Irreducible Polynomials

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An Algorithm to Find the Irreducible Polynomials Over Galois Field GF(pm)

An Algorithm to Find the Irreducible Polynomials Over Galois Field GF(pm)

The computation is undertaken for monic as well as non- monic irreducible polynomials for the first six prime moduli with m = 2, 3 and 4. It is observed that the results of monic irreducible polynomials for first four prime moduli are identical to what is given in [1], [2]. It is also observed that all the non-monic polynomials are in conformity with what is stated in [3]. For prime moduli 11 and 13, there is no mention of irreducible polynomials in literatures. The monic irreducible polynomials for p = 11 and 13 with m = 2, 3 and 4 are obtained and their results for p = 11 and 13 with m = 2 and 3 are given in Appendix A while the whole list are uploaded in [18]. Their non-monic polynomials are also found to be in conformity with what is stated in [3].

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Efficient Indexing of Necklaces and Irreducible Polynomials over Finite Fields

Efficient Indexing of Necklaces and Irreducible Polynomials over Finite Fields

Recent independent work. A preliminary version of this paper appeared as [16]. At about the same time, similar results were published by Kociumaka, Radoszewski and Rytter [15]. The work in these two papers was done independently. The papers both have polynomial-time algorithms for indexing necklaces; the authors in [15] exercised more care in designing the algorithm to obtain a better polynomial running time. Their approach to alphabets of size greater than 2 is cleaner than ours. On the other hand, we put the results in a broader context and have some additional applications (indexing irreducible polynomials and explicit constructions).

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Impact of 16 Primitive Irreducible Polynomials Over Erection of High Performance S Boxes Emanate from Linear Fractional Transformation

Impact of 16 Primitive Irreducible Polynomials Over Erection of High Performance S Boxes Emanate from Linear Fractional Transformation

Primitive irreducible polynomials (PIPs) which are defined on smallest finite field 2 plays a greater part in 2 . Also primitive polynomials have convenient applications in the field of computers. Digital computers utilize a binary number system to operate on given instructions, so for this purpose primitive polynomials defined on a finite field 0,1 . Furthermore, there exists 29 irreducible polynomials in 2 but only 16 polynomials are primitive which provides a surety that elements can be generated for 2 . The design methodology of proposed S-boxes is classified as follows,

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Infinite Parametric Families of Irreducible Polynomials with a Prescribed Number of Complex Roots

Infinite Parametric Families of Irreducible Polynomials with a Prescribed Number of Complex Roots

Theorem 1. For any pair of natural numbers ( ) n k , , n ≥ 3 , k ≥ 1 , and 2k n < , there exists an infinite parametric family of irreducible polynomials in ℤ [ X ] of degree n , which have exactly n − 2 k complex non-real roots if n is even and exactly n − 2 k − 1 complex non-real roots if n is odd .

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Irreducible Polynomials in Ζ[x] That Are Reducible Modulo All Primes

Irreducible Polynomials in Ζ[x] That Are Reducible Modulo All Primes

x + is irreducible in  [ ] x but is locally reducible, that is, it factors modulo p for all primes p . In this paper we investigate this phe- nomenon and prove that for any composite natural number N there are monic irreducible polynomials in  [ ] x which are reducible modulo every prime.

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Search for Monic Irreducible Polynomials with Decimal Equivalents of Polynomials over Galois Field GF(pq)

Search for Monic Irreducible Polynomials with Decimal Equivalents of Polynomials over Galois Field GF(pq)

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 1 st IP over Galois field GF (2 8 ) by adding

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Search for irreducible polynomials over Galois Field GF(pq)

Search for irreducible polynomials over Galois Field GF(pq)

decimal equivalents of each monic elemental polynomial (ep), two at a time, are split into the p-nary coefficients of each term, of those two monic elemental polynomials. From those coefficients, the p-nary coefficients of the resultant monic basic polynomials (BP) have been obtained. The decimal equivalents of resultant basic polynomials with p-nary coefficients are treated as decimal equivalents of the monic reducible polynomials, since monic reducible polynomials must have two monic elemental polynomials as its factor. The decimal equivalents of polynomials belonging to the list of reducible polynomials are cancelled leaving behind the monic irreducible polynomials. A non-monic irreducible polynomial is computed by multiplying a monic irreducible polynomial by α where α  GF(p q ) and assumes values from 2 to (p-1).

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Descending Central Series of Free Pro-p-Groups

Descending Central Series of Free Pro-p-Groups

We saw that the minimal number of generators for the quotient of the lower p-central series of a free pro-p-group is given by the Witt numbers. In this chapter we call these generators basic commutators. The Witt numbers are also counting the number of irreducible polynomials over certain finite fields. In this chapter we show the explicit connection between basic commutators and irreducible polynomials of a fixed degree with coefficients in F p .

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VLSI Architecture for Systolic Like Modular Multipliers over GF (2m) Build on Irreducible All One Polynomials

VLSI Architecture for Systolic Like Modular Multipliers over GF (2m) Build on Irreducible All One Polynomials

With the rapid expansion of the Internet and wireless communications, more and more digital systems are becoming increasingly equipped with some form of cryptosystems to provide various kinds of data security. Many such cryptosystems rely on computations in very large finite fields and it requires fast computation. Finite fields arithmetic multiplication over GF (2 m ) has gained very high importance to obtain secure communication by integrating elliptical curve cryptography (ECC) and error control systems. Among the different basis of multipliers polynomial basis are relatively easy to design, and subjects to scalability for the higher order fields. The real-time applications are hard-ware efficient with polynomial-based multiplication [1].Multipliers with different basis of representations are normal basis, dual basis and polynomial basis used for several applications in earlier cases. Based on a number of significant classes, irreducible polynomials for the field are all-one polynomials can be defined [4]. 1-equally spaced polynomials (or) All-one polynomials (AOP) form a special class, which can be used for simpler and more efficient implementation compared to trinomials and pentanomial-based multipliers. The AOP-based representations of elements are expected to have potential application in elliptic curve cryptosystems and error control coding procures efficient hardware implementation. The first multiplier for GF (2 m ) generated by AOP which was followed by some bit-parallel architectures[8].The bit-parallel designs are useful for low-latency realization, but due to their large critical path, they cannot provide high throughput rate and involve high average computation time which increases rapidly with the field order m.

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Breaking  the  Hidden  Irreducible  Polynomials  Scheme

Breaking the Hidden Irreducible Polynomials Scheme

In this paper we have shown a full break of the Hidden Irreducible Polynomials scheme introduced by Gómez in [1]. We have shown that the private key is publicly known by the design of the system. Moreover, we have shown that due to the construction of the private map, namely univariate polynomial multiplication, one can even easily read off the transformation matrix for the system of multivariate quadratic polynomial equations such that not even linear algebra is needed for attacking the scheme.

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4-bit  crypto  S-boxes:  Generation  with  irreducible  polynomials  over  Galois  field  GF(24)   and  cryptanalysis.

4-bit crypto S-boxes: Generation with irreducible polynomials over Galois field GF(24) and cryptanalysis.

Abstract. 4-bit crypto S-boxes play a significant role in encryption and decryption of many cipher algorithms from last 4 decades. Generation and cryptanalysis of generated 4-bit crypto S-boxes is one of the major concerns of modern cryptography till now. In this paper 48, 4-bit crypto S-boxes are generated with addition of all possible additive constants to the each element of crypto S-box of corresponding multiplicative inverses of all elemental polynomials (EPs) under the concerned irreducible polynomials (IPs) over Galois field GF(2 4 ). Cryptanalysis of 48 generated 4-bit crypto S-boxes is done with all relevant cryptanalysis algorithms of 4-bit crypto S-boxes. The result shows the better security of generated 4-bit crypto S-boxes.

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Improving  the  Polynomial  time  Precomputation  of  Frobenius  Representation  Discrete  Logarithm  Algorithms -  Simplified  Setting  for  Small  Characteristic  Finite  Fields

Improving the Polynomial time Precomputation of Frobenius Representation Discrete Logarithm Algorithms - Simplified Setting for Small Characteristic Finite Fields

given in (4). Assuming that both g 1 and g 2 are not equal to v + w then all three values (g 1 , g 2 , g 3 ) are well-defined. With this specific choice, the right-hand side that now appear in Equation (5) or (6) gains a new systematic degree 2 factor θh 1 + h 0 + (v + w)h 1 = (1 + u) θ 2 + (1 + u)(v + w)θ + vw + v 2 as given in Lemma 4. Again, the remaining factor in the right-hand side when considering these groups is of degree 4. Since the probability of a degree 4 polynomial to factor in terms of degree at most 2 is higher than 1/3, we can recover all the discrete logarithms of the irreducible polynomials of P g 1 ,G(g 1 ),G(G(g 1 )) .

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Computing in Algebraic Closures of Finite Fields

Computing in Algebraic Closures of Finite Fields

We present algorithms to construct and perform computations in algebraic closures of finite fields. Inspired by algorithms for constructing irreducible polynomials, our approach for con- structing closures consists of two phases; First, extension towers of prime power degree are built, and then they are glued together using composita techniques. To be able to move elements around in the closure we give efficient algorithms for computing isomorphisms and embeddings. In most cases, our algorithms which are based on polynomial arithmetic, rather than linear algebra, have quasi-linear complexity.

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Anchors of irreducible characters

Anchors of irreducible characters

I. M. Isaacs and G. Navarro provided us with an example of a p-special char- acter of a p-solvable group none of whose irreducible Brauer constituents have degree prime to p. Proposition 5.5 can be used to prove that the anchors of the Isaacs-Navarro example, which we give below, are strictly contained in the Sylow p- subgroups of the ambient group (so in particular, these characters are not afforded by any lattice with full vertex).

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THE THEORY OF REPRESENTATIONS OF GROUPS SO0(2; 1) AND ISO(2; 1). WIGNER COEFFICIENTS OF THE GROUP SO0(2; 1)

THE THEORY OF REPRESENTATIONS OF GROUPS SO0(2; 1) AND ISO(2; 1). WIGNER COEFFICIENTS OF THE GROUP SO0(2; 1)

Abstract. This paper is devoted to the representations of the groups SO(2, 1) and ISO(2, 1). Those groups have an important role in cosmology, elementary particle theory and mathematical physics. Irreducible unitary representations of the principal continuous and supplementary as well as discrete series were obtained. Explicit expressions for spherical functions of the group SO 0 (2, 1) are obtained through the Gauss hypergeometric

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Primitive irreducible linear groups

Primitive irreducible linear groups

over an algebraically closed field, it is known (see Suprunenko 1963, p60) that irreducible nilpotent linear groups are never primitive: this fact lends interest to the discussion. In fact we shall prove that primitive irreducible groups can be nilpotent in the case of certain finite fields. The first such example was shown to me by M.P, Newman.

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The Irreducible Representations of D2n

The Irreducible Representations of D2n

In the last section we discussed a very important correspondence between KG- modules and pairs (V, φ), where V is a vector space over K and φ is a representation of G. That is, there is a bijection between KG-modules and representations of G or we say a KG module affords a representation. In this section we will let K = C, unless otherwise stated, and we will compute the irreducible representations of D 8 . Along the way we

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10. Some series identities for some special classes of Apostol-Bernoulli and Apostol-Euler polynomials related to generalized power and alternating sums

10. Some series identities for some special classes of Apostol-Bernoulli and Apostol-Euler polynomials related to generalized power and alternating sums

The generalized Apostol-Bernoulli polynomials B (α) n (x; λ) of order α ∈ C , the generalized Apostol-Euler polynomials E (α) n (x; λ) of order α ∈ C , the generalized Apostol-Genocchi polynomials G (α) n (x; λ) of order α ∈ C are defined respectively by the following generating functions

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Perron-Frobenius theory and KMS states on higher-rank graph C*-Algebras

Perron-Frobenius theory and KMS states on higher-rank graph C*-Algebras

Let T ∈ M n ( R ) be a non-negative matrix. Define a directed graph E associated with T in such a way that the indices of T are the vertices of E and T (u, v) is the number of edges from the vertex v to the vertex u. In fact, T is called the coordi- nate matrix of E. In Chapter 3, we give a classification of indices of a non-negative matrix into different self-communicating classes. The directed graphs associated to sub-matrices corresponding to self-communicating classes turn out to be strongly con- nected; such sub-matrices are termed as irreducible matrices. It comes into notice that primitive matrices are a particular case of irreducible matrices. We give the Perron-Frobenius theorem for irreducible matrices (see Theorem 3.4.1) which is a weaker version of the Perron-Frobenius theorem for primitive matrices. Some differ- ences between these two theorems are also mentioned in the last theorem of Chapter 3.

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Weakly irreducible ideals

Weakly irreducible ideals

Abstract. Let R be a commutative ring. The purpose of this article is to introduce a new class of ideals of R called weakly irreducible ideals. This class could be a generalization of the fam- ilies quasi-primary ideals and strongly irreducible ideals. The re- lationships between the notions primary, quasi-primary, weakly ir- reducible, strongly irreducible and irreducible ideals, in different rings, has been given. Also the relations between weakly irre- ducible ideals of R and weakly irreducible ideals of localizations of the ring R are also studied.

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