A Curious Identity on Multiple Sums over Fields with Applications

2021 ◽  
Vol 28 (02) ◽  
pp. 295-308
Author(s):  
Yongchao Xu ◽  
Shaofang Hong
Keyword(s):  
Finite Field ◽  
Cross Classification ◽  
The One

Let [Formula: see text] be a field, and let [Formula: see text] be integers such that [Formula: see text] and [Formula: see text]. We show that for any subset [Formula: see text], the curious identity [Formula: see text] holds with [Formula: see text] being the set of nonnegative integers. As an application, we prove that for any subset [Formula: see text] with [Formula: see text] being the finite field of [Formula: see text] elements and [Formula: see text] being integers such that [Formula: see text] and [Formula: see text], [Formula: see text] Using this identity and providing an extension of the principle of cross-classification that slightly generalizes the one obtained by Hong in 1996, we show that if [Formula: see text] is an integer with [Formula: see text], then for any subset [Formula: see text] we have [Formula: see text] This implies [Formula: see text].

scholarly journals Traces, high powers and one level density for families of curves over finite fields

2017 ◽  
Vol 165 (2) ◽  
pp. 225-248 ◽  
Author(s):  
ALINA BUCUR ◽  
EDGAR COSTA ◽  
CHANTAL DAVID ◽  
JOÃO GUERREIRO ◽  
DAVID LOWRY–DUDA
Keyword(s):  
Finite Field ◽  
Moduli Spaces ◽  
Function Field ◽  
Level Density ◽  
Zeta Functions ◽  
Hyperelliptic Curves ◽  
Unitary Matrix ◽  
Moduli Spaces Of Curves ◽  
A New Technique ◽  
The One

AbstractThe zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix ΘC. We develop and present a new technique to compute the expected value of tr(ΘCn) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalising and extending the work of Rudnick [Rud10] and Chinis [Chi16]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [BDF+16] and [Zha]. We extend [BDF+16] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet L-functions L(1/2 + it, χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic ℓ-covers, and cubic non-Galois covers.

scholarly journals Partitions in the S-Box of Streebog and Kuznyechik

2019 ◽  
pp. 302-329
Author(s):  
Léo Perrin
Keyword(s):  
Finite Field ◽  
Design Process ◽  
Algebraic Structure ◽  
Matrix Multiplication ◽  
Visual Pattern ◽  
Cryptographic Primitives ◽  
Number Of Components ◽  
Linear Layer ◽  
The One ◽  
The Impact

Streebog and Kuznyechik are the latest symmetric cryptographic primitives standardized by the Russian GOST. They share the same S-Box, π, whose design process was not described by its authors. In previous works, Biryukov, Perrin and Udovenko recovered two completely different decompositions of this S-Box.We revisit their results and identify a third decomposition of π. It is an instance of a fairly small family of permutations operating on 2m bits which we call TKlog and which is closely related to finite field logarithms. Its simplicity and the small number of components it uses lead us to claim that it has to be the structure intentionally used by the designers of Streebog and Kuznyechik.The 2m-bit permutations of this type have a very strong algebraic structure: they map multiplicative cosets of the subfield GF(2m)* to additive cosets of GF(2m)*. Furthermore, the function relating each multiplicative coset to the corresponding additive coset is always essentially the same. To the best of our knowledge, we are the first to expose this very strong algebraic structure.We also investigate other properties of the TKlog and show in particular that it can always be decomposed in a fashion similar to the first decomposition of Biryukov et al., thus explaining the relation between the two previous decompositions. It also means that it is always possible to implement a TKlog efficiently in hardware and that it always exhibits a visual pattern in its LAT similar to the one present in π. While we could not find attacks based on these new results, we discuss the impact of our work on the security of Streebog and Kuznyechik. To this end, we provide a new simpler representation of the linear layer of Streebog as a matrix multiplication in the exact same field as the one used to define π. We deduce that this matrix interacts in a non-trivial way with the partitions preserved by π.

A Novel RFID Anti-Counterfeiting Based on Bisectional Multivariate Quadratic Equations

2018 ◽  
Vol 6 (2) ◽  
pp. 1-9
Author(s):  
Xiaoyi Zhou ◽  
Jixin Ma ◽  
Xiaoming Yao ◽  
Honglei Li
Keyword(s):  
Finite Field ◽  
Data Encryption ◽  
Public Key ◽  
Generation Process ◽  
Key Generation ◽  
Quadratic Equations ◽  
Private Key ◽  
Message Digest ◽  
The One ◽  
N Vector

This article proposes a novel scheme for RFID anti-counterfeiting by applying bisectional multivariate quadratic equations (BMQE) system into an RF tag data encryption. In the key generation process, arbitrarily choose two matrix sets (denoted as A and B) and a base RAB such that [(AB) ⃗ ]=λ〖R_AB〗^T, and generate 2n BMQ polynomials (denoted as ρ) over finite field F_q. Therefore, (F_q, ρ) is taken as a public key and (A,B,λ) as a private key. In the encryption process, the EPC code is hashed into a message digest d_m. Then d_m is padded to d_m^' which is a non-zero 2n×2n matrix over F_q. With (A,B,λ)and d_m^', s_m is formed as an n-vector over F_2. Unlike the existing anti-counterfeit scheme, the one the authors proposed is based on quantum cryptography, thus it is robust enough to resist the existing attacks and has high security.

CORRELATION FUNCTIONS IN ONE-DIMENSIONAL LARGE-U HUBBARD MODEL: ZERO-FIELD AND FINITE-FIELD CASES

1991 ◽  
Vol 05 (01n02) ◽  
pp. 31-44 ◽  
Author(s):  
Hiroyuki SHIBA ◽  
Masao OGATA
Keyword(s):  
Wave Function ◽  
Finite Field ◽  
Hubbard Model ◽  
Slater Determinant ◽  
Direct Calculation ◽  
Spin Correlation ◽  
Zero Field ◽  
Property A ◽  
One Dimensional ◽  
The One

A recent study of the large-U limit of the one-dimensional (1D) Hubbard model is presented and discussed. It is pointed out first that the wave function has a simple structure in this limit. Namely, it is a product of Slater determinant of noninteracting spinless fermions and the wave function of 1D S=1/2 Heisenberg antiferromagnet. Secondly, by using this property, a direct calculation of momentum distribution and spin correlation function is carried out for the ground state at zero-field and finite-field cases. The results show various singularities exactly at the same position where one expects for the small-U case in both zero-field and finite-field cases. The critical exponents estimated from the size dependence are in reasonable agreement with those predicted by the Tomonaga-Luttinger-liquid and the conformal field theory.

scholarly journals Number of Directions Determined by a Set in $$\mathbb {F}_{q}^{2}$$ and Growth in $$\mathrm {Aff}(\mathbb {F}_{q})$$

2021 ◽  
Author(s):  
Daniele Dona
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Upper Bound ◽  
Finite Plane ◽  
Structural Theorem ◽  
Prime Fields ◽  
The One

AbstractWe prove that a set A of at most q non-collinear points in the finite plane $$\mathbb {F}_{q}^{2}$$ F q 2 spans more than $${|A|}/\!{\sqrt{q}}$$ | A | / q directions: this is based on a lower bound by Fancsali et al. which we prove again together with a different upper bound than the one given therein. Then, following the procedure used by Rudnev and Shkredov, we prove a new structural theorem about slowly growing sets in $$\mathrm {Aff}(\mathbb {F}_{q})$$ Aff ( F q ) for any finite field $$\mathbb {F}_{q}$$ F q , generalizing the analogous results by Helfgott, Murphy, and Rudnev and Shkredov over prime fields.

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.

Nucleation of Disorder in Fe3Al Alloys

1967 ◽  
Vol 25 ◽  
pp. 312-313
Author(s):  
P. R. Swann ◽  
W. R. Duff ◽  
R. M. Fisher
Keyword(s):  
Electron Microscopy ◽  
Transmission Electron Microscopy ◽  
Annealing Temperature ◽  
Antiphase Domain ◽  
Effect Of Temperature ◽  
Domain Structures ◽  
Transmission Electron ◽  
Domain Boundaries ◽  
Higher Temperature ◽  
The One

Recently we have investigated the phase equilibria and antiphase domain structures of Fe-Al alloys containing from 18 to 50 at.% Al by transmission electron microscopy and Mössbauer techniques. This study has revealed that none of the published phase diagrams are correct, although the one proposed by Rimlinger agrees most closely with our results to be published separately. In this paper observations by transmission electron microscopy relating to the nucleation of disorder in Fe-24% Al will be described. Figure 1 shows the structure after heating this alloy to 776.6°C and quenching. The white areas are B2 micro-domains corresponding to regions of disorder which form at the annealing temperature and re-order during the quench. By examining specimens heated in a temperature gradient of 2°C/cm it is possible to determine the effect of temperature on the disordering reaction very precisely. It was found that disorder begins at existing antiphase domain boundaries but that at a slightly higher temperature (1°C) it also occurs by homogeneous nucleation within the domains. A small (∼ .01°C) further increase in temperature caused these micro-domains to completely fill the specimen.

Fundamental Research into Thin Films in a Developing Country

1972 ◽  
Vol 30 ◽  
pp. 500-501
Author(s):  
J.A. Eades ◽  
E. Grünbaum
Keyword(s):  
Thin Films ◽  
Growth Process ◽  
Empirical Process ◽  
Nucleation And Growth ◽  
Research Groups ◽  
Film Research ◽  
Fundamental Research ◽  
Controlled Deposition ◽  
The One ◽  
The University

In the last decade and a half, thin film research, particularly research into problems associated with epitaxy, has developed from a simple empirical process of determining the conditions for epitaxy into a complex analytical and experimental study of the nucleation and growth process on the one hand and a technology of very great importance on the other. During this period the thin films group of the University of Chile has studied the epitaxy of metals on metal and insulating substrates. The development of the group, one of the first research groups in physics to be established in the country, has parallelled the increasing complexity of the field.The elaborate techniques and equipment now needed for research into thin films may be illustrated by considering the plant and facilities of this group as characteristic of a good system for the controlled deposition and study of thin films.

STM studies of crystal growth

1991 ◽  
Vol 49 ◽  
pp. 374-375
Author(s):  
M. G. Lagally
Keyword(s):  
Crystal Growth ◽  
Molecular Beam ◽  
Chemical Vapor ◽  
Sputter Deposition ◽  
Field Of View ◽  
Scanning Tunneling ◽  
Wide Field ◽  
Tunneling Microscopy ◽  
Wide Field Of View ◽  
The One

It has been recognized since the earliest days of crystal growth that kinetic processes of all Kinds control the nature of the growth. As the technology of crystal growth has become ever more refined, with the advent of such atomistic processes as molecular beam epitaxy, chemical vapor deposition, sputter deposition, and plasma enhanced techniques for the creation of “crystals” as little as one or a few atomic layers thick, multilayer structures, and novel materials combinations, the need to understand the mechanisms controlling the growth process is becoming more critical. Unfortunately, available techniques have not lent themselves well to obtaining a truly microscopic picture of such processes. Because of its atomic resolution on the one hand, and the achievable wide field of view on the other (of the order of micrometers) scanning tunneling microscopy (STM) gives us this opportunity. In this talk, we briefly review the types of growth kinetics measurements that can be made using STM. The use of STM for studies of kinetics is one of the more recent applications of what is itself still a very young field.

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