scholarly journals Aperiodic Photonics of Elliptic Curves

Crystals ◽  
2019 ◽  
Vol 9 (9) ◽  
pp. 482 ◽  
Author(s):  
Luca Dal Dal Negro ◽  
Yuyao Chen ◽  
Fabrizio Sgrignuoli
Keyword(s):  
Light Scattering ◽  
Elliptic Curves ◽  
Random Media ◽  
Scattering Problem ◽  
Discrete Logarithm ◽  
Structural Complexity ◽  
Structure Property ◽  
Wave Localization ◽  
Multiple Wave ◽  
Aperiodic Order

In this paper we propose a novel approach to aperiodic order in optical science and technology that leverages the intrinsic structural complexity of certain non-polynomial (hard) problems in number theory and cryptography for the engineering of optical media with novel transport and wave localization properties. In particular, we address structure-property relationships in a large number (900) of light scattering systems that physically manifest the distinctive aperiodic order of elliptic curves and the associated discrete logarithm problem over finite fields. Besides defining an extremely rich subject with profound connections to diverse mathematical areas, elliptic curves offer unprecedented opportunities to engineer light scattering phenomena in aperiodic environments beyond the limitations of traditional random media. Our theoretical analysis combines the interdisciplinary methods of point patterns spatial statistics with the rigorous Green’s matrix solution of the multiple wave scattering problem for electric and magnetic dipoles and provides access to the spectral and light scattering properties of novel deterministic aperiodic structures with enhanced light-matter coupling for nanophotonics and metamaterials applications to imaging and spectroscopy.

Colloidal Plasmonic Nanostar Antennas with Wide Range Resonance Tunability

2019 ◽  
Author(s):  
Theodoros Tsoulos ◽  
Supriya Atta ◽  
Maureen Lagos ◽  
Michael Beetz ◽  
Philip Batson ◽  
...  
Keyword(s):  
Single Particle ◽  
Structural Complexity ◽  
Field Enhancement ◽  
Hot Electron ◽  
Structure Property ◽  
Plasmon Band ◽  
Gold Nanostars ◽  
Wide Range ◽  
Particle Level ◽  
Spectrally Resolved

<div>Gold nanostars display exceptional field enhancement properties and tunable resonant modes that can be leveraged to create effective imaging tags or phototherapeutic agents, or to design novel hot-electron based photocatalysts. From a fundamental standpoint, they represent important tunable platforms to study the dependence of hot carrier energy and dynamics on plasmon band intensity and position. Toward the realization of these platforms, holistic approaches taking into account both theory and experiments to study the fundamental behavior of these</div><div>particles are needed. Arguably, the intrinsic difficulties underlying this goal stem from the inability to rationally design and effectively synthesize nanoparticles that are sufficiently monodispersed to be employed for corroborations of the theoretical results without the need of single particle experiments. Herein, we report on our concerted computational and experimental effort to design, synthesize, and explain the origin and morphology-dependence of the plasmon modes of a novel gold nanostar system, with an approach that builds upon the well-known plasmon hybridization model. We have synthesized monodispersed samples of gold nanostars with finely tunable morphology employing seed-mediated colloidal protocols, and experimentally observed narrow and spectrally resolved harmonics of the primary surface plasmon resonance mode both at the single particle level (via electron energy loss spectroscopy) and in ensemble (by UV-Vis and ATR-FTIR spectroscopies). Computational results on complex anisotropic gold nanostructures are validated experimentally on samples prepared colloidally, underscoring their importance as ideal testbeds for the study of structure-property relationships in colloidal nanostructures of high structural complexity.</div>

PERSPECTIVES OF ELLIPTICAL CRYPTOGRAPHY TO ENSURE THE INTEGRITY AND CONFIDENTIALITY OF INFORMATION

2019 ◽  
Author(s):  
Anna ILYENKO ◽  
Sergii ILYENKO ◽  
Yana MASUR
Keyword(s):  
Information Systems ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Computational Efficiency ◽  
Digital Signature ◽  
Discrete Logarithm ◽  
Algebraic Groups ◽  
Schnorr Signature ◽  
Stability Of Algorithms

In this article, the main problems underlying the current asymmetric crypto algorithms for the formation and verification of electronic-digital signature are considered: problems of factorization of large integers and problems of discrete logarithm. It is noted that for the second problem, it is possible to use algebraic groups of points other than finite fields. The group of points of the elliptical curve, which satisfies all set requirements, looked attractive on this side. Aspects of the application of elliptic curves in cryptography and the possibilities offered by these algebraic groups in terms of computational efficiency and crypto-stability of algorithms were also considered. Information systems using elliptic curves, the keys have a shorter length than the algorithms above the finite fields. Theoretical directions of improvement of procedure of formation and verification of electronic-digital signature with the possibility of ensuring the integrity and confidentiality of information were considered. The proposed method is based on the Schnorr signature algorithm, which allows data to be recovered directly from the signature itself, similarly to RSA-like signature systems, and the amount of recoverable information is variable depending on the information message. As a result, the length of the signature itself, which is equal to the sum of the length of the end field over which the elliptic curve is determined, and the artificial excess redundancy provided to the hidden message was achieved.

scholarly journals On the discrete logarithm problem for prime-field elliptic curves

2018 ◽  
Vol 51 ◽  
pp. 168-182 ◽  
Author(s):  
Alessandro Amadori ◽  
Federico Pintore ◽  
Massimiliano Sala
Keyword(s):  
Elliptic Curves ◽  
Discrete Logarithm ◽  
Discrete Logarithm Problem ◽  
Prime Field

Application of the Solution of the Inverse Light Scattering Problem for Characterization of Mononuclear Cells

2009 ◽  
Vol 4 (2) ◽  
pp. 61-68
Author(s):  
Dmitriy Strokotov ◽  
Yuriy Pichugin ◽  
Maxim Yurkin ◽  
Mariya Gridina ◽  
Oleg Serov ◽  
...  
Keyword(s):  
Stem Cells ◽  
Light Scattering ◽  
Mononuclear Cells ◽  
Human Lymphocytes ◽  
Scattering Problem ◽  
Refractive Indices ◽  
Single Particles ◽  
Differentiation Status

In this manuscript we propose two methods to solve inverse light scattering problem for single particles, which can be described as a coated sphere. The efficiency of the methods is illustrated by characterization of lymphocytes and stem cells using light scattering patterns obtained with scanning flow cytometer. Both methods, spectral and global optimization, were used to obtain diameters and refractive indices of the cytoplasm and the nucleus of mice embryo stem cells and human lymphocytes. These results agree with data obtained from other studies. Determination of these parameters is important for diagnostics of pathological states of lymphocytes and differentiation status of embryo stem cells. Moreover, methods described in this manuscript are applicable to all mononuclear cells. We also considered limitations of these methods and their possible improvements.

scholarly journals Using Symmetries in the Index Calculus for Elliptic Curves Discrete Logarithm

2013 ◽  
Vol 27 (4) ◽  
pp. 595-635 ◽  
Author(s):  
Jean-Charles Faugère ◽  
Pierrick Gaudry ◽  
Louise Huot ◽  
Guénaël Renault
Keyword(s):  
Elliptic Curves ◽  
Discrete Logarithm ◽  
Index Calculus

scholarly journals Improved lower bound for Diffie–Hellman problem using multiplicative group of a finite field as auxiliary group

2018 ◽  
Vol 12 (2) ◽  
pp. 101-118 ◽  
Author(s):  
Prabhat Kushwaha
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Elliptic Curves ◽  
Multiplicative Group ◽  
Discrete Logarithm ◽  
Discrete Logarithm Problem ◽  
Reduction Algorithm ◽  
Practical Applications ◽  
Diffie Hellman ◽  
Estimate Lower

Abstract In 2004, Muzereau, Smart and Vercauteren [A. Muzereau, N. P. Smart and F. Vercauteren, The equivalence between the DHP and DLP for elliptic curves used in practical applications, LMS J. Comput. Math. 7 2004, 50–72] showed how to use a reduction algorithm of the discrete logarithm problem to Diffie–Hellman problem in order to estimate lower bound for the Diffie–Hellman problem on elliptic curves. They presented their estimates on various elliptic curves that are used in practical applications. In this paper, we show that a much tighter lower bound for the Diffie–Hellman problem on those curves can be achieved if one uses the multiplicative group of a finite field as auxiliary group. The improved lower bound estimates of the Diffie–Hellman problem on those recommended curves are also presented. Moreover, we have also extended our idea by presenting similar estimates of DHP on some more recommended curves which were not covered before. These estimates of DHP on these curves are currently the tightest which lead us towards the equivalence of the Diffie–Hellman problem and the discrete logarithm problem on these recommended elliptic curves.

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