Prime Order Research Articles

Evaluate All The Order of Every Element in The Higher Even, Odd, and Prime Order of Group for Composition

This paper aims to treat a study on the order of every element in the higher even, odd and prime order of group for composition. In fact, express order of a group and order of an element of a group in real numbers. Here we discuss the higher order of groups in different types of order, which will give us practical knowledge to see the applications of the composition. In order to find out the order of an element am∈G in which an= e= identity element, then find the least common multiple (i.e.(LCM))= λ) of m and n. The least common multiple of two numbers is the "smallest non-zero common number," which is a multiple of both the numbers. So O(am )= λ/m. Also, if G is a finite group, n is a positive integer, and a∈G then the order of the products na. When G is a finite group, every element must have finite order, but the converse is false. There are infinite groups where each element has finite order. Finally, find out the order of every element of a group in different types of the higher even, odd and prime order of group for composition.

Constructing group inclusions with arbitrary depth via wreath products

It was proved in a paper by Burciu, Kadison and Külshammer in 2011 that the ordinary depth d(S_n,S_{n+1}) of the symmetric group S_n in S_{n+1} is 2n-1, so arbitrarily large odd numbers can occur as subgroup depth. Lars Kadison in 2011 posed the question if subgroups of even ordinary depth bigger than 6 can occur. Recently in a paper with Breuer we constructed a series (G_n,H_n) of groups and subgroups where the depth d(H_n,G_n) was 2n, thus answering the question of Kadison. Here we generalize the method of that proof. The main result of this paper is that for every positive integer n there are infinitely many pairs (G, H) of finite groups such that d(H,G)=n. As a corollary of its proof we get that for every positive integer n there are infinitely many triples (H, N, G) of finite solvable groups Htriangleleft Ntriangleleft G such that G/N is cyclic of order lceil {n/2} rceil , N/H is cyclic of arbitrarily large prime order and d(H,G)=n. We investigate the series d(H_n,G_n) in the cases when the depth, d(H_1,G_1), is 1, 2 or 3, where H_n:=H_1times G^{n-1} and G_n:=G_1wr C_n. We also prove that if H_1=S_k and G_1=S_{k+1} then d(H_n,G_n)=2nk-1.

On groups in which every element has a prime power order and which satisfy some boundedness condition

In this paper, we shall deal with periodic groups, in which each element has a prime power order. A group [Formula: see text] will be called a [Formula: see text]-group if each element of [Formula: see text] has a prime power order and for each [Formula: see text] there exists a positive integer [Formula: see text] such that each [Formula: see text]-element of [Formula: see text] is of order [Formula: see text]. A group [Formula: see text] will be called a [Formula: see text]-group if each element of [Formula: see text] has a prime power order and for each [Formula: see text] there exists a positive integer [Formula: see text] such that each finite [Formula: see text]-subgroup of [Formula: see text] is of order [Formula: see text]. Here, [Formula: see text] denotes the set of all primes dividing the order of some element of [Formula: see text]. Our main results are the following four theorems. Theorem 1: Let [Formula: see text] be a finitely generated [Formula: see text]-group. Then [Formula: see text] has only a finite number of normal subgroups of finite index. Theorem 4: Let [Formula: see text] be a locally graded [Formula: see text]-group. Then [Formula: see text] is a locally finite group. Theorem 7: Let [Formula: see text] be a locally graded [Formula: see text]-group. Then [Formula: see text] is a finite group. Theorem 8: Let [Formula: see text] be a [Formula: see text]-group satisfying [Formula: see text]. Then [Formula: see text] is a locally finite group.

Hyperelliptic Covers of Different Degree for Elliptic Curves

In elliptic curve cryptography (ECC) and hyperelliptic curve cryptography (HECC), the size of cipher-text space defined by the cardinality of Jacobian is a significant factor to measure the security level. Counting problems on Jacobians of elliptic curve can be solved in polynomial time by Schoof–Elkies–Atkin (SEA) algorithm. However, counting problems on Jacobians of hyperelliptic curves are solved less satisfactorily than those on elliptic curves. So, we consider the construction of the cover map from the hyperelliptic curves to the elliptic curves to convert point counting problems on hyperelliptic curves to those on elliptic curves. We can also use the cover map as a kind of cover attacks. Given an elliptic curve over an extension field of degree n , one might try to use the cover attack to reduce the discrete logarithm problem (DLP) in the group of rational points of the elliptic curve to DLPs in the Jacobian of a curve of genus g ≥ n over the base field. An algorithm has been proposed for finding genus 3 hyperelliptic covers as a cover attack for elliptic curves with cofactor 2. Our algorithms are about the cover map from hyperelliptic curves of genus 2 to elliptic curves of prime order. As an application, an example of an elliptic curve whose order is a 256-bit prime vulnerable to our algorithms is given.

On classification of finite groups with four generators three of which having prime orders $p,q,q$ $(p$ < $q)$. II.

On classification of finite groups with four generators three of which having prime orders $p,q,q$ $(p$ < $q)$. II.

On finite groups with four independent generators three of which being of odd prime order

On finite groups with four independent generators three of which being of odd prime order

On finite groups with three independent generators two of which being of odd prime order

On finite groups with three independent generators two of which being of odd prime order

Split logarithm problem and a candidate for a post-quantum signature scheme

A new form of the hidden discrete logarithm problem, called split logarithm problem, is introduced as primitive of practical post-quantum digital signature schemes, which is characterized in using two non-permutable elements $A$ and $B$ of a finite non-commutative associative algebra, which are used to compute generators $Q=AB$ and $G=BQ$ of two finite cyclic groups of prime order $q$. The public key is calculated as a triple of vectors $(Y,Z,T)$: $Y=Q^x$, $Z=G^w$, and $T=Q^aB^{-1}G^b$, where $x$, $w$, $a$, and $b$ are random integers. Security of the signature scheme is defined by the computational difficulty of finding the pair of integers $(x,w)$, although, using a quantum computer, one can easily find the ratio $x/w\bmod q$.

On finite groups with three independent generators two of which have distinct odd prime orders

On finite groups with three independent generators two of which have distinct odd prime orders

Finite groups with Hall normally embedded subgroups

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is called Hall normally embedded in [Formula: see text] if [Formula: see text] is a Hall subgroup of the normal closure [Formula: see text]. In this paper, we investigate the structure of a finite group [Formula: see text] under the assumption that certain subgroups of prime power order are Hall normally embedded in [Formula: see text].

Kummer versus Montgomery Face-off over Prime Order Fields

This paper makes a comprehensive comparison of the efficiencies of vectorized implementations of Kummer lines and Montgomery curves at various security levels. For the comparison, nine Kummer lines are considered, out of which eight are new, and new assembly implementations of all nine Kummer lines have been made. Seven previously proposed Montgomery curves are considered and new vectorized assembly implementations have been made for three of them. Our comparisons show that for all security levels, Kummer lines are consistently faster than Montgomery curves, though the speed-up gap is not much.

Regularity and inverse theorems for uniformity norms on compact abelian groups and nilmanifolds

Abstract We prove a general form of the regularity theorem for uniformity norms, and deduce an inverse theorem for these norms which holds for a class of compact nilspaces including all compact abelian groups, and also nilmanifolds; in particular we thus obtain the first non-abelian versions of such theorems. We derive these results from a general structure theorem for cubic couplings, thereby unifying these results with the Host–Kra Ergodic Structure Theorem. A unification of this kind had been propounded as a conceptual prospect by Host and Kra. Our work also provides new results on nilspaces. In particular, we obtain a new stability result for nilspace morphisms. We also strengthen a result of Gutman, Manners and Varjú, by proving that a k-step compact nilspace of finite rank is a toral nilspace (in particular, a connected nilmanifold) if and only if its k-dimensional cube set is connected. We also prove that if a morphism from a cyclic group of prime order into a compact finite-rank nilspace is sufficiently balanced (i.e. equidistributed in a certain quantitative and multidimensional sense), then the nilspace is toral. As an application of this, we obtain a new proof of a refinement of the Green–Tao–Ziegler inverse theorem.

An efficient and access policy-hiding keyword search and data sharing scheme in cloud-assisted IoT

By leveraging the cloud computing paradigm, the cloud-assisted Internet of Things (IoT) is able to offer the massive storage and highly available computation services for end-users. When the data collected from IoT devices is endlessly gathering to the cloud center, the data security become new challenges. Encrypting data is an effective measure to guarantee data confidentiality. However, traditional encryption techniques make the data searching and access control inoperative. Current attribute-based keyword search (ABKS) techniques provide a feasibility to achieve data searching and access control over encrypted data simultaneously. However, existing ABKS schemes leak sensitive information via clear access policy and may be unsuitable for certain IoT applications, where the access policies in data might contain private information of data owners such as the security number or the residential area of a resident in smart grids. In this work, we design an efficient and policy-hiding attribute-based keyword search and data sharing scheme (PH-ABKS-DS) in cloud-assisted IoT. To the best of our knowledge, this construction is the first PH-ABKS-DS with practical efficiency that is constructed on prime order group. We provide detailed correctness and security analyses for PH-ABKS-DS. Extensive experiments demonstrate that our proposed PH-ABKS-DS is correct and practical.

Computing normalisers of intransitive groups

The normaliser problem takes as input subgroups G and H of the symmetric group Sn, and asks one to compute NG(H). The fastest known algorithm for this problem is simply exponential, whilst more efficient algorithms are known for restricted classes of groups. In this paper, we will focus on groups with many orbits. We give a new algorithm for the normaliser problem for these groups that performs many orders of magnitude faster than previous implementations in GAP. We also prove that the normaliser problem for the special case G=Sn is at least as hard as computing the group of monomial automorphisms of a linear code over any field of fixed prime order.

Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group II

Let G be an additive finite abelian group. Let F(G) denote the monoid of all sequences over G. For Ω⊂F(G), let dΩ(G) be the smallest integer t such that every sequence S over G of length |S|≥t has a subsequence in Ω. A sequence S over G is a weak-regular sequence if vg(S)≤ord(g) for every g∈G. Let N(G) be the smallest integer t such that every weak-regular sequence S over G of length |S|≥t has a nontrivial zero-sum subsequence T with vg(T)=vg(S)>0 for some g|S. The invariant N(G) was formulated in a recent paper [4] to study dΩ(G) and has its own interest, and N(G) has been determined for the cyclic groups of prime order there. In this paper, among other results, we shall determine the precise value of N(G) for cyclic groups of prime power order and for elementary abelian groups of rank two.

On alpha -points of q-analogs of the Fano plane

Arguably, the most important open problem in the theory of q-analogs of designs is the question regarding the existence of a q-analog D of the Fano plane. As of today, it remains undecided for every single prime power order q of the base field. A point P is called an alpha -point of D if the derived design of D in P is a geometric spread. In 1996, Simon Thomas has shown that there always exists a non-alpha -point. For the binary case q = 2, Olof Heden and Papa Sissokho have improved this result in 2016 by showing that the non-alpha -points must form a blocking set with respect to the hyperplanes. In this article, we show that a hyperplane consisting only of alpha -points implies the existence of a partition of the symplectic generalized quadrangle W(q) into spreads. As a consequence, the statement of Heden and Sissokho is generalized to all primes q and all even values of q.

Tamagawa numbers of elliptic curves with torsion points

Let K be a global field and let E/K be an elliptic curve with a K-rational point of prime order p. In this paper, we are interested in how often the (global) Tamagawa number c(E/K) of E/K is divisible by p. This is a natural question to consider in view of the fact that the fraction $$c(E/K)/ |E(K)_{\text {tors}}|$$ appears in the second part of the Birch and Swinnerton-Dyer conjecture. We focus on elliptic curves defined over global fields, but we also prove a result for higher dimensional abelian varieties defined over $${\mathbb {Q}}$$ .

CGP17Pat: Automated Schizophrenia Detection Based on a Cyclic Group of Prime Order Patterns Using EEG Signals.

Background and Purpose: Machine learning models have been used to diagnose schizophrenia. The main purpose of this research is to introduce an effective schizophrenia hand-modeled classification method. Method: A public electroencephalogram (EEG) signal data set was used in this work, and an automated schizophrenia detection model is presented using a cyclic group of prime order with a modulo 17 operator. Therefore, the presented feature extractor was named as the cyclic group of prime order pattern, CGP17Pat. Using the proposed CGP17Pat, a new multilevel feature extraction model is presented. To choose a highly distinctive feature, iterative neighborhood component analysis (INCA) was used, and these features were classified using k-nearest neighbors (kNN) with the 10-fold cross-validation and leave-one-subject-out (LOSO) validation techniques. Finally, iterative hard majority voting was employed in the last phase to obtain channel-wise results, and the general results were calculated. Results: The presented CGP17Pat-based EEG classification model attained 99.91% accuracy employing 10-fold cross-validation and 84.33% accuracy using the LOSO strategy. Conclusions: The findings and results depicted the high classification ability of the presented cryptologic pattern for the data set used.

Nonsymplectic automorphisms of prime order on O’Grady’s sixfolds

We classify nonsymplectic automorphisms of prime order on irreducible holomorphic symplectic manifolds of O’Grady’s 6-dimensional deformation type. More precisely, we give a classification of the invariant and coinvariant sublattices of the second integral cohomology group.

Applications of group theory in crystallography

Group theory is a powerful tool for studying symmetric physical systems. Such systems include, in particular, molecules and crystals with symmetry. Group theory serves to explain the most important characteristics of atomic spectra. Group theory is also applied to the problems of atomic and nuclear physics. This paper gives examples of the use of the apparatus of group theory in research on crystallography, quantum mechanics, elementary particle physics. In particular, in these studies matrix groups and representations of unitary groups are actively used. For such groups we give an overview of the results on their recognition by the spectrum (by the orders of the elements of the group). This direction has been intensively developed in recent years both in our country and abroad. Recognition of finite simple non-Abelian groups by spectrum has been studied for last thirty years in Yekaterinburg at the Institute of Mathematics and Mechanics of the Ural Division of the Russian Academy of Sciences, in Chelyabinsk Federal University and in the Novosibirsk Institute of Mathematics of Siberian Division of the Russian Academy of Sciences. Some simple non-Abelian groups are not recognizable by their spectra. We have proposed an approach for recognizing groups by the bottom layer. The bottom layer of a group is the set of its elements of prime orders. A group is called recognizable by the bottom layer under additional conditions if it is uniquely restored by the bottom layer under these conditions. The paper considers some examples of simple non-Abelian finite groups that are not recognizable by spectra. For these examples, simultaneous recognition by spectrum and by the bottom layer is proved.