4,454 research outputs found
Still Wrong Use of Pairings in Cryptography
Several pairing-based cryptographic protocols are recently proposed with a
wide variety of new novel applications including the ones in emerging
technologies like cloud computing, internet of things (IoT), e-health systems
and wearable technologies. There have been however a wide range of incorrect
use of these primitives. The paper of Galbraith, Paterson, and Smart (2006)
pointed out most of the issues related to the incorrect use of pairing-based
cryptography. However, we noticed that some recently proposed applications
still do not use these primitives correctly. This leads to unrealizable,
insecure or too inefficient designs of pairing-based protocols. We observed
that one reason is not being aware of the recent advancements on solving the
discrete logarithm problems in some groups. The main purpose of this article is
to give an understandable, informative, and the most up-to-date criteria for
the correct use of pairing-based cryptography. We thereby deliberately avoid
most of the technical details and rather give special emphasis on the
importance of the correct use of bilinear maps by realizing secure
cryptographic protocols. We list a collection of some recent papers having
wrong security assumptions or realizability/efficiency issues. Finally, we give
a compact and an up-to-date recipe of the correct use of pairings.Comment: 25 page
Distortion maps for genus two curves
Distortion maps are a useful tool for pairing based cryptography. Compared
with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more
complicated since the full torsion subgroup has rank 2g. In this paper we prove
that distortion maps always exist for supersingular curves of genus g>1 and we
construct distortion maps in genus 2 (for embedding degrees 4,5,6 and 12).Comment: 16 page
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in his unpublished manuscript.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
Solving discrete logarithms on a 170-bit MNT curve by pairing reduction
Pairing based cryptography is in a dangerous position following the
breakthroughs on discrete logarithms computations in finite fields of small
characteristic. Remaining instances are built over finite fields of large
characteristic and their security relies on the fact that the embedding field
of the underlying curve is relatively large. How large is debatable. The aim of
our work is to sustain the claim that the combination of degree 3 embedding and
too small finite fields obviously does not provide enough security. As a
computational example, we solve the DLP on a 170-bit MNT curve, by exploiting
the pairing embedding to a 508-bit, degree-3 extension of the base field.Comment: to appear in the Lecture Notes in Computer Science (LNCS
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