves
equipped with fast endomorphisms, with almost-prime-order twists, over
\FF_{p^2} for and
Easy decision-Diffie-Hellman groups
The decision-Diffie-Hellman problem (DDH) is a central computational problem
in cryptography. It is known that the Weil and Tate pairings can be used to
solve many DDH problems on elliptic curves. Distortion maps are an important
tool for solving DDH problems using pairings and it is known that distortion
maps exist for all supersingular elliptic curves. We present an algorithm to
construct suitable distortion maps. The algorithm is efficient on the curves
usable in practice, and hence all DDH problems on these curves are easy. We
also discuss the issue of which DDH problems on ordinary curves are easy
A Survey Report On Elliptic Curve Cryptography
The paper presents an extensive and careful study of elliptic curve cryptography (ECC) and its applications. This paper also discuss the arithmetic involved in elliptic curve and how these curve operations is crucial in determining the performance of cryptographic systems. It also presents different forms of elliptic curve in various coordinate system , specifying which is most widely used and why. It also explains how isogenenies between elliptic curve provides the secure ECC. Exentended form of elliptic curve i.e hyperelliptic curve has been presented here with its pros and cons. Performance of ECC and HEC is also discussed based on scalar multiplication and DLP. Keywords: Elliptic curve cryptography (ECC), isogenies, hyperelliptic curve (HEC) , Discrete Logarithm Problem (DLP), Integer Factorization , Binary Field, Prime FieldDOI:http://dx.doi.org/10.11591/ijece.v1i2.8
Stopping time signatures for some algorithms in cryptography
We consider the normalized distribution of the overall running times of some
cryptographic algorithms, and what information they reveal about the
algorithms. Recent work of Deift, Menon, Olver, Pfrang, and Trogdon has shown
that certain numerical algorithms applied to large random matrices exhibit a
characteristic distribution of running times, which depends only on the
algorithm but are independent of the choice of probability distributions for
the matrices. Different algorithms often exhibit different running time
distributions, and so the histograms for these running time distributions
provide a time-signature for the algorithms, making it possible, in many cases,
to distinguish one algorithm from another. In this paper we extend this
analysis to cryptographic algorithms, and present examples of such algorithms
with time-signatures that are indistinguishable, and others with
time-signatures that are clearly distinct.Comment: 20 page
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