326 research outputs found
Self-pairings on Hyperelliptic Curves
A self-pairing is a pairing computation where both inputs are the same group element. Self-pairings are used in some cryptographic schemes and protocols. In this paper, we show how to compute the Tate-Lichtenbaum pairing (D,\phi(D)) on a curve more efficiently than the general case. The speedup is obtained by requiring a simpler final exponentiation. We also discuss how to use this pairing in cryptographic applications
Pairing-based algorithms for jacobians of genus 2 curves with maximal endomorphism ring
Using Galois cohomology, Schmoyer characterizes cryptographic non-trivial
self-pairings of the -Tate pairing in terms of the action of the
Frobenius on the -torsion of the Jacobian of a genus 2 curve. We apply
similar techniques to study the non-degeneracy of the -Tate pairing
restrained to subgroups of the -torsion which are maximal isotropic with
respect to the Weil pairing. First, we deduce a criterion to verify whether the
jacobian of a genus 2 curve has maximal endomorphism ring. Secondly, we derive
a method to construct horizontal -isogenies starting from a
jacobian with maximal endomorphism ring
Generalized Jacobians and explicit descents
We develop a cohomological description of various explicit descents in terms
of generalized Jacobians, generalizing the known description for hyperelliptic
curves. Specifically, given an integer dividing the degree of some reduced
effective divisor on a curve , we show that multiplication by
on the generalized Jacobian factors through an isogeny
whose kernel is
naturally the dual of the Galois module
. By geometric class
field theory, this corresponds to an abelian covering of of exponent
unramified outside . The -coverings of parameterized
by explicit descents are the maximal unramified subcoverings of the -forms
of this ramified covering. We present applications of this to the computation
of Mordell-Weil groups of Jacobians.Comment: to appear in Math. Com
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
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