3,685 research outputs found
Elliptic Curves and Hyperdeterminants in Quantum Gravity
Hyperdeterminants are generalizations of determinants from matrices to
multi-dimensional hypermatrices. They were discovered in the 19th century by
Arthur Cayley but were largely ignored over a period of 100 years before once
again being recognised as important in algebraic geometry, physics and number
theory. It is shown that a cubic elliptic curve whose Mordell-Weil group
contains a Z2 x Z2 x Z subgroup can be transformed into the degree four
hyperdeterminant on a 2x2x2 hypermatrix comprising its variables and
coefficients. Furthermore, a multilinear problem defined on a 2x2x2x2
hypermatrix of coefficients can be reduced to a quartic elliptic curve whose
J-invariant is expressed in terms of the hypermatrix and related invariants
including the degree 24 hyperdeterminant. These connections between elliptic
curves and hyperdeterminants may have applications in other areas including
physics.Comment: 7 page
The sixth Painleve transcendent and uniformization of algebraic curves
We exhibit a remarkable connection between sixth equation of Painleve list
and infinite families of explicitly uniformizable algebraic curves. Fuchsian
equations, congruences for group transformations, differential calculus of
functions and differentials on corresponding Riemann surfaces, Abelian
integrals, analytic connections (generalizations of Chazy's equations), and
other attributes of uniformization can be obtained for these curves. As
byproducts of the theory, we establish relations between Picard-Hitchin's
curves, hyperelliptic curves, punctured tori, Heun's equations, and the famous
differential equation which Apery used to prove the irrationality of Riemann's
zeta(3).Comment: Final version. Numerous improvements; English, 49 pages, 1 table, no
figures, LaTe
Large J expansion in ABJM theory revisited
Recently there has been progress in the computation of the anomalous
dimensions of gauge theory operators at strong coupling by making use of
AdS/CFT correspondence. On string theory side they are given by dispersion
relations in the semiclassical regime. We revisit the problem of large charges
expansion of the dispersion relations for simple semiclassical strings in
background. We present the calculation of the
corresponding anomalous dimensions of the gauge theory operators to an
arbitrary order using three different methods. Although the results of the
three methods look different, power series expansions show their consistency.Comment: 24 pages, 2 figure
Efficient Unified Arithmetic for Hardware Cryptography
The basic arithmetic operations (i.e. addition, multiplication, and inversion) in finite fields, GF(q), where q = pk and p is a prime integer, have several applications in cryptography, such as RSA algorithm, Diffie-Hellman key exchange algorithm [1], the US federal Digital Signature Standard [2], elliptic curve cryptography [3, 4], and also recently identity based cryptography [5, 6]. Most popular finite fields that are heavily used in cryptographic applications due to elliptic curve based schemes are prime fields GF(p) and binary extension fields GF(2n). Recently, identity based cryptography based on pairing operations defined over elliptic curve points has stimulated a significant level of interest in the arithmetic of ternary extension fields, GF(3^n)
Cognitive Architecture and the Epistemic Gap : Defending Physicalism without Phenomenal Concepts
The novel approach presented in this paper accounts for the occurrence of the epistemic gap and defends physicalism against anti-physicalist arguments without relying on so-called phenomenal concepts. Instead of concentrating on conceptual features, the focus is shifted to the special characteristics of experiences themselves. To this extent, the account provided is an alternative to the Phenomenal Concept Strategy. It is argued that certain sensory representations, as accessed by higher cognition, lack constituent structure. Unstructured representations could freely exchange their causal roles within a given system which entails their functional unanalysability. These features together with the encapsulated nature of low level complex processes giving rise to unstructured sensory representations readily explain those peculiarities of phenomenal consciousness which are usually taken to pose a serious problem for contemporary physicalism. I conclude that if those concepts which are related to the phenomenal character of conscious experience are special in any way, their characteristics are derivative of and can be accounted for in terms of the cognitive and representational features introduced in the present paper
Classical elliptic hypergeometric functions and their applications
General theory of elliptic hypergeometric series and integrals is outlined.
Main attention is paid to the examples obeying properties of the "classical"
special functions. In particular, an elliptic analogue of the Gauss
hypergeometric function and some of its properties are described. Present
review is based on author's habilitation thesis [Spi7] containing a more
detailed account of the subject.Comment: 42 pages, typos removed, references update
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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