Irreducible Polynomials Research Articles

High-Speed Variable Polynomial Toeplitz Hash Algorithm Based on FPGA.

In the Quantum Key Distribution (QKD) network, authentication protocols play a critical role in safeguarding data interactions among users. To keep pace with the rapid advancement of QKD technology, authentication protocols must be capable of processing data at faster speeds. The Secure Hash Algorithm (SHA), which functions as a cryptographic hash function, is a key technology in digital authentication. Irreducible polynomials can serve as characteristic functions of the Linear Feedback Shift Register (LFSR) to rapidly generate pseudo-random sequences, which in turn form the foundation of the hash algorithm. Currently, the most prevalent approach to hardware implementation involves performing block computations and pipeline data processing of the Toeplitz matrix in the Field-Programmable Gate Array (FPGA) to reach a maximum computing rate of 1 Gbps. However, this approach employs a fixed irreducible polynomial as the characteristic polynomial of the LFSR, which results in computational inefficiency as the highest bit of the polynomial restricts the width of parallel processing. Moreover, an attacker could deduce the irreducible polynomials utilized by an algorithm based on the output results, creating a serious concealed security risk. This paper proposes a method to use FPGA to implement variational irreducible polynomials based on a hashing algorithm. Our method achieves an operational rate of 6.8 Gbps by computing equivalent polynomials and updating the Toeplitz matrix with pipeline operations in real-time, which accelerates the authentication protocol while also significantly enhancing its security. Moreover, the optimization of this algorithm can be extended to quantum randomness extraction, leading to a considerable increase in the generation rate of random numbers.

Efficient Decoding Algorithm for Binary Quadratic Residue Codes Using Reduced Permutation Sets

The Quadratic Residue (QR) codes have a rich mathematic structure. Unfortunately, their Algebraic Decoding (AD) is not generalizable for all QR codes. In this study, an efficient hard decoding algorithm is proposed to generalize the decoding of the binary systematic Quadratic Residue (QR) codes. The proposed decoder corrects t erroneous bits or less, in the received word, based on a reduced set of permutations derived from the large automorphism group of QR codes. This set of permutations is applied to the received word to move the error positions and trap all of them in redundancy. Then, to evaluate the proposed method, we applied it to many binary QR codes of moderate code length starting with 17 until 113 with reducible and irreducible generator polynomials. The proposed decoder was validated by inserting all possible error patterns, that have t or less erroneous positions, as input of the proposed decoder and the output is always a correct codeword. The complexity study, in terms of the number of operations used, reveals that the light permutation decoding LPD algorithm significantly decreases decoding complexity without performance loss. So, it is qualified to be a good competitor to decode QR codes with lower lengths but is the best for QR codes with higher lengths.

A Novel Cipher-Based Data Encryption with Galois Field Theory

Both the act of keeping information secret and the research on how to achieve it are included in the broad category of cryptography. When people refer to "information security," they are referring to the study and use of methods that make data transfers harder to intercept. When we talk about "information security," this is what we have in mind. Using private keys to encrypt and decode messages is a part of this procedure. Because of its vital role in modern information theory, computer security, and engineering, cryptography is now considered to be a branch of both mathematics and computer science. Because of its mathematical properties, the Galois field may be used to encrypt and decode information, making it relevant to the subject of cryptography. The ability to encrypt and decode information is one such use. In this case, the data may be encoded as a Galois vector, and the scrambling process could include the application of mathematical operations that involve an inverse. While this method is unsafe when used on its own, it forms the foundation for secure symmetric algorithms like AES and DES when combined with other bit shuffling methods. A two-by-two encryption matrix is used to protect the two data streams, each of which contains 25 bits of binary information which is included in the proposed work. Each cell in the matrix represents an irreducible polynomial of degree 6. Fine-tuning the values of the bits that make up each of the two 25-bit binary data streams using the Discrete Cosine Transform (DCT) with the Advanced Encryption Standard (AES) Method yields two polynomials of degree 6. Optimization is carried out using the Black Widow Optimization technique is used to tune the key generation in the cryptographic processing. By doing so, we can produce two polynomials of the same degree, which was our original aim. Users may also use cryptography to look for signs of tampering, such as whether a hacker obtained unauthorized access to a patient's medical records and made any changes to them. Cryptography also allows people to look for signs of tampering with data. Indeed, this is another use of cryptography. It also has the added value of allowing users to check for indications of data manipulation. Users may also positively identify faraway people and objects, which is especially useful for verifying a document's authenticity since it lessens the possibility that it was fabricated. The proposed work achieves higher accuracy of 97.24%, higher throughput of 93.47%, and a minimum decryption time of 0.0047 s.

The Darboux Polynomials and Integrability of Polynomial Levinson–Smith Differential Equations

We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations. These equations generalize Liénard equations and are used to describe self-sustained oscillations. Our results are valid for arbitrary degrees of the polynomials arising in the equations. We find a number of novel Liouvillian integrable subfamilies. We derive an upper bound with respect to one of the variables on the degrees of irreducible Darboux polynomials in the case of nondegenerate or algebraically degenerate near infinity polynomial Levinson–Smith equations. We perform the complete classification of Liouvillian first integrals for the nondegenerate or algebraically degenerate near infinity Rayleigh–Duffing–van der Pol equation that is a cubic Levinson–Smith equation.

Polycyclic codes over [formula omitted]: Classification, Hamming distance, and annihilators

Let p be prime integer, s and m positive integers; f(x) be a monic irreducible polynomial over R where R:=Fpm[u]/〈u2〉. This paper gives a complete classification of polycyclic codes over R associated to f(x)ps, and the other hand, when f(x)∈Fpm[x], their Hamming distances and their annihilators are explicitly given.

Three proofs of an observation on irreducible polynomials over GF(2)

We present three proofs of an observation of Ahmadi on the number of irreducible polynomials over GF(2) with certain traces and cotraces, the most interesting of which uses an explicit natural bijection. We also present two proofs of a related observation.

GALOIS GROUPS OF RECIPROCAL SEXTIC POLYNOMIALS

AbstractLet F be a subfield of the complex numbers and $f(x)=x^6+ax^5+bx^4+cx^3+bx^2+ax+1 \in F[x]$ an irreducible polynomial. We give an elementary characterisation of the Galois group of $f(x)$ as a transitive subgroup of $S_6$ . The method involves determining whether three expressions involving a, b and c are perfect squares in F and whether a related quartic polynomial has a linear factor. As an application, we produce one-parameter families of reciprocal sextic polynomials with a specified Galois group.

Creation of Database of Irreducible Polynomial Over A Finite Field GF(2) Based on FPGA

Creation of Database of Irreducible Polynomial Over A Finite Field GF(2) Based on FPGA

Incorporating a New Pseudorandom Number Generator in AES Algorithm to Improve Its Security Level

The implementation of the multiplicative inverses of elemental polynomials under an irreducible polynomial over GF(pn) played an important role in cryptography. In the AES algorithm, the multiplicative inverses under the first irreducible polynomial over GF(28) have been used for the first time in 1999 to design its substitution box. The new PRNG RC4-MI in which the multiplicative inverses under the two irreducible polynomials over GF(73) are being used accepts the AES key as its own key and generates random bytes. Undertaking exclusive OR operation of RC4-MI bytes with AES cipher bytes, one gets a new sequence of cipher bytes thatexhibits randomness quantitatively better than that of the original sequence of AES cipher bytes.

Diminution of Extended Euclidean Algorithm for Finding Multiplicative Inverse in Galois Field

This manuscript deals with the theorem on diminution of the Extended Euclidean Algorithm for finding the multiplicative inverse of non-zero elemental polynomials of Galois field with respect to a monic irreducible polynomial over , where is a prime and is any positive integer. This method became successful in finding the multiplicative inverse of all those non-zero polynomials for which the Extended Euclidean Algorithm fails. We find the inverse of all 342 non-zero elemental polynomials of using an irreducible polynomial We also used Cayley Hamilton’s theorem for finding the multiplicative inverse of the non-zero elemental polynomials of a finite field.

Computation of Multiplicative Inverses of non-zero Polynomials of GF(73) by Gauss-Jordan method.

The present manuscript deals with the derivation of multiplicative inverses of all non-zero elemental polynomials of the Galois field with respect to the monic irreducible polynomials of degree 3 over the prime field by Gauss-Jordan method as the Extended Euclidean Algorithm (EEA) is not always successful over the prime field , when prime .

Two classes of quasi-cyclic codes via irreducible polynomials

Two classes of quasi-cyclic codes via irreducible polynomials

On diamond products ensuring irreducibility of the associated composed product

Let f and g be two irreducible polynomials of coprime degrees m and n whose zeroes lie in a set . Let be a diamond product on G. We define the weaker cancelation property of and show that it is sufficient to conclude that the composed product of f and g derived from is an irreducible polynomial of degree mn. We also prove that a wide class of diamond products on finite fields satisfy the weaker cancelation property. These results extend the corresponding results of Brawley and Carlitz (1987), and Munemasa and Nakamura (2016).

On fixed divisors of the values of the minimal polynomials over Z of algebraic numbers

Let $K$ be a number field of degree $n$, $A$ be its ring of integers, and $A_n$ (resp. $K_n$) be the set of elements of $A$ ( resp. $K$) which are primitive over $\mathbb Q$. For any $\gamma \in {K_n}$, let $F_{\gamma} (x)$ be the unique irreducible polynomial in $\mathbb Z[x]$, such that its leading coefficient is positive and $F_{\gamma} ({\gamma}) = 0$. Let $i(\gamma)=\gcd_{x\in\mathbb Z}F_{\gamma}(x)$, $i(K)=\lcm_{\theta\in{A_n}}i(\theta)$ and $\hat{\imath}(K) = \lcm_{\gamma\in{K_n}}i(\gamma)$. For any $\gamma \in {K_n}$, there exists a unique pair $(\theta,d)$, where $\theta\in A_n$ and $d$ is a positive integer such that $\gamma=\theta/d$ and $\theta\not\equiv 0\pmod{p}$ for any prime divisor $p$ of $d$. In this paper, we study the possible values of $\nu_{p}(d)$ when $p | i(\gamma)$. We introduce and study a new invariant of $K$ defined using $\nu_{p}(d)$, when $\gamma$ describes $K_n$. In the last theorem of this paper, we establish a generalisation of a theorem of MacCluer.

Integral Bases and Monogenity of Pure Number Fields with Non-Square Free Parameters up to Degree 9

AbstractLetKbe a pure number field generated by a rootαof a monic irreducible polynomialf(x)=xn−mwithma rational integer and 3≤n≤ 9 an integer. In this paper, we calculate an integral basis of ℤK, and we study the monogenity ofK, extending former results to the case whenmis not necessarily square-free. Collecting and completing the corresponding results in this more general case, our purpose is to provide a parallel to [Gaál, I.—Remete, L.:Power integral bases and monogenity of pure fields,J.Number Theory,173(2017), 129–146], where only square-free values ofmwere considered.

On The Geometric Determination of Extensions of Non-Archimedean Absolute Values

Abstract 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] whose reduction modulo 𝓜 is irreducible, the φ-Newton polygon N φ ¯ ( F ) $N\overline \phi \left( F \right)$ has a single side of negative slope λ, and the residual polynomial R λ(F )(y) has no multiple factors in 𝔽 φ [y]. In this paper, we describe all absolute values of L extending | |. The problem is classical but our approach uses new ideas. Some useful remarks and computational examples are given to highlight some improvements due to our results.

ON NONMONOGENIC ALGEBRAIC NUMBER FIELDS

Let q be a prime number and f(x)=xqs−axm−b be a monic irreducible polynomial of degree qs having integer coefficients. Let K=ℚ(𝜃) be an algebraic number field with 𝜃 a root of f(x). We give some explicit conditions involving only a,b,m,q,s for which K is not monogenic. As an application, we show that if p is a prime number of the form 32k+1, k∈ℤ and 𝜃 is a root of a monic polynomial f(x)=x2s−32cpx2r−p∈ℤ[x] with s>4,2∤c,s≠5+r, then ℚ(𝜃) is not monogenic.

Equivalence and reduction of bivariate polynomial matrices to their Smith forms

This paper is concerned with Smith forms of bivariate polynomial matrices. For a bivariate polynomial square matrix with the determinant being the product of two distinct and irreducible univariate polynomials, we prove that it is equivalent to its Smith form. We design an algorithm to reduce this class of bivariate polynomial matrices to their Smith forms, and an example is given to illustrate the algorithm. Furthermore, we extend the above class of matrices to a more general case, and derive a necessary and sufficient condition for the equivalence of a matrix and one of its all possible existing Smith forms.

Non-monogenity of some number fields generated by binomials or trinomials of prime-power degree

Let [Formula: see text] be a prime number and [Formula: see text] be an algebraic number field with [Formula: see text] a root of an irreducible polynomial [Formula: see text] having integer coefficients. In this paper, we provide some explicit conditions involving only [Formula: see text] for which [Formula: see text] is non-monogenic. As an application, in the special case of [Formula: see text] and [Formula: see text], we show that if [Formula: see text] and [Formula: see text] divides [Formula: see text], then [Formula: see text] is not monogenic. We illustrate our results through examples.

Towards van der Waerden’s conjecture

How often is a quintic polynomial solvable by radicals? We establish that the number of such polynomials, monic and irreducible with integer coefficients in [ − H , H ] [-H,H] , is O ( H 3.91 ) O(H^{3.91}) . More generally, we show that if n ⩾ 3 n \geqslant 3 and n ∉ { 7 , 8 , 10 } n \notin \{ 7, 8, 10 \} then there are O ( H n − 1.017 ) O(H^{n-1.017}) monic, irreducible polynomials of degree n n with integer coefficients in [ − H , H ] [-H,H] and Galois group not containing A n A_n . Save for the alternating group and degrees 7 , 8 , 10 7,8,10 , this establishes a 1936 conjecture of van der Waerden.