Irreducible Polynomial Of Degree Research Articles

Upper bound on the number of inequivalent extended binary irreducible Goppa codes

In this paper, we always assume that p and n>3 are two distinct odd primes, m≥2 is a positive integer, and gcd⁡(pm,2n(22n−1))≠1. Set r=p2 or r=p3, we determine upper bounds on the number of inequivalent extended binary irreducible Goppa codes Γ(L,g) of degree r and length 2n+1, where L=F2n∪{∞} is the support set and each g(x) is an irreducible polynomial of degree r over F2n.

Modeling and simulation of a Tamil language encoder for advanced encryption technologies

This paper presents a Tamil language (TL) encoder that benefits advanced encryption technologies such as the advanced encryption standard (AES). It defines a product set of vowel and consonant sounds of the Tamil language and reveals its connection to Hardy-Ramanujan prime factors and Tamil letters as a one-to-one mapping. It also reveals that the Tamil letters, combined with the digits from 1 to 9, form a Galois field of over an irreducible polynomial of degree 8. Additionally, it implements these two mathematical properties, models the TL encoder, and replaces the pre-round transformation of AES with the model to enhance cryptographic strengths. The cryptographic strengths are measured by the runs test scores of the bit sequences of the ciphers of AES and compared with that of English language. The modeling and simulation conclude that the TL encoder enhances the cryptographic strength of AES at every step of the encryption flow.

On the least common multiple of polynomial sequences at prime arguments

Cilleruelo conjectured that if [Formula: see text] is an irreducible polynomial of degree [Formula: see text] then, [Formula: see text] In this paper, we investigate the analog of prime arguments, namely, [Formula: see text] where [Formula: see text] denotes a prime and obtain nontrivial lower bounds on it. Further, we also show some results regarding the greatest prime divisor of [Formula: see text]

The Effect of the Primitive Irreducible Polynomial on the Quality of Cryptographic Properties of Block Ciphers

Substitution boxes are the only nonlinear component of the symmetric key cryptography and play a key role in the cryptosystem. In block ciphers, the S-boxes create confusion and add valuable strength. The majority of the substitution boxes algorithms focus on bijective Boolean functions and primitive irreducible polynomial that generates the Galois field. For binary field F2, there are exactly 16 primitive irreducible polynomials of degree 8 and it prompts us to construct 16 Galois field extensions of order 256. Conventionally, construction of affine power affine S-box is based on Galois field of order 256, depending on a single degree 8 primitive irreducible polynomial over ℤ2. In this manuscript, we study affine power affine S-boxes for all the 16 distinct degree 8 primitive irreducible polynomials over ℤ2 to propose 16 different 8×8 substitution boxes. To perform this idea, we introduce 16 affine power affine transformations and, for fixed parameters, we obtained 16 distinct S-boxes. Here, we thoroughly study S-boxes with all possible primitive irreducible polynomials and their algebraic properties. All of these boxes are evaluated with the help of nonlinearity test, strict avalanche criterion, bit independent criterion, and linear and differential approximation probability analyses to measure the algebraic and statistical strength of the proposed substitution boxes. Majority logic criterion results indicate that the proposed substitution boxes are well suited for the techniques of secure communication.

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.

S-box Construction Based on Linear Fractional Transformation and Permutation Function

Substitution boxes (S-box) with strong and secure cryptographic properties are widely used for providing the key property of nonlinearity in block ciphers. This is critical to be resistant to a standard attack including linear and differential cryptanalysis. The ability to create a cryptographically strong S-box depends on its construction technique. This work aims to design and develop a cryptographically strong 8 × 8 S-box for block ciphers. In this work, the construction of the S-box is based on the linear fractional transformation and permutation function. Three steps involved in producing the S-box. In step one, an irreducible polynomial of degree eight is chosen, and all roots of the primitive irreducible polynomial are calculated. In step two, algebraic properties of linear fractional transformation are applied in Galois Field GF (28). Finally, the produced matrix is permuted to add randomness to the S-box. The strength of the S-box is measured by calculating its potency to create confusion. To analyze the security properties of the S-box, some well-known and commonly used algebraic attacks are used. The proposed S-box is analyzed by nonlinearity test, algebraic degree, differential uniformity, and strict avalanche criterion which are the avalanche effect test, completeness test, and strong S-box test. S-box analysis is done before and after the application of the permutation function and the analysis result shows that the S-box with permutation function has reached the optimal properties as a secure S-box.

On the number of irreducible polynomials of special kinds in finite fields

Let $\mathbb{F}_q$ be the finite field of order $q$ and $f(x)$ be an irreducible polynomial of degree $n$ over $\mathbb{F} _q$. For a positive divisor $n_1$ of $n$, define the $n_1$-traces of $f(x)$ to be $\mathrm{Tr}(\alpha; n_1) = \alpha+\alpha^q+\cdots+\alpha^{q^{n_1-1}}$ where $\alpha$'s are the roots of $f(x)$. Let $N_q^*(n; n_1)$ denote the number of monic irreducible polynomials of degree $n$ over $\mathbb{F} _q$ with nozero $n_1$-traces. Ruskey, Miers and Sawada have found the formula for $N_q^*(n; n)$. Based on the properties of linearized polynomials, we obtain the formula for $N_q^*(n; n_1)$ in the general case, including a new proof to the result by Ruskey, Miers and Sawada.

Is every irreducible polynomial reducible after a polynomial substitution?

Let K be a non-algebraically closed field and f∈K[x] be an irreducible polynomial of degree d≥3. In this paper we study the existence of a polynomial h∈K[x] of degree ≤d−1 such that the polynomial f(h(x)) is reducible in K[x]. We prove the existence of such polynomial in case of d≤4. A similar (and quite general) result is true in the case when f represents an even function, i.e., f(x)=f(−x). In particular, if char(K)=2, then we prove that the existence of a required polynomial h.

Polynomials of degree 4 defining units

If $x$ is the generator of a cyclic group of order $n$ then every element of the group ring $\\mathbb Z \\langle x \\rangle$ is the result of evaluating $x$ at a polynomial of degree smaller than $n$ with integral coefficients. When such an evaluation result into a unit we say that the polynomial defines a unit on order $n$. Marciniak and Sehgal have classified the polynomials of degree at most 3 defining units. The number of such polynomials is finite. However the number of polynomials of degree 4 defining units on order 5 is infinite and we give the full list of such polynomials. We prove that (up to a sign) every irreducible polynomial of degree 4 defining a unit on an order greater than 5 is of the form $a(X^4+1)+b(X^3+X)+(1-2a-2b)X^2$ and obtain conditions for a polynomial of this form to define a unit. As an application we prove that if $n$ is greater than 5 then the number of polynomials of degree 4 defining units on order $n$ is finite and for $n\\le 10$ we give explicitly all the polynomials of degree 4 defining units on order $n$. We also include a conjecture on what we expect to be the full list of polynomials of degree 4 defining units, which is based on computer aided calculations.

Factoring polynomials of the form [formula omitted

Let f(x)∈Fq[x] be an irreducible polynomial of degree m and exponent e. For each positive integer n, such that νp(q−1)≥νp(e)+νp(n) for all prime divisors p of n, we show a fast algorithm to determine the irreducible factors of f(xn). Using this algorithm, we give the complete factorization of xn−1 into irreducible factors in the case where n=dpt, p is an odd prime, q is a generator of the group Zp2⁎ and either d=2m with m≤ν2(q−1) or d=ra, where r is a prime dividing q−1 but not p−1.

Lower bounds for discriminants of polynomials

Let [Formula: see text] be a monic irreducible polynomial of degree [Formula: see text] having exactly [Formula: see text] real roots and [Formula: see text] complex roots with integer coefficients and [Formula: see text] be its polynomial discriminant. Suppose that [Formula: see text] is square-free. In this paper, we will give a lower bound for polynomial discriminants [Formula: see text] for given [Formula: see text].

An explicit polynomial analogue of Romanoff's theorem

Given a polynomial g of positive degree over a finite field, we show that the proportion of polynomials of degree n, which can be written as h+gk, where h is an irreducible polynomial of degree n and k is a nonnegative integer, has order of magnitude 1/deg⁡g.

Irreducible polynomials with several prescribed coefficients

We study the number of irreducible polynomials over Fq with some coefficients prescribed. Using the technique developed by Bourgain, we show that there is an irreducible polynomial of degree n with r coefficients prescribed in any location when r≤[(1/4−ϵ)n] for any ϵ>0 and q is large; and when r≤δn for some δ>0 and for any q. The result improves earlier work of Pollack stating that a similar result holds for r≤[(1−ϵ)n].

Extension of the field, generates a primitive space matrix Galois

The paper deals with the formation of extended fields, elements of which are Galois matrix representing the spatial non-degenerate matrix synthesized by forming elements – one-dimensional vectors and irreducible polynomials of degree by the method of successive rows of filling. The essence of the method of successive filling option for two-dimensional matrix is reduced to the placement of elements in the lower row of the matrix in which the following lines (bottom to top) fit shifted by one bit to the left vectors lying in the previous line. In the case where a shift length of the vector is greater than the order of the matrix , this vector provides the residue modulo . Introduced Galois conjugate matrix and unambiguously associated right-hand base and conjugate transpose matrix Fibonacci. Discussed the possibility of building advanced fields on the basis of spatial matrices formed by two-dimensional matrix Galois.

Centralizers of Transitive Permutation Groups and Applications to Galois Theory

Let $f(x)$ be an irreducible polynomial of degree $n$ defined over a field $F$, and let $G$ be the Galois group of $f$, identified as a transitive subgroup of $S_n$. Let $K/F$ be the stem field of $f$. We show the automorphism group of $K/F$ is isomorphic to the centralizer of $G$ in $S_n$. We include two applications to computing Galois groups; one in the case $F$ is the rational numbers, the other when $F$ is the 5-adic numbers.

Power‐free values of polynomials

Let be an irreducible polynomial of degree with no fixed th power prime divisor. We derive an asymptotic formula for the number of primes such that is ‐free.

Counting Irreducible Polynomials of Degree r over Fqn and Generating Goppa Codes Using the Lattice of Subfields of Fqnr

The problem of finding the number of irreducible monic polynomials of degree r over Fqn is considered in this paper. By considering the fact that an irreducible polynomial of degree r over Fqn has a root in a subfield Fqs of Fqnr if and only if (nr/s,r)=1, we show that Gauss’s formula for the number of monic irreducible polynomials can be derived by merely considering the lattice of subfields of Fqnr . We also use the lattice of subfields of Fqnr to determine if it is possible to generate a Goppa code using an element lying in a proper subfield of Fqnr.

Groups Acting on Necklaces and Sandpile Groups

We introduce a group naturally acting on aperiodic necklaces of length n with two colors using a one-to-one correspondence between such necklaces and irreducible polynomials of degree n over the field F2 of two elements. We notice that this group is isomorphic to the quotient group of nondegenerate circulant matrices of size n over that field modulo a natural cyclic subgroup. Our groups turn out to be isomorphic to the sandpile groups for a special sequence of directed graphs. Bibliography: 15 titles.

Construction of infinite sequences of irreducible polynomials using Kloosterman sum

We construct infinite sequences of irreducible polynomials over prime field having characteristic 3 using R-transform of the irreducible polynomial. The sequences start from a monic irreducible polynomial of degree n. The initial polynomial is free from any restrictions on its coefficients.

Distribution of values of polynomial Fermat quotients

Let P be an irreducible polynomial of degree n over Fq. For A∈Fq[X] with gcd(A,P)=1 the polynomial Fermat quotient qP(A) is defined byqP(A)≡Aqn−1−1P(modP)anddegqP(A)<n. We study several properties of polynomial Fermat quotients including the number of fixed points, the image size, and multiplicity of values in the image.