arXiv subjects

Rajendra Kumar Sharma

Publications and source records attributed to Rajendra Kumar Sharma.

8 recordsLinked to original sources

arXiv2025

The Exact Enumeration of $4$-nomial and $5$-nomial Multiples of the Product of Primitive Polynomials over GF(2)

Linear feedback shift registers (LFSRs) are used to generate secret keys in stream cipher cryptosystems. There are different kinds of key-stream generators like filter generators, combination generators, clock-controlled generators, etc. For a combination generator, the connection polynomial is the product of the connection polynomials of constituent LFSRs. For better cryptographic properties, the connection polynomials of the constituent LFSRs should be primitive with coprime degrees. The cryptographic systems using LFSRs as their components are vulnerable to correlation attacks. The attack heavily depends on the $t$-nomial multiples of the connection polynomial for small values of $t$. In 2005, Maitra, Gupta, and Venkateswarlu provided a lower bound for the number of $t$-nomial multiples of the product of primitive polynomials over GF(2). The lower bound is exact when $t=3$. In this article, we provide the exact number of $4$-nomial and $5$-nomial multiples of the product of primitive polynomials. This helps us to choose a more suitable connection polynomial to resist the correlation attacks. Next, we disprove a conjecture by Maitra, Gupta, and Venkateswarlu.

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arXiv2024

$({\sigma}, {\tau})$-Derivations of Number Rings with Coding Theory Applications

In this article, we study $(\sigma, \tau)$-derivations of number rings by considering them as commutative unital $\mathbb{Z}$-algebras. We begin by characterizing all $(\sigma, \tau)$-derivations and inner $(\sigma, \tau)$-derivations of the ring of algebraic integers of a quadratic number field. Then we characterize all $(\sigma, \tau)$-derivations of the ring of algebraic integers $\mathbb{Z}[\zeta]$ of a $p^{\text{th}}$-cyclotomic number field $\mathbb{Q}(\zeta)$ ($p$ odd rational prime and $\zeta$ a primitive $p^{\text{th}}$-root of unity). We also conjecture (using SageMath and MATLAB) an \enquote{if and only if} condition for a $(\sigma, \tau)$-derivation $D$ on $\mathbb{Z}[\zeta]$ to be inner. We further characterize all $(\sigma, \tau)$-derivations and inner $(\sigma, \tau)$-derivations of the bi-quadratic number ring $\mathbb{Z}[\sqrt{m}, \sqrt{n}]$ ($m$, $n$ distinct square-free rational integers). In each of the above cases, we also determine the rank and an explicit basis of the derivation algebra consisting of all $(\sigma, \tau)$-derivations of the number ring. As a consequence, we solve the twisted derivation problem in the ring of algebraic integers of a quadratic number field and in a bi-quadratic number ring, and we conjecture a solution of the twisted derivation problem in the ring of algebraic integers of a $p^{\text{th}}$-cyclotomic number field. Finally, we give the applications of our work in coding theory by constructing Hom-IDD codes.

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arXiv2024

Twisted Derivations in Algebraic Number Fields

Let $A$ be a commutative ring with unity and $B = A[\theta]$ be an integral extension of $A$. Assume that $B$ is an integral domain with quotient field $\mathbb{K}$ and $\mathbb{E}$ is the minimal splitting field of $\theta$ over $\mathbb{K}$. Suppose $\sigma, \tau: B \rightarrow \mathbb{E}$ are two different ring homomorphisms that fix $A$ element-wise. In this article, we classify all $A$-linear maps $D: B \rightarrow \mathbb{E}$ which are $(\sigma, \tau)$-derivations. Consequently, we classify all $(\sigma, \tau)$-derivations in certain field extensions, algebraic number fields, and their ring of algebraic integers. For the ring of algebraic integers, $O_{\mathbb{K}} = \mathbb{Z}[\zeta]$ of the cyclotomic number field $\mathbb{K} = \mathbb{Q}(\zeta)$ ($\zeta$ an $n^{\text{th}}$ primitive root of unity), and a pair $(\sigma, \tau)$ of two different $\mathbb{Z}$-algebra endomorphisms of $O_{\mathbb{K}}$, we conjecture (using SageMath) a necessary and sufficient condition for a $(\sigma, \tau)$-derivation $D:O_{\mathbb{K}} \rightarrow O_{\mathbb{K}}$ to be inner. This is done for two different forms of $n$: (i) $n = 2^{r}p$ ($r \in \mathbb{N}$ and $p$ an odd rational prime), and (ii) $n=p^{k}$ ($k \in \mathbb{N} \setminus \{1\}$ and $p$ any rational prime). As an application of our main result on classification of $(\sigma, \tau)$-derivations $D:B \rightarrow \mathbb{E}$ and also the conjectures on inner $(\sigma, \tau)$-derivations of $O_{\mathbb{K}}$, we also conjecture the existence and non-existence of non-zero outer derivations of $O_{\mathbb{K}}$ for the above two forms of $n$, thus answering the twisted derivation problem in $O_{\mathbb{K}}$. Finally, as another application of our main result on the classification of $(\sigma, \tau)$-derivations $D:B \rightarrow \mathbb{E}$, we construct some binary Hom-IDD codes in coding theory.

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arXiv2024

$(σ, τ)$-Derivations of Group Rings with Applications

Leo Creedon and Kieran Hughes in [18] studied derivations of a group ring $RG$ (of a group $G$ over a commutative unital ring $R$) in terms of generators and relators of group $G$. In this article, we do that for $(σ, τ)$-derivations. We develop a necessary and sufficient condition such that a map $f:X \rightarrow RG$ can be extended uniquely to a $(σ, τ)$-derivation $D$ of $RG$, where $R$ is a commutative ring with unity, $G$ is a group having a presentation $\langle X \mid Y \rangle$ ($X$ the set of generators and $Y$ the set of relators) and $(σ, τ)$ is a pair of $R$-algebra endomorphisms of $RG$ which are $R$-linear extensions of the group endomorphisms of $G$. Further, we classify all inner $(σ, τ)$-derivations of the group algebra $RG$ of an arbitrary group $G$ over an arbitrary commutative unital ring $R$ in terms of the rank and a basis of the corresponding $R$-module consisting of all inner $(σ, τ)$-derivations of $RG$. We obtain several corollaries, particularly when $G$ is a $(σ, τ)$-FC group or a finite group $G$ and when $R$ is a field. We also prove that if $R$ is a unital ring and $G$ is a group whose order is invertible in $R$, then every $(σ, τ)$-derivation of $RG$ is inner. We apply the results obtained above to study $σ$-derivations of commutative group algebras over a field of positive characteristic and to classify all inner and outer $σ$-derivations of dihedral group algebras $\mathbb{F}D_{2n}$ ($D_{2n} = \langle a, b \mid a^{n} = b^{2} = 1, b^{-1}ab = a^{-1}\rangle$, $n \geq 3$) over an arbitrary field $\mathbb{F}$ of any characteristic. Finally, we give the applications of these twisted derivations in coding theory by giving a formal construction with examples of a new code called IDD code.

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arXiv2024

Inner and Outer Derivations of $\mathbb{F}V_{8n}$

Let $\mathbb{F}$ be a field of characteristic $0$ or an odd rational prime $p$. In this article, we give an explicit classification of all the inner and outer derivations of the group algebra $\mathbb{F}V_{8n}$, where $V_{8n}$ is a group of order $8n$ ($n$ a positive integer) with presentation $\langle a, b \mid a^{2n} = b^{4} = 1, ba = a^{-1}b^{-1}, b^{-1}a = a^{-1}b \rangle$. First, we explicitly classify all the $\mathbb{F}$-derivations of $\mathbb{F}V_{8n}$ by giving the dimension and a basis of the derivation algebra consisting of all $\mathbb{F}$-derivations of $\mathbb{F}V_{8n}$. Consequently, we classify all inner and outer derivations of $\mathbb{F}V_{8n}$ when $\mathbb{F}$ is an algebraic extension of a prime field. Thus, we establish that all the derivations of $\mathbb{F}V_{8n}$ are inner when the characteristic of $\mathbb{F}$ is $0$ or $p$ with $p$ relatively prime to $n$, and that non-zero outer derivations exist only in the case when the characteristic of $\mathbb{F}$ is $p$ with $p$ dividing $n$.

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arXiv2024

$\mathbb{F}_q$-primitive points on varieties over finite fields

Let $r$ be a positive divisor of $q-1$ and $f(x,y)$ a rational function of degree sum $d$ over $\mathbb{F}_q$ with some restrictions, where the degree sum of a rational function $f(x,y) = f_1(x,y)/f_2(x,y)$ is the sum of the degrees of $f_1(x,y)$ and $f_2(x,y)$. In this article, we discuss the existence of triples $(α, β, f(α, β))$ over $\mathbb{F}_q$, where $α, β$ are primitive and $f(α, β)$ is an $r$-primitive element of $\mathbb{F}_q$. In particular, this implies the existence of $\mathbb{F}_q$-primitive points on the surfaces of the form $z^r = f(x,y)$. As an example, we apply our results on the unit sphere over $\mathbb{F}_q$.

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arXiv2024

Inner and Outer Twisted Derivations of Cyclic Group Rings

In this article, we study twisted derivations of cyclic group rings. Let $R$ be a commutative ring with unity, $G$ be a finite cyclic group, and ($σ, τ$) be a pair of $R$-algebra endomorphisms of the group algebra $RG$, which are $R$-linear extensions of the group endomorphisms of $G$. In this article, we give two characterizations concerning $(σ, τ)$-derivations of the group ring $RG$. First, we develop a necessary and sufficient condition for a $(σ, τ)$-derivation of $RG$ to be inner. Second, we provide a necessary and sufficient condition for an $R$-linear map $D: RG \rightarrow RG$ with $D(1) = 0$ to be a $(σ, τ)$-derivation. We also illustrate our theorems with the help of examples. As a consequence of these two characterizations, we answer the well-known twisted derivation problem for $RG$: Under what conditions are all $(σ, τ)$-derivations of $RG$ inner? Or is the space of outer $(σ, τ)$-derivations trivial? More precisely, we give a sufficient condition under which all $(σ, τ)$-derivations of $RG$ are inner and a sufficient condition under which $RG$ has non-trivial outer $(σ, τ)$-derivations. Our result helps in generating several examples of non-trivial outer derivations.

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arXiv2023

Derivations of Non-Commutative Group Algebras

In this article, we study the derivations of group algebras of some important groups, namely, dihedral ($D_{2n}$), Dicyclic ($T_{4n}$) and Semi-dihedral ($SD_{8n}$). First, we explicitly classify all inner derivations of a group algebra $\mathbb{F}G$ of a finite group $G$ over an arbitrary field $\mathbb{F}$. Then we classify all $\mathbb{F}$-derivations of the group algebras $\mathbb{F}D_{2n}$, $\mathbb{F}T_{4n}$ and $\mathbb{F}(SD_{8n})$ when $\mathbb{F}$ is a field of characteristic $0$ or an odd rational prime $p$ by giving the dimension and an explicit basis of these derivation algebras. We explicitly describe all inner derivations of these group algebras over an arbitrary field. Finally, we classify all derivations of the above group algebras when $\mathbb{F}$ is an algebraic extension of a prime field.

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