Polynomial irreducibility via shifts

Originally irreducible polynomials were obtained essentially by checking the irreducibility of a randomly chosen polynomial. Recently, irreducible polynomials of a specified form have been produced and a transformation of the variable has been applied to produce a systematic method of deriving irreducible polynomials. In this paper, we consider irreducible polynomials from the viewpoint of cyclotomic polynomials. We define conditions for the existence of irreducible cyclotomic polynomials and show that these polynomials have some specified forms. Additionally, it is shown that an infinite number of irreducible polynomials can be generated which satisfy those conditions.

Let $X^3+AX+B$ be an irreducible abelian cubic polynomial in $Z[X]$.\nWe determine explicitly integers $a_1,\\ldots,a_t$, $F$ such that, except for finitely many primes $p$,\n\\[ x^3+Ax+B\\equiv 0\\pmod{p} \\text{ has three solutions} \\Leftrightarrow p\\equiv a_1,\\ldots,a_t\\pmod{F}. \\]

Let be an arbitrary subset of a unique factorization domain R and be the field of fractions of R. The ring of integer-valued polynomials over S is the set This article is an effort to study the irreducibility of integer-valued polynomials over arbitrary subsets of a unique factorization domain. We give a method to construct special kinds of sequences, which we call d-sequences. We then use these sequences to obtain a criteria for the irreducibility of the polynomials in In some special cases, we explicitly construct these sequences and use these sequences to check the irreducibility of some polynomials in At the end, we suggest a generalization of our results to an arbitrary subset of a Dedekind domain.

On the construction of irreducible and primitive polynomials from [formula omitted] to [formula omitted

In this paper, we present a specific type of irreducible polynomial which is an irreducible m-term polynomial of degree m. Designing the parallel multiplier over GF(2m) by the quadrinomial obtained from this irreducible polynomial, its critical delay path is smaller than that of conventional multipliers for some degree m.

The Massey-Omura multiplier of GF(2/sup m/) uses a normal basis and its bit parallel version is usually implemented using m identical combinational logic blocks whose inputs are cyclically shifted from one another. In the past, it was shown that, for a class of finite fields defined by irreducible all-one polynomials, the parallel Massey-Omura multiplier had redundancy and a modified architecture of lower circuit complexity was proposed. In this article, it is shown that, not only does this type of multiplier contain redundancy in that special class of finite fields, but it also has redundancy in fields GF(2/sup m/) defined by any irreducible polynomial. By removing the redundancy, we propose a new architecture for the normal basis parallel multiplier, which is applicable to any arbitrary finite field and has significantly lower circuit complexity compared to the original Massey-Omura normal basis parallel multiplier. The proposed multiplier structure is also modular and, hence, suitable for VLSI realization. When applied to fields defined by the irreducible all-one polynomials, the multiplier's circuit complexity matches the best result available in the open literature.

1. Introduction and Loewy’s theorem. By a classical theorem the number of real roots of an irreducible polynomial f(X) of odd prime degree p over a real number eld K is either 1 or p if the Galois group of f(X) over K is solvable. This result was generalized by A. Loewy in the following way: For a polynomial f(X) we let r(f) denote the number of real roots of f(X). Loewy’s theorem. Let K be a real number eld and f(X) an irreducible polynomial in K[X] of odd degree n. If p is the smallest prime divisor of n and the Galois group of f(X) over K is solvable, then r(f) = 1 or n or satises the inequalities p r(f) n p + 1. When the degree of f(X) is a prime number the above theorem is an immediate corollary to the following Galois’ theorem. Let f(X) be an irreducible separable polynomial over a eld K having a solvable Galois group over K. If the degree of f(X) is a prime number, then any two roots of f(X) generate the splitting eld of f(X) over K. Galois’ theorem, which is basically a group-theoretic result, cannot be generalized to yield a proof of Loewy’s theorem. Indeed, for any odd prime number p and any t, 1 t p, there exists an irreducible polynomial f(X) in Q[X] of degree p 2 with solvable Galois group having t roots 1;:::; t

Recurrent Methods for Constructing Irreducible Polynomials over GF(2s)

Irreducible multivariate polynomials obtained from polynomials in fewer variables

In most of the methods of public key cryptography devised in recent years, a finite field of a large order is used as the field of definition. In contrast, there are many studies in which a higher‐degree extension field of characteristic 2 is fast implemented for easier hardware realization. There are also many reports of the generation of the required higher‐degree irreducible polynomial, and of the construction of a basis suited to fast implementation, such as an optimal normal basis (ONB). For generating higher‐degree irreducible polynomials, there is a method in which a 2m‐th degree self‐reciprocal irreducible polynomial is generated from an m‐th degree irreducible polynomial by a simple polynomial transformation (called the self‐reciprocal transformation). This paper considers this transformation and shows that when the set of zeros of the m‐th degree irreducible polynomial forms a normal basis, the set of zeros of the generated 2m‐th order self‐reciprocal irreducible polynomial also forms a normal base. Then it is clearly shown that there is a one‐to‐one correspondence between the transformed irreducible polynomial and the generated self‐reciprocal irreducible polynomial. Consequently, the inverse transformation of the self‐reciprocal transformation (self‐reciprocal inverse transformation) can be applied to a self‐reciprocal irreducible polynomial. It is shown that an m‐th degree irreducible polynomial can always be generated from a 2m‐th degree self‐reciprocal irreducible polynomial by the self‐reciprocal inverse transformation. We can use this fact for generating 1/2‐degree irreducible polynomials. As an application of 1/2‐degree irreducible polynomial generation, this paper proposes a method which generates a prime degree irreducible polynomial with a Type II ONB as its zeros. © 2005 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 88(7): 23–32, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.20151

An irreducible polynomial can be derived in a stochastic way by examining the irreducibility of randomly generated polynomials. On the other hand, a systematic method of derivation for the irreducible polynomial has recently been introduced by presenting a class of irreducible polynomials of particular forms.This paper shows clearly that a higher‐order irreducible polynomial can be derived by applying a suitable variable transformation to the given irreducible polynomial. It is shown also that, when an irreducible polynomial is given where the coefficient is the element of a finite field with odd prime characteristic, an infinite number of higher‐order irreducible polynomials can be derived from that polynomial. A precise algorithm for the derivation is shown.

In this paper, we study the irreducibility of polynomials of the form \( f(X) + p^k g(X) \), where \( f(X) \) and \( g(X) \) are polynomials with integer coefficients, \( p \) is a prime number, and \( k \) is a positive integer. Unlike previous results, we do not require \( f(X) \) and \( g(X) \) to be relatively prime or impose any conditions on \( \gcd(k, \deg g) \). We prove that, for all but finitely many primes \( p \), the polynomial \( f(X) + p^k g(X) \) is either irreducible over \( \mathbb{Q} \) or factors into polynomials whose degrees are multiples of \( \gcd(k, \deg g) \). This generalizes and extends earlier work on the irreducibility of such polynomials.

Substitution boxes or S-boxes play a significant role in encryption and decryption of bit level plaintext and cipher-text respectively. Irreducible Polynomials (IPs) have been used to construct 4-bit or 8-bit substitution boxes in many cryptographic block ciphers. In Advance Encryption Standard the 8-bit the elements S-box have been obtained from the Multiplicative Inverse (MI) of elemental polynomials (EPs) of the 1st IP over Galois field GF(28) by adding an additive element. In this paper a mathematical method and the algorithm of the said method with the discussion of the execution time of the algorithm, to obtain monic IPs over Galois field GF(pq) have been illustrated with example. The method is very similar to polynomial multiplication of two polynomials over Galois field GF(pq) but has a difference in execution. The decimal equivalents of polynomials have been used to identify Basic Polynomials (BPs), EPs, IPs and Reducible polynomials (RPs). The monic RPs have been determined by this method and have been cancelled out to produce monic IPs. The non-monic IPs have been obtained with multiplication of α where α GF(pq) and assume values from 2 to (p-1) to monic IPs.

Substitution boxes or S-boxes play a significant role in encryption and de-cryption of bit level plaintext and cipher-text respectively. Irreducible Poly-nomials (IPs) have been used to construct 4-bit or 8-bit substitution boxes in many cryptographic block ciphers. In Advance Encryption Standard, the ele-ments of 8-bit S-box have been obtained from the Multiplicative Inverse (MI) of elemental polynomials (EPs) of the 1st IP over Galois field GF(28) by adding an additive element. In this paper, a mathematical method and the algorithm of the said method with the discussion of the execution time of the algorithm, to obtain monic IPs over Galois field GF(pq) have been illustrated with example. The method is very similar to polynomial multiplication of two polynomials over Galois field GF(pq) but has a difference in execution. The decimal equivalents of polynomials have been used to identify Basic Polynomials (BPs), EPs, IPs and Reducible polynomials (RPs). The monic RPs have been determined by this method and have been cancelled out to produce monic IPs. The non-monic IPs have been obtained with multiplication of α where α∈ GF(pq) and assume values from 2 to (p − 1) to monic IPs.

An irreducible polynomial is one of the main components in building an S-box with an algebraic technique approach. The selection of the precise irreducible polynomial will determine the quality of the S-box produced. One method for determining good S-box quality is strict avalanche criterion will be perfect if it has a value of 0.5. Unfortunately, in previous studies, the strict avalanche criterion value of the S-box produced still did not reach perfect value. In this paper, we will discuss S-box construction using selected irreducible polynomials. This selection is based on the number of elements of the least amount of irreducible polynomials that make it easier to construct S-box construction. There are 17 irreducible polynomials that meet these criteria. The strict avalanche criterion test results show that the irreducible polynomial p17(x) =x8 + x7 + x6 + x + 1 is the best with a perfect SAC value of 0.5. One indicator that a robust S-box is an ideal strict avalanche criterion value of 0.5