Natural Boundaries II: Algebraic Groups

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.

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Cohomology of Arithmetic Subgroups of Algebraic Groups: II

Annals of Mathematics ◽

10.2307/1970585 ◽

1968 ◽

Vol 87 (2) ◽

pp. 279 ◽

Cited By ~ 8

Author(s):

M. S. Raghunathan

Keyword(s):

Algebraic Groups

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An Algebraic Brascamp–Lieb Inequality

Journal of Geometric Analysis ◽

10.1007/s12220-021-00638-9 ◽

2021 ◽

Author(s):

Jennifer Duncan

Keyword(s):

General Class ◽

Recent Progress ◽

Algebraic Groups ◽

Hölder’S Inequality ◽

Duke Math ◽

Rational Maps ◽

Affine Invariant ◽

Linear Maps ◽

Nonlinear Maps ◽

Funct Anal

AbstractThe Brascamp–Lieb inequalities are a very general class of classical multilinear inequalities, well-known examples of which being Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. Conventionally, a Brascamp–Lieb inequality is defined as a multilinear Lebesgue bound on the product of the pullbacks of a collection of functions $$f_j\in L^{q_j}(\mathbb {R}^{n_j})$$ f j ∈ L q j ( R n j ) , for $$j=1,\ldots ,m$$ j = 1 , … , m , under some corresponding linear maps $$B_j$$ B j . This regime is now fairly well understood (Bennett et al. in Geom Funct Anal 17(5):1343–1415, 2008), and moving forward there has been interest in nonlinear generalisations, where $$B_j$$ B j is now taken to belong to some suitable class of nonlinear maps. While there has been great recent progress on the question of local nonlinear Brascamp–Lieb inequalities (Bennett et al. in Duke Math J 169(17):3291–3338, 2020), there has been relatively little regarding global results; this paper represents some progress along this line of enquiry. We prove a global nonlinear Brascamp–Lieb inequality for ‘quasialgebraic’ maps, a class that encompasses polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that both compensates for local degeneracies and yields a constant with minimal dependence on the underlying maps. We then show that this inequality generalises Young’s convolution inequality on algebraic groups with suboptimal constant.

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Which algebraic groups are Picard varieties?

Science China Mathematics ◽

10.1007/s11425-014-4882-3 ◽

2014 ◽

Vol 58 (3) ◽

pp. 461-478 ◽

Cited By ~ 2

Author(s):

Michel Brion

Keyword(s):

Algebraic Groups

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Short- and long-term movement patterns of six temperate reef fishes (Families Labridae and Monacanthidae)

Marine and Freshwater Research ◽

10.1071/mf9950853 ◽

1995 ◽

Vol 46 (5) ◽

pp. 853 ◽

Cited By ~ 67

Author(s):

NS Barrett

Keyword(s):

Natural Habitat ◽

Movement Patterns ◽

Reef Fishes ◽

Long Term Results ◽

Home Ranges ◽

Temperate Reef ◽

Effective Deterrent ◽

Short And Long Term ◽

Natural Boundaries

Movement patterns were studied on a 1-ha isolated reef surrounding Arch Rock in southern Tasmania. Short-term movements were identified from diver observations, and interpretation of long-term movements involved multiple recaptures of tagged individuals. Visual observations indicated that the sex-changing labrids Notolabrus tetricus, Pictilabrus laticlavius and Pseudolabrus psittaculus were all site-attached, with females having overlapping home ranges and males being territorial. In the non-sex-changing labrid Notolabrus fucicola and in the monacanthids Penicipelta vittiger and Meuschenia australis, there was no evidence of territorial behaviour and 1-h movements were in excess of the scale of the study. The long-term results indicated that all species were permanent reef residents, with most individuals of all species except M. australis always being recaptured within a home range of 100 m × 25 m or less. Only 15% of individuals of M. australis were always recaptured within this range category. The natural habitat boundary of open sand between the Arch Rock reef and adjacent reefs appeared to be an effective deterrent to emigration. The use of natural boundaries should be an important consideration in the design of marine reserves where the aim is to minimize the loss of protected species to adjacent fished areas.

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