1,850 research outputs found
Constructing Elliptic Curves with Prescribed Embedding Degrees
Pairing-based cryptosystems depend on the existence of groups where
the Decision Diffie-Hellman problem is easy to solve, but the
Computational Diffie-Hellman problem is hard. Such is the case of
elliptic curve groups whose embedding degree is large enough to
maintain a good security level, but small enough for arithmetic
operations to be feasible. However, the embedding degree is usually
enormous, and the scarce previously known suitable elliptic groups
had embedding degree . In this note, we examine
criteria for curves with larger that generalize prior work by
Miyaji et al. based on the properties of cyclotomic
polynomials, and propose efficient representations for the
underlying algebraic structures
More Discriminants with the Brezing-Weng Method
The Brezing-Weng method is a general framework to generate families of
pairing-friendly elliptic curves. Here, we introduce an improvement which can
be used to generate more curves with larger discriminants. Apart from the
number of curves this yields, it provides an easy way to avoid endomorphism
rings with small class number
Construction of noncommutative surfaces with exceptional collections of length 4
Recently de Thanhoffer de V\"olcsey and Van den Bergh classified the Euler
forms on a free abelian group of rank 4 having the properties of the Euler form
of a smooth projective surface. There are two types of solutions: one
corresponding to (and noncommutative
quadrics), and an infinite family indexed by the natural numbers. For
there are commutative and noncommutative surfaces having this Euler form,
whilst for there are no commutative surfaces. In this paper we
construct sheaves of maximal orders on surfaces having these Euler forms,
giving a geometric construction for their numerical blowups.Comment: 24 pages, see also companion paper arXiv:1811.0881
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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