Irreducible Polynomials Research Articles

Linear fractional transformation based S-boxes design by Galois fields over irreducible polynomials

Linear fractional transformation based S-boxes design by Galois fields over irreducible polynomials

A Family of Fractal-Based Irreducible Polynomials

Summary We provide some fractal-based irreducibility criteria for integer polynomials. The results are obtained using the self-similar pattern of Pascal’s triangle modulo two and by applying a restatement of the Schönemann–Eisenstein’s irreducibility criterion.

NONMONOGENITY OF NUMBER FIELDS DEFINED BY TRUNCATED EXPONENTIAL POLYNOMIALS

Abstract Let p be a prime number. Let $n\geq 2$ be an integer given by $n = p^{m_1} + p^{m_2} + \cdots + p^{m_r}$ , where $0\leq m_1 < m_2 < \cdots < m_r$ are integers. Let $a_0, a_1, \ldots , a_{n-1}$ be integers not divisible by p. Let $K = \mathbb Q(\theta )$ be an algebraic number field with $\theta \in {\mathbb C}$ a root of an irreducible polynomial $f(x) = \sum _{i=0}^{n-1}a_i{x^i}/{i!} + {x^n}/{n!}$ over the field $\mathbb Q$ of rationals. We prove that p divides the common index divisor of K if and only if $r>p$ . In particular, if $r>p$ , then K is always nonmonogenic. As an application, we show that if $n \geq 3$ is an odd integer such that $n-1\neq 2^s$ for $s\in {\mathbb Z}$ and K is a number field generated by a root of a truncated exponential Taylor polynomial of degree n, then K is always nonmonogenic.

Synchronization of M-sequences based on fast Нadamard transform

Options have been developed for constructing circulant matrices of any M-sequence (MS) based on automorphic multiplicative groups of the extended Galois field, constructed using an irreducible primitive polynomial, on the basis of which the original MS is formed. The result of this approach is the identification of new methods for transforming MS circulant matrices to a matrix of Walsh functions, ordered by the powers of the antiderivative element of the field. It is shown for the first time that, depending on the initial conditions of the transformation, a set of any number of any cyclic shifts of the MP, shifted relative to each other by one symbol, can be transformed to any rows of the ordered matrix of Walsh functions, following one another. This circumstance makes it possible to simplify the MS synchronization algorithm for a known range of its cyclic shifts, especially in the case of large periods of its repetition, and also to reduce the computational complexity of the processing algorithm when working in a truncated basis of Walsh–Hadamard functions.

An elementary problem in Galois theory about the roots of irreducible polynomials

An elementary problem in Galois theory about the roots of irreducible polynomials

On the construction of certain odd degree irreducible polynomials over finite fields

On the construction of certain odd degree irreducible polynomials over finite fields

Rational solutions of the matrix equation $p(X) = A$

We extend Theorem 1 of R. Reams, A Galois approach to m-th roots of matrices with rational entries, LAA, 258:187-194, 1997. Let $p(\lambda)$ be any polynomial over $\mathbb{Q}$, and let $A\in M_n(\mathbb{Q})$ have irreducible characteristic polynomial $f(\lambda)$ with degree $n$. We provide necessary and sufficient conditions for the existence of a solution $X\in M_n(\mathbb{Q})$ of the polynomial matrix equation $p(X) = A.$ Specifically, we find necessary and sufficient conditions for $f(p(\lambda))$ to have a factor of degree $n$ over $\mathbb{Q}.$

Separable Polynomials and Separable Extensions

Summary We continue the formalization of field theory in Mizar [2], [3], [4]. We introduce separability of polynomials and field extensions: a polynomial is separable, if it has no multiple roots in its splitting field; an algebraic extension E of F is separable, if the minimal polynomial of each a ∈ E is separable. We prove among others that a polynomial q(X) is separable if and only if the gcd of q(X) and its (formal) derivation equals 1 – and that a irreducible polynomial q(X) is separable if and only if its derivation is not 0 – and that q(X) is separable if and only if the number of q(X)’s roots in some field extension equals the degree of q(X). A field F is called perfect if all irreducible polynomials over F are separable, and as a consequence every algebraic extension of F is separable. Every field with characteristic 0 is perfect [13]. To also consider separability in fields with prime characteristic p we define the rings Rp = { ap | a ∈ R} and the polynomials Xn − a for a ∈ R. Then we show that a field F with prime characteristic p is separable if and only if F = F p and that finite fields are perfect. Finally we prove that for fields F ⊆ K ⊆ E where E is a separable extension of F both E is separable over K and K is separable over F .

On 2-superirreducible polynomials over finite fields

We investigate k-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most k. Let F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in F[t] when p=2 and that no such polynomials of odd degree exist when p is odd. We address the remaining case in which p is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree d. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.

COBLAH: A chaotic OBL initialized hybrid algebraic-heuristic algorithm for optimal S-box construction

The Substitution box (S-box) is the main nonlinear component responsible for the cryptographic strength of any Substitution-Permutation Network (SPN) based block cipher. Generating the S-box with optimal cryptographic properties is one of cryptography's most challenging combinatorial problems because of its enormous search space, lack of guidance, and conflicting performance criteria. This paper introduces a novel Chaotic Opposition-based Learning Initialized Hybrid Algebraic-Heuristic (COBLAH) algorithm, combining the favorable traits of Algebraic and heuristics methods based on Galois field inversion, affine mapping, and Genetic Algorithm (GA). The Galois field inversion and affine mapping are used to construct the S-box, while the GA guides the algebraic construction to find the best bit-matrix and additive vector based on any irreducible polynomial for GF(28). GA initializes with a random population generated using a newly constructed cosine-cubic map incorporated with binarization and Opposition-based Learning (OBL). Further, Multi-Objective Optimization Ratio Analysis (MOORA) is utilized to identify the best S-box from the final optimized population. The performance of the proposed algorithm is evaluated by comparing the generated COBLAH S-box with more than twenty state-of-the-art S-boxes, including Advanced Encryption Standard (AES), Skipjack, Gray, and Affine Power Affine (APA). The COBLAH S-box has nonlinearity 112, Strict Avalanche Criterion (SAC) offset 0.0202, Distance to SAC (DSAC) 332, Differential Approximation Probability (DP) 0.0625, Linear Approximation Probability (LP) 0.0156, Bit Independence Criterion-Strict Avalanche Criterion (BIC-SAC) 0.50006, and Bit Independence Criterion-Nonlinearity (BIC-NL) 112, which stands as the optimal observed thus far. The absence of fixed and opposite fixed points and the fact that it adheres to a single cycle aligns the COBLAH S-box with an ideal S-box. In addition, an image encryption mechanism is utilized to encrypt and decrypt the different images sourced from the standard USC-SIPI image dataset using COBLAH S-box and compared against different state-of-the-art S-boxes based on various image characteristics.

Monogenity of iterates of irreducible binomials

One of the oldest problems of algebraic number theory is to find a method to determine if the ring of integers of a number field K is Z [ θ ] for some θ ∈ K ; a field for which the answer to this question is affirmative is referred to as a monogenic field. Suppose f ( x ) = x m − a ∈ Z [ x ] is a monic irreducible polynomial and α n ∈ C is a root of f n ( x ) , the n-fold composition of f. In this article, we prove a necessary and sufficient condition for K n = Q ( α n ) to be monogenic for every n ∈ N and m ≥ 2.

On the ideal discriminant of some relative pure extensions

Let L = K(α) be an extension of a number field K where α satisfies the monic irreducible polynomial P(X) = Xp −a ∈ R[X] of prime degree p and such that a is pth power free in R := OK (the ring of integers of K). The purpose of this paper is to give an explicit formula for the ideal discriminant DL/K of L over K involving only the prime ideals dividing the principal ideals aR and pR. As an illustration, we compute the discriminant DL/K of a family of septic and quintic pure fields over quadratic fields. Hence a slightly simpler computation of discriminant DL/K is obtained.

The On power integral bases of certain pure number fields defined by $x^{120}-m $

Let $K$ be a pure number field with $\alpha$ a complex root of a monic irreducible polynomial $F(x) = x^{120}-m \in \mathbb{Z}[x]$ with $ m \neq \pm 1 $. In this paper, we study the monogenity of $K$. More precisely, we prove that if $m$ is square-free, $m \not \equiv 1\md{4}$, $m \not \equiv \pm 1 \md{9} $, and $\ol{m}\not \in \{ \mp 1, 7, 18 \} \md {25}$, then $K$ is monogenic. On the other hand, if $m \equiv 1\md{4}$, $m \equiv 1 \md{9} $, or $m \equiv 1 \md{25}$, then $K$ is not monogenic. Our results are illustrated by some computational examples.

Kubiliaus nelygybės analogas polinomų pusgrupėje

Let P be a set of primary irreducible polynomials and Qm = {p + 1; p ∈ P , ∂(p) = m}. Kubilius inequality for additive functions f : Qm → C is proved.

On common index divisors and monogenity of certain number fields defined by x^{5}+ax+b

Let K = Q(α) be a number field, where α satisfies the monic irreducible polynomial F (x) = x5 + ax + b belonging to Z[x]. The purpose of this paper is to caracterise when a prime p is a common index divisor of K. More precisely, we give explicitly a necessary and sufficient conditions, on a and b for which K is not monogenic. Some useful examples are also given.

An Enhanced AES-ECC model with Key Dependent Dynamic S-Box for the Security of Mobile Applications using Cloud Computing

The spectacular growth of computational clouds has drawn attention and enabled intensive computing on client devices with constrained resources. Smart phones mostly use the data centre demand service model to provide data to computationally intensive applications. It is challenging to outsource sensitive and private information to distant data centres due to growing concerns about data privacy and security. Therefore, conventional security algorithms have to be upgraded to address the brand-new security concerns that have emerged in the cloud environment. The Advanced Encryption Standard (AES) algorithm encrypts and decrypts messages using the four steps, namely Sub-Bytes, Shift-Rows, Mix-Columns, and Add Round Key. It has been changed so that the Sub-byte transformation depends on the round key. Making the S box round key dependent is intended to make it such that even a little change in the key will have a big impact on the cipher text. The avalanche effect and execution time of the standard and modified AES algorithms are implemented and assessed. In order to suggest a safe and efficient solution, the study also aims to modify the shift rows and the mix column phases of AES. Using cryptographic techniques such as elliptic curve cryptography (ECC), several protocols and algorithms have been developed to guarantee the security and integrity of the data. The proposed solution combines the enhanced and updated The Advanced Encryption Standard (AES) technology with the ECC to guarantee data secrecy. The avalanche effect approach is used in this study to explore and analyze the strength and quality of the new s-box. The results demonstrated that each encryption operation generates a unique cipher text. A strong avalanche effect was also achieved with the redesigned dynamic s-box using irreducible polynomials, surpassing the strict avalanche criterion (SAC). The results show that the proposed method ECC-EAESKDS Elliptic Curve Cryptography-Enhanced AES with Key Dependent S box is efficient and yields better results than the alternatives. According to the suggested security architecture, cloud users may control data protection and integrity in a secure manner.

CHARACTERISATION OF PRIMES DIVIDING THE INDEX OF A CLASS OF POLYNOMIALS AND ITS APPLICATIONS

Abstract Let ${\mathbb {Z}}_{K}$ denote the ring of algebraic integers of an algebraic number field $K = {\mathbb Q}(\theta )$ , where $\theta $ is a root of a monic irreducible polynomial $f(x) = x^n + a(bx+c)^m \in {\mathbb {Z}}[x]$ , $1\leq m<n$ . We say $f(x)$ is monogenic if $\{1, \theta , \ldots , \theta ^{n-1}\}$ is a basis for ${\mathbb {Z}}_K$ . We give necessary and sufficient conditions involving only $a, b, c, m, n$ for $f(x)$ to be monogenic. Moreover, we characterise all the primes dividing the index of the subgroup ${\mathbb {Z}}[\theta ]$ in ${\mathbb {Z}}_K$ . As an application, we also provide a class of monogenic polynomials having non square-free discriminant and Galois group $S_n$ , the symmetric group on n letters.

Distribution of constant terms of irreducible polynomials in ℤ_p[x] whose degree is a product of two distinct odd primes

We obtain explicit formulas for the number of monic irreducible polynomials with prescribed constant term and degree $q_1q_2$ over a finite field, where $q_1$ and $q_2$ are distinct odd~primes. These formulas are derived from work done by Yucas. We show that the number of polynomials of a given constant term depends only on whether the constant term is a $q_1$-residue and/or a $q_2$-residue in the underlying field. We further show that as $k$ becomes large, the proportion of irreducible polynomials having each constant term is asymptotically equal. This paper continues work done in [1].

The product of a quartic and a sextic number cannot be octic

Abstract In this article, we prove that the product of two algebraic numbers of degrees 4 and 6 over Q {\mathbb{Q}} cannot be of degree 8. This completes the classification of so-called product-feasible triplets ( a , b , c ) ∈ N 3 \left(a,b,c)\in {{\mathbb{N}}}^{3} with a ≤ b ≤ c a\le b\le c and b ≤ 7 b\le 7 . The triplet ( a , b , c ) \left(a,b,c) is called product-feasible if there are algebraic numbers α , β \alpha ,\beta , and γ \gamma of degrees a , b a,b , and c c over Q {\mathbb{Q}} , respectively, such that α β = γ \alpha \beta =\gamma . In the proof, we use a proposition that describes all monic quartic irreducible polynomials in Q [ x ] {\mathbb{Q}}\left[x] with four roots of equal moduli and is of independent interest. We also prove a more general statement, which asserts that for any integers n ≥ 2 n\ge 2 and k ≥ 1 k\ge 1 , the triplet ( a , b , c ) = ( n , ( n − 1 ) k , n k ) \left(a,b,c)=\left(n,\left(n-1)k,nk) is product-feasible if and only if n n is a prime number. The choice ( n , k ) = ( 4 , 2 ) \left(n,k)=\left(4,2) recovers the case ( a , b , c ) = ( 4 , 6 , 8 ) \left(a,b,c)=\left(4,6,8) as well.

Exceptional biases in counting primes over function fields

AbstractWe study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call “complete bias,” “lower order bias,” and “reversed bias,” occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in as , a power of a fixed prime, goes to infinity. The bounds given extend a previous result of Kowalski, who studied a similar question along particular one‐parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret–Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry.