23,369 research outputs found
Constructing suitable ordinary pairing-friendly curves: A case of elliptic curves and genus two hyperelliptic curves
One of the challenges in the designing of pairing-based cryptographic protocols is to construct suitable pairing-friendly curves: Curves which would provide e�cient implementation without compromising the security of the protocols. These curves have small embedding degree and large prime order subgroup. Random curves are likely to have large embedding degree and hence are not practical for implementation of pairing-based protocols.
In this thesis we review some mathematical background on elliptic and hyperelliptic curves in relation to the construction of pairing-friendly hyper-elliptic curves. We also present the notion of pairing-friendly curves. Furthermore, we construct new pairing-friendly elliptic curves and Jacobians of genus two hyperelliptic curves which would facilitate an efficient implementation in pairing-based protocols. We aim for curves that have smaller values than ever before reported for di�erent embedding degrees. We also discuss optimisation of computing pairing in Tate pairing and its variants. Here we show how to e�ciently multiply a point in a subgroup de�ned on a twist curve by a large cofactor. Our approach uses the theory of addition chains. We also show a new method for implementation of the computation of the hard part of the �nal exponentiation in the calculation of the Tate pairing and its varian
An Efficient hardware implementation of the tate pairing in characteristic three
DL systems with bilinear structure recently became an important base for cryptographic protocols such as identity-based encryption (IBE). Since the main
computational task is the evaluation of the bilinear pairings over elliptic curves, known to be prohibitively expensive, efficient implementations are required to render them applicable in real life scenarios. We present an efficient accelerator for computing the Tate Pairing in characteristic 3, using the Modified Duursma-Lee algorithm. Our accelerator shows that it is possible to improve the area-time product by 12 times on FPGA, compared to estimated values from one of the best known hardware architecture [6] implemented on the same type of FPGA. Also the computation time is improved upto 16 times compared to software applications reported in [17]. In addition, we present the result of an ASIC implementation of the algorithm, which is the first hitherto
Efficient Pairings on Various Platforms
Pairings have found a range of applications in many areas of cryptography. As such, to
utilize the enormous potential of pairing-based protocols one needs to efficiently compute
pairings across various computing platforms. In this thesis, we give an introduction to
pairing-based cryptography and describe the Tate pairing and its variants. We then describe
some recent work to realize efficient computation of pairings. We further extend
these optimizations and implement the O-Ate pairing on BN-curves on ARM and x86-64
platforms. Specifically, we extend the idea of lazy reduction to field inversion, optimize
curve arithmetic, and construct efficient tower extensions to optimize field arithmetic. We
also analyze the use of affine coordinates for pairing computation leading us to the conclusion
that they are a competitive choice for fast pairing computation on ARM processors,
especially at high security level. Our resulting implementation is more than three
times faster than any previously reported implementation on ARM processors
Secure pairing-free two-party certificateless authenticated key agreement protocol with minimal computational complexity
Key agreement protocols play a vital role in maintaining security in many critical applications due to the importance of the secret key. Bilinear pairing was commonly used in designing secure protocols for the last several years; however, high computational complexity of this operation has been the main obstacle towards its practicality. Therefore, implementation of Elliptic-curve based operations, instead of bilinear pairings, has become popular recently, and pairing-free key agreement protocols have been explored in many studies. A considerable amount of literatures has been published on pairing-free key agreement protocols in the context of Public Key Cryptography (PKC). Simpler key management and non-existence of key escrow problem make certificateless PKC more appealing in practice. However, achieving certificateless pairing-free two-party authenticated key agreement protocols (CL-AKA) that provide high level of security with low computational complexity, remains a challenge in the research area. This research presents a secure and lightweight pairingfree CL-AKA protocol named CL2AKA (CertificateLess 2-party Authenticated Key Agreement). The properties of CL2AKA protocol is that, it is computationally lightweight while communication overhead remains the same as existing protocols of related works. The results indicate that CL2AKA protocol is 21% computationally less complex than the most efficient pairing-free CL-AKA protocol (KKC-13) and 53% less in comparison with the pairing-free CL-AKA protocol with highest level of security guarantee (SWZ-13). Security of CL2AKA protocol is evaluated based on provable security evaluation method under the strong eCK model. It is also proven that the CL2AKA supports all of the security requirements which are necessary for authenticated key agreement protocols. Besides the CL2AKA as the main finding of this research work, there are six pairing-free CL-AKA protocols presented as CL2AKA basic version protocols, which were the outcomes of several attempts in designing the CL2AKA
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
Computational and Energy Costs of Cryptographic Algorithms on Handheld Devices
Networks are evolving toward a ubiquitous model in which heterogeneous
devices are interconnected. Cryptographic algorithms are required for developing security
solutions that protect network activity. However, the computational and energy limitations
of network devices jeopardize the actual implementation of such mechanisms. In this
paper, we perform a wide analysis on the expenses of launching symmetric and asymmetric
cryptographic algorithms, hash chain functions, elliptic curves cryptography and pairing
based cryptography on personal agendas, and compare them with the costs of basic operating
system functions. Results show that although cryptographic power costs are high and such
operations shall be restricted in time, they are not the main limiting factor of the autonomy
of a device
Faster computation of the Tate pairing
This paper proposes new explicit formulas for the doubling and addition step
in Miller's algorithm to compute the Tate pairing. For Edwards curves the
formulas come from a new way of seeing the arithmetic. We state the first
geometric interpretation of the group law on Edwards curves by presenting the
functions which arise in the addition and doubling. Computing the coefficients
of the functions and the sum or double of the points is faster than with all
previously proposed formulas for pairings on Edwards curves. They are even
competitive with all published formulas for pairing computation on Weierstrass
curves. We also speed up pairing computation on Weierstrass curves in Jacobian
coordinates. Finally, we present several examples of pairing-friendly Edwards
curves.Comment: 15 pages, 2 figures. Final version accepted for publication in
Journal of Number Theor
Efficient Design of Triplet Based Spike-Timing Dependent Plasticity
Spike-Timing Dependent Plasticity (STDP) is believed to play an important
role in learning and the formation of computational function in the brain. The
classical model of STDP which considers the timing between pairs of
pre-synaptic and post-synaptic spikes (p-STDP) is incapable of reproducing
synaptic weight changes similar to those seen in biological experiments which
investigate the effect of either higher order spike trains (e.g. triplet and
quadruplet of spikes), or, simultaneous effect of the rate and timing of spike
pairs on synaptic plasticity. In this paper, we firstly investigate synaptic
weight changes using a p-STDP circuit and show how it fails to reproduce the
mentioned complex biological experiments. We then present a new STDP VLSI
circuit which acts based on the timing among triplets of spikes (t-STDP) that
is able to reproduce all the mentioned experimental results. We believe that
our new STDP VLSI circuit improves upon previous circuits, whose learning
capacity exceeds current designs due to its capability of mimicking the
outcomes of biological experiments more closely; thus plays a significant role
in future VLSI implementation of neuromorphic systems
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