scholarly journals Free ℤ-module

2012 ◽  
Vol 20 (4) ◽  
pp. 275-280
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Finite Rank ◽  
Reduction Algorithm ◽  
Selection Of

Summary In this article we formalize a free ℤ-module and its rank. We formally prove that for a free finite rank ℤ-module V , the number of elements in its basis, that is a rank of the ℤ-module, is constant regardless of the selection of its basis. ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [15]. Some theorems in this article are described by translating theorems in [21] and [8] into theorems of Z-module.

scholarly journals Matrix of ℤ-module

2015 ◽  
Vol 23 (1) ◽  
pp. 29-49 ◽  
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Linear Transformation ◽  
Bilinear Form ◽  
Finite Rank ◽  
Coding Theory ◽  
Reduction Algorithm ◽  
Gramian Matrix ◽  
Selection Of

Summary In this article, we formalize a matrix of ℤ-module and its properties. Specially, we formalize a matrix of a linear transformation of ℤ-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free ℤ-module V, determinant of its Gramian matrix is constant regardless of selection of its basis. ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattices [22] and coding theory [14]. Some theorems in this article are described by translating theorems in [24], [26] and [19] into theorems of ℤ-module.

scholarly journals Torsion Z-module and Torsion-free Z-module

2014 ◽  
Vol 22 (4) ◽  
pp. 277-289
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Finite Rank ◽  
Coding Theory ◽  
Reduction Algorithm ◽  
Finitely Generated ◽  
Torsion Free

Summary In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].

scholarly journals Torsion Part of ℤ-module

2015 ◽  
Vol 23 (4) ◽  
pp. 297-307 ◽  
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Finite Rank ◽  
Rational Numbers ◽  
Reduction Algorithm ◽  
Definition Of ◽  
Torsion Part ◽  
Torsion Free

Summary In this article, we formalize in Mizar [7] the definition of “torsion part” of ℤ-module and its properties. We show ℤ-module generated by the field of rational numbers as an example of torsion-free non free ℤ-modules. We also formalize the rank-nullity theorem over finite-rank free ℤ-modules (previously formalized in [1]). ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [23] and cryptographic systems with lattices [24].

Computational Cryptography

2021 ◽  
Keyword(s):  
Number Theory ◽  
Lattice Reduction ◽  
Reduction Algorithm ◽  
65Th Birthday ◽  
Integer Factorization ◽  
Cryptographic Keys ◽  
Mathematical Problems ◽  
The Impact ◽  
Selection Of ◽  
Scientific Achievements

The area of computational cryptography is dedicated to the development of effective methods in algorithmic number theory that improve implementation of cryptosystems or further their cryptanalysis. This book is a tribute to Arjen K. Lenstra, one of the key contributors to the field, on the occasion of his 65th birthday, covering his best-known scientific achievements in the field. Students and security engineers will appreciate this no-nonsense introduction to the hard mathematical problems used in cryptography and on which cybersecurity is built, as well as the overview of recent advances on how to solve these problems from both theoretical and practical applied perspectives. Beginning with polynomials, the book moves on to the celebrated Lenstra–Lenstra–Lovász lattice reduction algorithm, and then progresses to integer factorization and the impact of these methods to the selection of strong cryptographic keys for usage in widely used standards.

Cognitive gadgets and cognitive priors

2019 ◽  
Vol 42 ◽  
Author(s):  
Gian Domenico Iannetti ◽  
Giorgio Vallortigara
Keyword(s):  
Cognitive Neuroscience ◽  
Selection Of

Abstract Some of the foundations of Heyes’ radical reasoning seem to be based on a fractional selection of available evidence. Using an ethological perspective, we argue against Heyes’ rapid dismissal of innate cognitive instincts. Heyes’ use of fMRI studies of literacy to claim that culture assembles pieces of mental technology seems an example of incorrect reverse inferences and overlap theories pervasive in cognitive neuroscience.

Note on Selection of Coordinate Systems for Geodynamics

1975 ◽  
Vol 26 ◽  
pp. 395-407
Author(s):  
S. Henriksen
Keyword(s):  
Sea Level ◽  
The Other ◽  
Coordinate Systems ◽  
Mean Sea Level ◽  
The Earth ◽  
The One ◽  
Static Systems ◽  
Selection Of

The first question to be answered, in seeking coordinate systems for geodynamics, is: what is geodynamics? The answer is, of course, that geodynamics is that part of geophysics which is concerned with movements of the Earth, as opposed to geostatics which is the physics of the stationary Earth. But as far as we know, there is no stationary Earth – epur sic monere. So geodynamics is actually coextensive with geophysics, and coordinate systems suitable for the one should be suitable for the other. At the present time, there are not many coordinate systems, if any, that can be identified with a static Earth. Certainly the only coordinate of aeronomic (atmospheric) interest is the height, and this is usually either as geodynamic height or as pressure. In oceanology, the most important coordinate is depth, and this, like heights in the atmosphere, is expressed as metric depth from mean sea level, as geodynamic depth, or as pressure. Only for the earth do we find “static” systems in use, ana even here there is real question as to whether the systems are dynamic or static. So it would seem that our answer to the question, of what kind, of coordinate systems are we seeking, must be that we are looking for the same systems as are used in geophysics, and these systems are dynamic in nature already – that is, their definition involvestime.

Measurements of the Cape Astrometric Survey of the Southern Hemisphere

1978 ◽  
Vol 48 ◽  
pp. 515-521
Author(s):  
W. Nicholson
Keyword(s):  
Southern Hemisphere ◽  
Automatic Measuring ◽  
Measuring Machine ◽  
Final Selection ◽  
Selection Of

SummaryA routine has been developed for the processing of the 5820 plates of the survey. The plates are measured on the automatic measuring machine, GALAXY, and the measures are subsequently processed by computer, to edit and then refer them to the SAO catalogue. A start has been made on measuring the plates, but the final selection of stars to be made is still a matter for discussion.

Application of Improved Voltage Compensation in the SEM to Imaging of Organic Materials

1973 ◽  
Vol 31 ◽  
pp. 444-445
Author(s):  
P.J. Killingworth ◽  
M. Warren
Keyword(s):  
Electron Beam ◽  
Organic Materials ◽  
Operating Parameters ◽  
Low Density ◽  
Beam Diameter ◽  
Incident Electron Beam ◽  
Specimen Material ◽  
Voltage Compensation ◽  
Selection Of ◽  
Scanning Electron

Ultimate resolution in the scanning electron microscope is determined not only by the diameter of the incident electron beam, but by interaction of that beam with the specimen material. Generally, while minimum beam diameter diminishes with increasing voltage, due to the reduced effect of aberration component and magnetic interference, the excited volume within the sample increases with electron energy. Thus, for any given material and imaging signal, there is an optimum volt age to achieve best resolution.In the case of organic materials, which are in general of low density and electric ally non-conducting; and may in addition be susceptible to radiation and heat damage, the selection of correct operating parameters is extremely critical and is achiev ed by interative adjustment.

Selection and Preparation of Specific Tissue Regions For Tem Using Large Epoxy-Embedded Blocks

1973 ◽  
Vol 31 ◽  
pp. 324-325 ◽  
Author(s):  
P. M. Lowrie ◽  
W. S. Tyler
Keyword(s):  
Electron Microscopy ◽  
Anatomical Structures ◽  
Ultrastructural Studies ◽  
Transmission Electron ◽  
Microscopic Evaluation ◽  
Embedded Blocks ◽  
Specific Tissue ◽  
Definition Of ◽  
Areas Of Interest ◽  
Selection Of

The importance of examining stained 1 to 2μ plastic sections by light microscopy has long been recognized, both for increased definition of many histologic features and for selection of specimen samples to be used in ultrastructural studies. Selection of specimens with specific orien ation relative to anatomical structures becomes of critical importance in ultrastructural investigations of organs such as the lung. The uantity of blocks necessary to locate special areas of interest by random sampling is large, however, and the method is lacking in precision. Several methods have been described for selection of specific areas for electron microscopy using light microscopic evaluation of paraffin, epoxy-infiltrated, or epoxy-embedded large blocks from which thick sections were cut. Selected areas from these thick sections were subsequently removed and re-embedded or attached to blank precasted blocks and resectioned for transmission electron microscopy (TEM).

The DQE of an electron image recording system

1978 ◽  
Vol 36 (1) ◽  
pp. 100-101
Author(s):  
K.-H. Herrmann ◽  
D. Krahl ◽  
H.-P Rust
Keyword(s):  
Primary Electron ◽  
Ideal Case ◽  
Image Recording ◽  
Main Requirement ◽  
High Detection ◽  
Electron Image ◽  
Video Amplifier ◽  
Detection Quantum Efficiency ◽  
Selection Of ◽  
The Ideal

The high detection quantum efficiency (DQE) is the main requirement for an imagerecording system used in electron microscopy of radiation-sensitive specimens. An electronic TV system of the type shown in Fig. 1 fulfills these conditions and can be used for either analog or digital image storage and processing [1], Several sources of noise may reduce the DQE, and therefore a careful selection of various elements is imperative.The noise of target and of video amplifier can be neglected when the converter stages produce sufficient target electrons per incident primary electron. The required gain depends on the type of the tube and also on the type of the signal processing chosen. For EBS tubes, for example, it exceeds 10. The ideal case, in which all impinging electrons create uniform charge peaks at the target, is not obtainable for several reasons, and these will be discussed as they relate to a system with a scintillator, fiber-optic and photo-cathode combination as the first stage.

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