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].

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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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,

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 .

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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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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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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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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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.

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.

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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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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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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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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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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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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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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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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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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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.