# Lhoussain El FadilSidi Mohamed Ben Abdellah University · Mathematics

Lhoussain El Fadil

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102

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## Publications

Publications (102)

Let K be a septic number field generated by a root, α, of an irreducible trinomial, x7+ax+b∈Z[x]. In this paper, for every prime integer, p, we calculate νp(i(K)); the highest power of p dividing the index, i(K), of the number field, K. In particular, we calculate the index, i(K). In application, when the index of K is not trivial, then K is not mo...

In this paper, we study the monogenity of any number field defined by a monic irreducible trinomial F(x)=x12+axm+b∈Z[x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F...

Let K be a number field generated by a root θ of a monic irreducible trinomial F ( x ) = x n + a x m + b ∈ ℤ [ x ] . In this paper, we study the problem of monogenity of K . More precisely, we provide some explicit conditions on a, b, n , and m for which K is not monogenic. As applications, we show that there are infinite families of non-monogenic...

Our goal in this paper is to compute the 2-rank of the class group of the fields K = ℚ ( − 2 , q , d ) \[\mathbb{K}=\mathbb{Q}\left( \sqrt{-2},\sqrt{q},\sqrt{d} \right)\] for any odd positive squarefree integer d and any prime integer q ≡ 5 (mod 8). Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic, of type (2,...

Let K be a pure number field generated by a root α of a monic irreducible polynomial f ( x )= x ⁿ − m with m a rational integer and 3 ≤ n ≤ 9 an integer. In this paper, we calculate an integral basis of ℤ K , and we study the monogenity of K , extending former results to the case when m is not necessarily square-free. Collecting and completing the...

Let | | be a discrete non-archimedean absolute value of a field K with valuation ring 𝒪, maximal ideal 𝓜 and residue field 𝔽 = 𝒪/𝓜 . Let L be a simple finite extension of K generated by a root α of a monic irreducible polynomial F ∈ O [ x ]. Assume that F ¯ = ϕ ¯ l $\overline F = \overline \varphi ^l$ in 𝔽[ x ] for some monic polynomial φ ∈ O [ x ]...

Let 𝔽 q be a finite field of q elements, where q is a power of an odd prime number. In this paper, we study the twisted Edwards curves denoted E E a,d over the local ring 𝔽 q [ e ], where e ² = 0. In the first time, we study the arithmetic of the ring 𝔽 q [ e ], e ² = 0. After that we define the twisted Edwards curves E E a,d over this ring and we...

The main goal of this paper is to provide a complete answer to the Problem 22 of Narkiewicz[24] for any quintic number field K generated by a complex root α of a monic irreducible trinomial F(x) = x⁵ + ax³ + b ∈ ℤ [x]. Namely we calculate the index of the field K. In particular, if the index is not trivial, then K is not mongenic. Finally, we illus...

The Eisenstein criterion is a particular case of the Schönemann’s irreducibility criterion stated in 1846. In 1906, based on Newton polygon techniques, Dumas gave a generalization of the Eisenstein criterion. In this paper, we extend this last generalization. Some applications on factorization of polynomials, and prime ideal factorization will be g...

In this paper, we study the monogenity of any number field defined by a monic irreducible trinomial F(x) = x12 + axm + b ∈ Z[x] with 1 ≤ m ≤ 11 {an integer. For every integer m, we give sufficient conditions on a and b so that the field index i(K) is not trivial. In particular, if i(K) ≠ 1, then K is not monogenic. For m=1, we give necessary and su...

Let be a finite field of q elements with for some odd prime integer p and a positive integer r. Let where The purpose of this paper is to investigate be the twisted Edwards curves over R, with . In the end of the paper, we study the complexity of this new addition law in and highlight some links of our results with elliptic curves cryptosystem.

In this paper, for any septic number field K generated by a root α of a monic irreducible trinomial F(x)=x7+ax3+b∈Z[x], we describe all prime power divisors of the index of K answering Problem 22 of Narkiewicz [ 26 Narkiewicz, W. (2004). Elementary and Analytic Theory of Algebraic Numbers, 3rd ed. Berlin: Springer.[Crossref] , [Google Scholar]]. In...

Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $F(x)=x^{2^r\cdot7^s}-m\in \mathbb{Z}[x]$, where $m\neq \pm 1$ is a square free integer, $r$ and $s$ are two positive integers. In this paper, we study the monogenity of $K$. We prove that if $m\not\equiv 1\md{4}$ and $\overline{m}\not\in\{\pm \overline{1},...

Let \(K = \mathbb {Q} (\alpha)\)
be a pure number field generated by a complex root \(\alpha \) of a monic irreducible polynomial
\(F(x) = x^{{2}^{u}.3^{v}} - m\), with
\(m \neq \pm 1 \) a square free rational integer,
u, and v two positive integers. In this paper, we study the monogenity
of K. The cases \(u = 0\) and
\(v=0\) have been previously s...

In this paper, we deal with the problem of monogenity of number fields defined by monic irreducible trinomials $F(x)=x^{12}+ax^m+b\in \mathbb{Z}[x]$ with $1\leq m\leq11$. We give sufficient conditions on $a$, $b$, and $m$ so that the number field $K$ is not monogenic. In particular, for $m=1$ and for every rational prime $p$, we characterize when $...

For a number field K defined by a trinomial F(x) = x⁶ + ax + b ∈ ℤ[x], Jakhar and Kumar gave some necessary conditions on a and b, which guarantee the non-monogenity of K [25]. In this paper, for every prime integer p, we characterize when p is a common index divisor of K. In particular, if any one of these conditions holds, then K is not monogenic...

Let K be a pure number field generated by a complex root of a monic irreducible polynomial F(x)=x60−m∈ℤ[x], with m≠ ± 1 a square free integer. In this paper, we study the monogenity of K. We prove that if m≢1 (mod 4), m≢ ± 1 (mod 9) and m¯∉{±1,±7}(mod25), then K is monogenic. But if m ≡ 1 (mod 4), m ≡± 1 (mod 9), or m ≡± 1 (mod 25), then K is not m...

The main goal of this paper is to provide a complete answer to the Problem 22 of Narkiewicz \cite{Nar} for any sextic number field $K$ generated by a complex root $\al$ of a monic irreducible trinomial $F(x) = x^6+ax^5+b \in \Z[x]$. Namely we calculate the index of the field $K$. In particular, if $i(K)\neq 1$, then $K$ is not mongenic. Finally, we...

We consider number fields $K$ generated by a root of an irreducible trinomial $x^4+ax^2+b\in \Bbb Z[x]$ and characterize when a prime $p$ is a common index divisor of $K$. The existence of a common index divisor $p$ implies that $K$ is not monogenic. We illustrate our statements with a series of examples.

Let $K = \mathbb{Q} (\alpha) $ be a pure number field generated by a root $\alpha$ of a monic irreducible polynomial $ F(x) = x^{2^u\cdot 3^v\cdot 5^t}-m$, with $ m \neq \pm 1 $ a square free rational integer, $u$, $v$ and $t$ three positive integers. In this paper, we study the monogenity of $K$. We prove that if $m\not\equiv 1\md4$, $m\not\equiv...

Let $K$ be a number field generated by a root $\th$ of a monic irreducible trinomial $F(x) = x^n+ax^{m}+b \in \Z[x]$. In this paper, we study the problem of $K$. More precisely, we provide some explicit conditions on $a$, $b$, $n$, and $m$ for which $K$ is not monogenic. As applications, we show that there are infinite families of non-monogenic num...

Let Fq be the finite field of q elements, where q is a prime power. In this paper, we study the Montgomery curves over the ring Fq[X]/(X^2−X), denoted by MA,B(Fq[X]/(X^2−X) ); (A,B) ∈ (Fq[X]/(X^2−X))^2. Using the Montgomery equation, we define the Montgomery curves MA,B(Fq[X]/X^2−X) and we give a bijection between this curve and product of two Mont...

For a number field $K$ defined by a trinomail $F(x) = x^6+ax+b \in \mathbb{Z}[x]$, Jakhar and Kumar gave some necessary conditions on $a$ and $b$, which guarantee the non-monogenity of $K$ \cite{A6}. In this paper, for every prime integer $p$, we characterize when $p$ is a common index divisor of $K$. In particular, if any one of these conditions h...

In all available papers, on power integral bases of pure octic number fields $K$, generated by a root $\alpha$ of a monic irreducible polynomial $f(x)=x^8-m\in\mathbf Z[x]$, it was assumed that $m\neq \pm 1$ is square free. In this paper, we investigate the monogenity of any pure octic number field, without the condition that $m$ is square free. We...

In their paper [1], Shahzad Ahmad et al. given a characterization on any pure sextic number field Q(m1/6) with square-free integers m satisfying m 6 ±1 (mod 9) to have a power integral bases or do not. In this paper, for these results, we give a new easier proof than that given in [1]. We further investigate the cases m 1 (mod 4) independently to t...

Let Fq[e] be a ﬁnite ﬁeld of q elements, where q is a power of a prime number p. In this paper, we study the Twisted Hessian curves over the ring Fq[e], where e2 = e, denoted by Ha,d(Fq[e]); (a,d) ∈ (Fq[e])2. Using the Twisted Hessian equation, we deﬁne the Twisted Hessian curves Ha,d(Fq[e]) and we will show that Hπ0(a),π0(d)(Fq) and Hπ1(a),π1(d)(F...

Let K=Q(α) be a number field generated by a complex root α of a monic irreducible trinomial F(x)=x6+ax3+b∈Z[x]. There are an extensive literature of monogenity of number fields defined by trinomials, Gaál studied the multi-monogenity of sextic number fields defined by x6+ax3+b with a and b two rational integers. Ben Yakkou and El Fadil studied mono...

Let K=Q(α) be a number field generated by a complex root α of a monic irreducible trinomial F(x)=x5+ax2+b∈Z[x]. In this paper, for every prime integer p, we give necessary and sufficient conditions on a and b so that p is a common index divisor of K. In particular, if any one of these conditions holds, then K is not monogenic.

Let $K=\Q(\theta)$ be a number field generated by a complex root $\th$ of a monic irreducible trinomial $F(x) = x^n+ax+b \in \Z[x]$. There is an extensive literature of monogenity of number fields defined by trinomials, Ga\'al studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja studied the integral closed...

Let $K=\Q(\alpha)$ be a number field generated by a complex root $\alpha$ of a monic irreducible trinomial $F(x) = x^5+ax^2+b \in \Z[x]$. In this paper, for every prime integer $p$, we give necessary and sufficient conditions on $a$ and $b$ so that $p$ is a common index divisor of $K$. In particular, when these conditions hold, then $K$ is not mono...

Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $F(x)=x^{60}-m\in \mathbb{Z}[x]$, with $m\neq \pm1$ a square free integer. In this paper, we study the monogeneity of $K$. We prove that if $m\not\equiv 1\md{4}$, $m\not\equiv \mp 1 \md{9} $ and $\overline{m}\not\in\{\mp 1,\mp 7\} \md{25}$, then $K$ is mono...

Let \(K = {\mathbb {Q}} (\alpha )\) be a pure number field generated by a complex root \(\alpha\) of a monic irreducible polynomial \(F(x) = x^{42} -m \in {{\mathbb {Z}}}[x]\), where \(m \ne \pm 1\) is a square-free rational integer. In this paper, we study the monogenity of K. We prove that if \(m\not \equiv 1\ \mathrm{(mod }{4})\), \(m\not \equiv...

Polynomial factorization over a field is very useful in algebraic number theory, in extensions of valuations, etc. For valued field extensions, the determination of irreducible polynomials was the focus of interest of many authors. In 1850, Eisenstein gave one of the most popular criterion to decide on irreducibility of a polynomial over Q. A crite...

In combining the value function approach and tangential subdifferentials, we establish necessary optimality conditions of a nonsmooth multiobjective bilevel programming problem under a suitable constraint qualification. The upper level objectives and constraint functions are neither assumed to be necessarily locally Lipschitz nor convex.

Let K = ℚ(α) be a number field generated by a complex root a of a monic irreducible polynomial ƒ (x) = x ³⁶ − m, with m ≠ ±1 a square free rational integer. In this paper, we prove that if m ≡ 2 or 3 (mod 4) and m ≠ ±1 (mod 9) then the number field K is monogenic. If m ≡ 1 (mod 4) or m ≡±1 (mod 9), then the number field K is not monogenic.

In this note, we show that the result [1, Proposition 5.2] is inaccurate. We further give and prove the correct modification of such a result. Some applications are also given.

Let $K=\Q(\theta)$ be a number field generated by a complex root $\th$ of a monic irreducible trinomial $F(x) = x^n+ax+b \in \Z[x]$. There is an extensive literature of monogenity of number fields defined by trinomials, Ga\'al studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja studied the integral closed...

Let K = Q(θ) be a number field generated by a complex root θ of a monic irreducible trinomial F (x) = x n + ax + b ∈ Z[x]. There is an extensive literature of monogenity of number fields defined by trinomials, Gaál studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja studied the integral closedness of Z[θ]...

Let \( (K,\nu) \) be an arbitrary-rank valued field, let \( R_{\nu} \) be the valuation ring of \( (K,\nu) \),
and let \( K(\alpha)/K \) be a separable finite field extension generated over \( K \) by a root of
a monic irreducible polynomial \( f\in R_{\nu}[X] \). We give some necessary and sufficient
conditions for \( R_{\nu}[\alpha] \) to be inte...

Let [Formula: see text] be a pure number field generated by a complex root [Formula: see text] of a monic irreducible polynomial [Formula: see text] where [Formula: see text] is a square free rational integer, [Formula: see text] is a rational prime integer, and [Formula: see text] is a positive integer. In this paper, we study the monogenity of [F...

Let \(K = \mathbb {Q} (\alpha )\) be a pure number field generated by a complex root \(\alpha\) of a monic irreducible polynomial \(F(x) = x^{20}-m\), with \(m \ne \mp 1\) a square free rational integer. In this paper, we study the monogenity of K. We prove that if \(m\not \equiv 1\ \text{(mod } {4})\) and \(\overline{m}\not \in \{\overline{1}, \ov...

Let $(K,\nu)$ be an arbitrary valued field with valuation ring $R_{\nu}$ and $L=K(\alpha)$, where $\alpha$ is a root of a monic irreducible polynomial $f\in R_{\nu}[x]$. In this paper, we characterize the integral closedness of $R_\nu[\alpha]$ in such a way that extend Dedekind's criterion. Without the assumption of separability of the extension $L...

Let $K = \mathbb{Q} (\alpha) $ be a pure number field generated by a complex root $\alpha$ of a monic irreducible polynomial $ F(x) = x^{2^u\cdot 3^v}-m$, with $m \neq \pm 1$ a square free rational integer, $u$, and $v$ two positive integers. In this paper, we study the monogenity of $K$. The case $u=0$ or $v=0$ has been previously studied. We prov...

Gassert's paper "A NOTE ON THE MONOGENEITY OF POWER MAPS" is cited at least by $17$ papers in the context of monogeneity of pure number fields despite some errors that it contains and remarks on it. In this note, we point out some of these errors, and make some improvements on it.

In all available papers, on power integral bases of any pure sextic number fields K generated by α a complex root of a monic irreducible polynomial f(x)=x6−m∈Z[x], it was assumed that the rational integer m≠∓1 is square free. In this paper, we investigate the monogeneity of any pure sextic number field, where the condition m is a square free ration...

Let K=Q(α) be a pure number field generated by a root α of a monic irreducible polynomial F(x)=x2r·5s−m, with m≠∓1 is a square free integer, r and s are two positive integers. In this article, we study the monogenity of K. We prove that if m≡1(mod 4) and m¯∉{1¯,7¯,18¯,24¯}(mod 25), then K is monogenic. We give also sufficient conditions on r, s, an...

Let $K = \mathbb{Q} (\alpha) $ be a pure number field generated by a complex root $\alpha$ a monic irreducible polynomial $ F(x) = x^{p^r} -m$, with $ m \neq 1 $ is a square free rational integer, $p$ is a rational prime integer, and $r$ is a positive integer. In this paper, we study the monogenity of $K$. We prove that if {{$\nu_p(m^p-m)=1$}}, the...

Let K = ℚ( α ) be a number field generated by a complex root α of a monic irreducible polynomial f ( x ) = x ²⁴ – m , with m ≠ 1 is a square free rational integer. In this paper, we prove that if m ≡ 2 or 3 (mod 4) and m ≢∓1 (mod 9), then the number field K is monogenic. If m ≡ 1 (mod 4) or m ≡ 1 (mod 9), then the number field K is not monogenic.

The goal of this chapter is to study some arithmetic proprieties of an elliptic curve defined by a Weierstrass equation on the local ring Rn=FqX/Xn, where n≥1 is an integer. It consists of, an introduction, four sections, and a conclusion. In the first section, we review some fundamental arithmetic proprieties of finite local rings Rn, which will b...

In this paper, we develop a new method based on Newton polygon and graded polynomials, similar to the known one based on Newton polygon and residual polynomials. This new method allows us the factorization of any monic polynomial in any henselian valued field. As applications, we give a new proof of Hensel’s lemma and a theorem on prime ideal facto...

Jakhar shown that for $f(x)=a_nx^n + a_{n-1}x^{n-1}+\cdot+ a_0$ ($a_0\neq 0$) is a polynomial with rational coefficients, if there exists a prime integer $p$ satisfying $\nu_p(a_n)=0$ and $n\nu_p(a_i)\ge (n-i)\nu_p(a_0)> 0$ for every $0\le i\le n-1$, then $f(x)$ has at most $gcd(\nu_p(a_0),n)$ irreducible factors over the field $\mathbb{Q}$ of rati...

We are interested in local quasi efficient solutions for nonsmooth vector optimization problems under new generalized approximate invexity assumptions. We formulate necessary and sufficient
optimality conditions based on Stampacchia and Minty types of vector variational inequalities involving Clarke’s generalized Jacobians. We also establish the re...

Let $K=\mathbb{Q}(\alpha)$ be a number field generated by a complex root $\alpha$ of a monic irreducible polynomial $f(x)=x^{12}-m$, with $m\neq 1$ is a square free rational integer. In this paper, we prove that if $m \equiv 2$ or $3$ (mod 4) and $m\not\equiv \mp 1$ (mod 9), then the number field $K$ is monogenic. If $m \equiv 1$ (mod 8) or $m\equi...

Interval-valued functions have been widely used to accommodate data inexactness in optimization and decision theory. In this paper, we study interval-valued vector optimization problems, and derive their relationships to interval variational inequality problems, of both Stampacchia and Minty types. Using the concept of interval approximate convexit...

We are interested in local quasi efficient solutions for nonsmooth vector optimization problems under new generalized approximate invexity assumptions. We formulate necessary and sufficient optimality conditions based on Stampacchia and Minty types of vector variational inequalities involving Clarke's generalized Jacobians. We also establish the re...

Let [Formula: see text] be a valued field, where [Formula: see text] is a rank-one discrete valuation, with valuation ring [Formula: see text]. The goal of this paper is to investigate some basic concepts of Newton polygon techniques of a monic polynomial [Formula: see text]; namely, theorem of the product, of the polygon, and of the residual polyn...

Let $(K,\nu)$ be an arbitrary-rank valued field, $R_\nu$ its valuation ring, $K(\alpha)/K$ a separable finite field extension generated over $K$ by a root of a monic irreducible polynomial $f\in R_\nu[X]$. We give necessary and sufficient conditions for $R_\nu[\alpha]$ to be integrally closed. We further characterize the integral closedness of $R_\...

Let R be a principal ideal domain with quotient field K , and L = K ( α ), where α is a root of a monic irreducible polynomial F ( x ) ∈ R [ x ]. Let ℤ L be the integral closure of R in L . In this paper, for every prime p of R , we give a new efficient version of Dedekind’s criterion in R , i.e., necessary and sufficient conditions on F ( x ) to h...

Multi-receiver cryptosystem enables any sender to encrypt a message and broadcast the cipher-text to a group of authorized users, while no one out this group can decrypt the message. Multi-receiver encryption is of great importance in many sectors such as broadcast communication, cloud computing, wireless communications, networking applications, an...

Let E be a Henselian valued field, D be a strongly tame central division algebra over E and K1(D) be the Whitehead group of D. We study in this paper the torsion subgroup of K1(D) by means of graded algebras and we use this to give a new proof to [9, Theorem 10].

In this paper, we give a new and simpler proof for Khanduja’s Theorem which characterizes tame field extensions over Henselian fields. We also give a new characterization of tame division algebras over Henselian fields.

Let L = ℚ(α) be a number field and ℤL its ring of integers, where α is a complex root of a monic irreducible polynomial F(X) ∈ ℤ[X]. In this paper, we give a new efficient version of Dedekind’s criterion, i.e., an efficient criterion to test either p divides or does not divide the index [ℤL: ℤ[α]]. As application, we study the integral closedness o...

Based on Newton polygon techniques, for every prime integer p, a p-integral basis of ℤK, and the factorization of the principal ideal pℤK into prime ideals of ℤK are given, where K is a quintic number field defined by an irreducible trinomial X5 + aX + b ∈ ℤ[X].

Multi-receiver encryption enables a sender to encrypt a message and transmit the ciphertext to a set of authorized users, while no one out this group of authorized users can decrypt the message. Multi-receiver encryption is of great importance in many sectors such as broadcast communication, cloud computing, wireless communications, networking appl...

Let $R$ be a Dedekind ring, $K$ its quotient field, and $L=K(\alpha)$ a finite field extension of $K$ defined by a monic irreducible polynomial $f(x)\in R[x]$. We give an easy version of Dedekind's criterion which computationally improves those versions know in the literature. We further use this result to give a sufficient condition for the integr...

Let (K, ν) be valued field, R its ring of valuation, m its maximal ideal, and L = K(α) a field defined by a root of a monic irre-ducible polynomial F (X) ∈ R[X]. Assume that F (X) factors modulo m into the product of powers of r distinct monic irreducible polynomi-als. We present in this paper a condition, weaker than the known ones, which guarante...

Let $K$ be a number field defined by a monic irreducible polynomial $F(X) \in \mathbb{Z}[X]$, $p$ a fixed rational prime, and $\nu_p$ the discrete valuation associated to $p$. Assume that $\overline{F}(X)$ factors modulo $p$ into the product of powers of $r$ distinct monic irreducible polynomials. We present in this paper a condition, weaker than t...

For every prime integer p, and for every number field K defined by a p-regular polynomial, the form of the factorization of the principal ideal pℤK into prime ideals of ℤK is given. To illustrate the potential applications of this factorization, we derive from this result an explicit description of the factorization of pℤK, where K is a quartic num...

Based on Newton polygon techniques, for every prime integer p, a p-integral basis of ℤK, and the factorization of the principal ideal pℤK into prime ideals of ℤK are given, where K is a quintic number field defined by an irreducible trinomial X5 + aX2 + b ∈ ℤ[X].

Based on Newton polygon techniques, for every prime integer p, a p-integral basis of ZK , and the factorization of the principal ideal pZ K into prime ideals of ZK are given, where K is a quintic number field defined by an irreducible trinomial X 5 + aX 2 + b ∈ Z[X].

This paper demonstrates new methodology to improve security and avoid data overlapping between patients records which are defined as Electronic Patient Records (EPR), a combination of digital watermarking techniques and cryptography are used to ensure the non-separation of EPR and medical images during communications within open networks. The EPR d...

In this paper, we investigate separability of CP-graded ring extensions. With restrictions neither to graded fields nor to grading by torsion–free groups, we show that some results on graded field extensions given in Hwang and Wadsworth [Commun Algebra 27(2):821–840, 1999] hold.

Let p be a prime number. In this paper we use an old technique of Ore, based on Newton polygons, to construct in an efficient way p-integral bases of number fields defined by a p-regular equation. To illustrate the potential applications of this construction, we show how this result yields a computation of a p-integral basis of an arbitrary quartic...

In this paper, we propose a novel method to embed Electronic Patient Records (EPR) data in medical images. Indeed, after liberating a zone by compressing the image Least Significant Bit plan using the Huffman coding, the EPR is encrypted by an Elliptic Curve Cryptosystem (ECC) and inserted into this zone. The proposed method improves medical securi...

In this paper, we propose a novel method to embed Electronic Patient Records (EPR) data in medical images. Indeed, after liberating a zone by compressing the image Least Significant Bit plan using the Huffman coding, the EPR is encrypted by an Elliptic Curve Cryptosystem (ECC) and inserted into this zone. The proposed method improves medical securi...

For every prime integer $p$, an explicit factorization of the principal ideal $p\z_K$ into prime ideals of $\z_K$ is given, where $K$ is a quartic number field defined by an irreducible polynomial $X^4+aX+b\in\z[X]$. Comment: submitted on 5.5.2010

In this paper, based on techniques of Newton polygons, a result which allows the computation of a p integral basis of every quartic number field is given. For each prime integer p, this result allows to compute a p-integral basis of a quartic number field K defined by an irreducible polynomial $P(X) = X4 + aX + b \in\z[X]$ in methodical and complet...

In \cite{Tm}, the authors give two public key encryptions based on third order linear sequences modulo $n^2$, where $n=pq$ is an RSA integer. In their scheme (3), there are two mistakes in the decryption procedure Comment: 3 pages, notes

Based on third order linear sequences, an improvement version of the Diffie-Hellman distribution key scheme and the El Gamal public key cryptosystem scheme are proposed, together with an implementation and computational cost. The security relies on the difficulty of factoring an RSA integer and on the difficulty of computing the discrete logarithm.

We study Galois theory of a graded field extensions. In particular, we take a closer look at the Galois theory studied by S.-Y. Hwang and A.R. Wadsworth [Commun. Algebra 27, No. 2, 821–840 (1999; Zbl 0964.12003)]. On the other hand, the Galois notions studied by M. Boulagouaz [in: Algebra and number theory. Lect. Notes Pure Appl. Math. 208, 21–31 (...

We study separability of graded field extensions. In particular, without restriction to torsion free of the grade group, we show that some separability results of graded field extensions given in [C. Greither, Cyclic Galois extensions of commutative rings, Berlin: Springer-Verlag (1992; Zbl 0788.13003)] hold. On the other hand, the separability not...

The Pohst-Zassenhaus process and the Dedekind criterion give an excellent tool to compute the index of the ℤ-order ℤ[α], then to compute the integral closure of ℤ in ℚ[α]. We extend these results, partially, to an extension of a Dedekind domain.

Let R be a commutative integrally closed domain. For any endomorphism u of a projective R-module M of constant rank, we define its characteristic polynomial and we give a finite sequence of monic polynomials {s i (u,X)} 1≤i≤t such that their product is the characteristic polynomial of u and s i (u,X) divides s i+i (u,X). Like the invariant factors...

In this paper we define the discriminant ideal of a projective algebra with constant rank. We show that such an ideal is an excellent tool to prove some deep results concerning the integral closure of a Dedekind domain in any extension, to characterize separable algebras and then to deduce important results concerning separable algebras.

"Vegeu el resum a l'inici del fitxer adjunt."