Odd Characteristics Research Articles

Learning Hamiltonian dynamics with reproducing kernel Hilbert spaces and random features

A method for learning Hamiltonian dynamics from a limited and noisy dataset is proposed. The method learns a Hamiltonian vector field on a reproducing kernel Hilbert space (RKHS) of inherently Hamiltonian vector fields, and in particular, odd Hamiltonian vector fields. This is done with a symplectic kernel, and it is shown how the kernel can be modified to an odd symplectic kernel to impose the odd symmetry. A random feature approximation is developed for the proposed odd kernel to reduce the problem size. The performance of the method is validated in simulations for three Hamiltonian systems. It is demonstrated that the use of an odd symplectic kernel improves prediction accuracy and data efficiency, and that the learned vector fields are Hamiltonian and exhibit the imposed odd symmetry characteristics.

ON TOPOLOGICAL COMPLEXITY OF GORENSTEIN SPACES

In this paper, using Sullivan's approach to rational homotopy theory of simply-connected finite type CW complexes, we endow the $\mathbb{Q}$-vector space $\mathcal{E}xt_{C^{\ast}(X;\mathbb{Q})}(\mathbb{Q}, C^{\ast}(X;\mathbb{Q}))$ with a graded commutative algebra structure. This leads us to introduce the $\mathcal{E}xt$-version of higher (resp. module, homology) topological complexity of $X_0$, the rationalization of $X$ (resp. of $X$ over $\mathbb{Q}$). We then compare these invariants and their respective ordinary ones for Gorenstein spaces.
 We also highlight, in this context, the benefit of Adams-Hilton models over a field of odd characteristics especially through two cases, the first one when the space is a $2$-cell CW-complex and the second one when it is a suspension.

Trivial source character tables of , part II

AbstractWe compute the trivial source character tables (also called species tables of the trivial source ring) of the infinite family of finite groups $\operatorname{SL}_{2}(q)$ for q even over a large enough field of odd characteristics. This article is a continuation of our article Trivial Source Character Tables of$\operatorname{SL}_{2}(q)$, where we considered, in particular, the case in which q is odd in non-defining characteristic.

Towards the Links of Cryptanalytic Methods on MPC/FHE/ZK-Friendly Symmetric-Key Primitives

Symmetric-key primitives designed over the prime field Fp with odd characteristics, rather than the traditional Fn2 , are becoming the most popular choice for MPC/FHE/ZK-protocols for better efficiencies. However, the security of Fp is less understood as there are highly nontrivial gaps when extending the cryptanalysis tools and experiences built on Fn2 in the past few decades to Fp.At CRYPTO 2015, Sun et al. established the links among impossible differential, zero-correlation linear, and integral cryptanalysis over Fn2 from the perspective of distinguishers. In this paper, following the definition of linear correlations over Fp by Baignères, Stern and Vaudenay at SAC 2007, we successfully establish comprehensive links over Fp, by reproducing the proofs and offering alternatives when necessary. Interesting and important differences between Fp and Fn2 are observed.- Zero-correlation linear hulls can not lead to integral distinguishers for some cases over Fp, while this is always possible over Fn2proven by Sun et al..- When the newly established links are applied to GMiMC, its impossible differential, zero-correlation linear hull and integral distinguishers can be increased by up to 3 rounds for most of the cases, and even to an arbitrary number of rounds for some special and limited cases, which only appeared in Fp. It should be noted that all these distinguishers do not invalidate GMiMC’s security claims.The development of the theories over Fp behind these links, and properties identified (be it similar or different) will bring clearer and easier understanding of security of primitives in this emerging Fp field, which we believe will provide useful guides for future cryptanalysis and design.

Towards the generalized purely wild inertia conjecture for product of alternating and symmetric groups

We obtain new evidence for the Purely Wild Inertia Conjecture posed by Abhyankar and for its generalization. We show that this generalized conjecture is true for any product of simple Alternating groups in odd characteristics, and for any product of certain Symmetric or Alternating groups in characteristic two. We also obtain important results towards the realization of the inertia groups which can be applied to more general set up. We further show that the Purely Wild Inertia Conjecture is true for any product of perfect quasi p-groups (groups generated by their Sylow p-subgroups) if the conjecture is established for individual groups.

Periodic Groups Saturated with Finite Simple Symplectic Groups of Dimension 6 over Fields of Odd Characteristics

Periodic Groups Saturated with Finite Simple Symplectic Groups of Dimension 6 over Fields of Odd Characteristics

The plethystic inverse of the odd Lie representations

The Frobenius characteristic of L i e n Lie_n , the representation of the symmetric group S n S_n afforded by the multilinear part of the free Lie algebra, is known to satisfy many interesting plethystic identities. In this paper we prove a conjecture of Richard Stanley establishing the plethystic inverse of the sum ∑ n ≥ 0 L i e 2 n + 1 \sum _{n\geq 0} Lie_{2n+1} of the odd Lie characteristics. We obtain an apparently new plethystic decomposition of the regular representation of S n S_n in terms of irreducibles indexed by hooks, and the Lie representations. We also determine the plethystic inverse of the alternating sum of the odd Lie characteristics.

Evolutionary relationships of anglerfishes (Lophiiformes) reconstructed using ultraconserved elements

The macroevolutionary consequences of evolving in the deep-sea remain poorly understood and are compounded by the fact that convergent adaptations for living in this environment makes elucidating phylogenetic relationships difficult. Lophiiform anglerfishes exhibit extreme habitat and predatory specializations, including the use of a fin-spine system as a luring device and unique reproductive strategies where parasitic males attach and fuse to females. Despite their notoriety for these odd characteristics, evolutionary relationships among these fishes remain unclear. We sought to clarify the evolutionary history of Lophiiformes using data from 1000 ultraconserved elements and phylogenomic inference methods with particular interest paid to the Ceratioidei (deep-sea anglerfishes) and Antennarioidei (frogfishes and handfishes). At the suborder level, we recovered similar topologies in separate phylogenomic analyses: The Lophioidei (monkfishes) are the sister group to the rest of the Lophiiformes, Ogcocephaloidei (batfishes) and Antennarioidei (frogfishes) form a sister group, and Chaunacioidei (coffinfishes) and Ceratioidei (deep-sea anglerfishes) form a clade. The relationships we recover within the ceratioids disagree with most previous phylogenetic investigations, which used legacy phylogenetic markers or morphology. We recovered non-monophyletic relationships in the Antennarioidei and proposed three new families based on molecular and morphological evidence: Histiophrynidae, Rhycheridae, and Tathicarpidae. Antennariidae was re-evaluated to include what was known as Antennariinae, but not Histiophryninae. Non-bifurcating signal in splits network analysis indicated reticulations among and within suborders, supporting the complicated history of the Lophiiformes previously found with morphological data. Although we resolve relationships within Antennarioidei, Ceratioidei relationships remain somewhat unclear without better taxonomic sampling.

On the pronormality of subgroups of odd index in some direct products of finite groups

A subgroup [Formula: see text] of a group [Formula: see text] is said to be pronormal in [Formula: see text] if [Formula: see text] and [Formula: see text] are conjugate in [Formula: see text] for each [Formula: see text]. Some problems in Finite Group Theory, Combinatorics and Permutation Group Theory were solved in terms of pronormality, therefore, the question of pronormality of a given subgroup in a given group is of interest. Subgroups of odd index in finite groups satisfy a native necessary condition of pronormality. In this paper, we continue investigations on pronormality of subgroups of odd index and consider the pronormality question for subgroups of odd index in some direct products of finite groups. In particular, in this paper, we prove that the subgroups of odd index are pronormal in the direct product [Formula: see text] of finite simple symplectic groups over fields of odd characteristics if and only if the subgroups of odd index are pronormal in each direct factor of [Formula: see text]. Moreover, deciding the pronormality of a given subgroup of odd index in the direct product of simple symplectic groups over fields of odd characteristics is reducible to deciding the pronormality of some subgroup [Formula: see text] of odd index in a subgroup of [Formula: see text], where each [Formula: see text] acts naturally on [Formula: see text], such that [Formula: see text] projects onto [Formula: see text]. Thus, in this paper, we obtain a criterion of pronormality of a subgroup [Formula: see text] of odd index in a subgroup of [Formula: see text], where each [Formula: see text] is a prime and each [Formula: see text] acts naturally on [Formula: see text], such that [Formula: see text] projects onto [Formula: see text].

Generalized Galois-Fibonacci Matrix Generators Pseudo-Random Sequences

The article discusses various options for constructing binary generators of pseudo-random numbers (PRN) based on the so-called generalized Galois and Fibonacci matrices. The terms "Galois matrix" and "Fibonacci matrix" are borrowed from the theory of cryptography, in which the linear feedback shift registers (LFSR) generators of the PRN according to the Galois and Fibonacci schemes are widely used. The matrix generators generate identical PRN sequences as the LFSR generators. The transition from classical to generalized matrix PRN generators (PRNG) is accompanied by expanding the variety of generators, leading to a significant increase in their cryptographic resistance. This effect is achieved both due to the rise in the number of elements forming matrices and because generalized matrices are synthesized based on primitive generating polynomials and polynomials that are not necessarily primitive. Classical LFSR generators of PRN (and their matrix equivalents) have a significant drawback: they are susceptible to Berlekamp-Messi (BM) attacks. Generalized matrix PRNG is free from BM attack. The last property is a consequence of such a feature of the BM algorithm. This algorithm for cracking classical LFSR generators of PRN solves the problem of calculating the only unknown – a primitive polynomial generating the generator. For variants of generalized matrix PRNG, it becomes necessary to determine two unknown parameters: both an irreducible polynomial and a forming element that produces a generalized matrix. This problem turns out to be unsolvable for the BM algorithm since it is designed to calculate only one unknown parameter. The research results are generalized for solving PRNG problems over a Galois field of odd characteristics.

Secondary constructions of (non)weakly regular plateaued functions over finite fields

Plateaued (vectorial) functions over finite fields have diverse applications in symmetric cryptography, coding theory, and sequence theory. Constructing these functions is an attractive research topic in the literature. We can distinguish two kinds of constructions of plateaued functions: secondary constructions and primary constructions. The first method uses already known functions to obtain new functions while the latter do not need to use previously constructed functions to obtain new functions. In this work, the first secondary constructions of (non)weakly regular plateaued (vectorial) functions are presented over the finite fields of odd characteristics. We also introduce some recursive constructions of (non)weakly regular plateaued $p$-ary functions by using already known such functions. We obtain nontrivial plateaued functions from the previously known trivial plateaued (partially bent) functions in the proposed construction methods.

Algorithms for quadratic forms over global function fields of odd characteristic

This paper presents an adaptation of recently developed algorithms for quadratic forms over number fields in [4] to global function fields of odd characteristics. First, we present algorithm for checking if a given non-degenerate quadratic form is isotropic or hyperbolic. Next we devise a method for computing the dimension of the anisotropic part of a quadratic form. Finally we present algorithms computing two field invariants: the level and the Pythagoras number.

Cyclic Codes from a Sequence over Finite Fields

A cyclic code has been one of the most active research topics in coding theory because they have many applications in data storage systems and communication systems. They have efficient encoding and decoding algorithms. This paper explains the construction of a family of cyclic codes from sequences generated by a trace of a monomial over finite fields of odd characteristics. The parameter and some examples of the codes are presented in this paper.

Construction of Binary Sequences With Low Correlation via Multiplicative Quadratic Character Over Finite Fields of Odd Characteristics

In literature, there are several methods to construct Gold sequences. One of the constructions is via the trace function from extension field of \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> . Estimation of correlation of this construction is based on number of rational points of elliptic curves. In this article, we generalize this construction from finite fields of even characteristic to odd characteristics by using multiplicative quadratic character. Again, estimation of correlation of this construction is based on number of rational points of elliptic curves. Thus, we obtain binary sequences which have more flexibility on length while still possessing low correlation property. Moreover, some of the sequences are optimally balanced.

Factoring polynomials over finite fields

In this paper, we describe a new polynomial factorization algorithm over finite fields with odd characteristics. The main ingredient of the algorithm is special singular curves. The algorithm relies on the extension of the Mumford representation and Cantor’s algorithm to these special singular curves.

Микросистемы с высоким коэффициентом преобразования слабого магнитного поля на основе магниторезистивных наноструктур

The magnetoresistive transducers of magnetic field have been claimed both for direct and indirect applications in various areas of industry, transport and special equipment. In the paper the special features of used terminology and dimensions in the main parameters of magnetic field magnetoresistive transformers have been revealed. The main results of experimental studies on the developed thin-film magnetoresistive nanostructures (TMN) with even and odd transfer characteristics have been presented. The TMN design on an anisotropic magnetoresistive effect has an odd transfer characteristic and conversion coefficient at the level of 8 mV/V/E. The TMN design based on a giant magnetoresistive effect has an even transfer characteristic and a conversion coefficient at the level of 27 mV/V/E. The results of promising design and technological solutions allowing reaching the values of conversion coefficient of TMN at the level exceeding 100 mV/V/E have been shown. The results of the study of the test spin-tunnel magnetoresistive nanostructure (spin-tunnel TMN) with giant magnetoresistive effect, exceeding 100% have been presented. The novelty of the obtained results has been reflected and the perspective of using highly sensitive TMN and creating the non-volatile MRAM on the basis of thin-film magnetoresistive structure with the spin-tunnel magnetoresistive effect have been determined.

The large sieve inequality with square moduli for quadratic extensions of function fields

In this paper, we establish a version of the large sieve inequality with square moduli for imaginary quadratic extensions of rational function fields of odd characteristics.

A recurrent construction of irreducible polynomials of fixed degree over finite fields

In this paper we consider in detail the composition of an irreducible polynomial with X^2 and suggest a recurrent construction of irreducible polynomials of fixed degree over finite fields of odd characteristics. More precisely, given an irreducible polynomial of degree n and order 2^rt with t odd, the construction produces ord_t(2) irreducible polynomials of degree n and order t. The construction can be used for example to search irreducible polynomials with specific requirements on its coefficients.

System Identification Method Based on a Disturbance Observer Using Symmetric Reference Trajectories in PMLSM-Based Motion Systems

This study is concerned with the problem of system identification for permanent magnet linear synchronous motor(PMLSM) based motion systems with disturbances such as parametric uncertainty, friction force, and force ripple. In order to identify the disturbances, a novel system identification method is proposed. The main idea of the proposed scheme is to separate and identify each element from a lumped disturbance by using even and odd characteristics of the state-dependent disturbances. The lumped disturbance is estimated by a linear disturbance observer(DOB), that includes the parametric errors, the friction force, and force ripple. To facilitate the decomposition of the lumped disturbance, the symmetric reference position trajectories are utilized, which make it useful to keep the even and odd characteristics of each element. The even and odd elements separated from the lumped disturbance by the decomposition method are represented as the force ripple multiplied by the mass error and the friction force multiplied by the viscous friction coefficient error, respectively. By comparing with experimental results based on different reference velocities and acceleration trajectories, the parametric errors are identified first, and then, nonlinear disturbances such as the force ripple and friction forces are identified by curve-fitting methods using the experimental data. The validity of the proposed scheme is illustrated by experimental tests. The effectiveness of the identified system elements is also verified by comparison with the position tracking performances of comparative studies.

Strongly regular graphs arising from non-weakly regular bent functions

In this paper, we study two special subsets of a finite field of odd characteristics associated with non-weakly regular bent functions. We show that those subsets associated to non-weakly regular even bent functions in the GMMF class (see Cesmelioglu et al. Finite Fields Appl. 24, 105–117 2013) are never partial difference sets (PDSs), and are PDSs if and only if they are trivial subsets. Moreover, we analyze the two known sporadic examples of non-weakly regular ternary bent functions given in Helleseth and Kholosha (IEEE Trans. Inf. Theory 52(5), 2018–2032 2006, Cryptogr. Commun. 3(4), 281–291 2011). We observe that corresponding subsets are non-trivial partial difference sets. We show that they are the union of some cyclotomic cosets and so correspond to 2-class fusion schemes of a cyclotomic scheme. We also present a further construction giving non-trivial PDSs from certain p-ary functions which are not bent functions.