On the construction of irreducible self-reciprocal polynomials over finite fields

Meyn, Helmut

doi:10.1007/bf01810846

Cited by 63 publications

(78 citation statements)

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“…The missing half possibilities for q odd, which occur when σ is not a square in F * q , can be covered with a simple extension of any of the various proofs for Carlitz's formula which are available, found in [Car67,Coh69,Mey90]. We have chosen a presentation close to that of [Mey90].…”

Section: Counting Irreducible Polynomials Obtained Through a Quadratimentioning

confidence: 99%

“…Various counting formulas for irreducible polynomials of certain types exist, on the model of Gauss's formula (1/n) d|n µ(d)q n/d for the total number of monic irreducible polynomials of degree n over the field F q of q elements. Carlitz proved in [Car67] that the number SRIM (2n, q) of srim polynomials of degree 2n over a finite field F q is given by Simpler proofs of Equation (1) were given by Cohen [Coh69] and Meyn [Mey90]. The latter proof applies Möbius inversion to the fact, of which our Theorem 4 below is a slight generalization, that the nonlinear srim are exactly the nonlinear irreducible factors of polynomials of the form x q n +1 −1.…”

Section: Introductionmentioning

confidence: 99%

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Generalizations of self-reciprocal polynomials

Mattarei

Pizzato²

2017

Finite Fields and Their Applications

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Abstract.A formula for the number of monic irreducible self-reciprocal polynomials, of a given degree over a finite field, was given by Carlitz in 1967. In 2011 Ahmadi showed that Carlitz's formula extends, essentially without change, to a count of irreducible polynomials arising through an arbitrary quadratic transformation. In the present paper we provide an explanation for this extension, and a simpler proof of Ahmadi's result, by a reduction to the known special case of self-reciprocal polynomials and a minor variation. We also prove further results on polynomials arising through a quadratic transformation, and through some special transformations of higher degree.

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Section: Counting Irreducible Polynomials Obtained Through a Quadratimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalizations of self-reciprocal polynomials

Mattarei

Pizzato²

2017

Finite Fields and Their Applications

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“…In this section we outline a connection between TSRs of order two (n = 2) and self-reciprocal polynomials and give a new proof of a theorem of Carlitz [2] (which has been reproved by Ahmadi [1], Cohen [4], Meyn [13], Meyn and Götz [14] and Miller [15]) on the number of self-reciprocal irreducible monic polynomials of a given degree over a finite field. In what follows, we denote the cardinality of S q (m, n) (as defined in Section 4) by N q (m, n).…”

Section: Irreducible Tsrs Of Order Twomentioning

confidence: 99%

Enumeration of linear transformation shift registers

Ram

2014

Des. Codes Cryptogr.

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Abstract. We consider the problem of counting the number of linear transformation shift registers (TSRs) of a given order over a finite field. We derive explicit formulae for the number of irreducible TSRs of order two. An interesting connection between TSRs and self-reciprocal polynomials is outlined. We use this connection and our results on TSRs to deduce a theorem of Carlitz on the number of self-reciprocal irreducible monic polynomials of a given degree over a finite field. IntroductionA linear feedback shift register (LFSR) is a mechanism for generating a sequence in a finite field. LFSRs have a plethora of practical applications and are frequently used in generating pseudorandom numbers, fast digital counters and stream ciphers. A generalization of LFSR called word-oriented feedback shift register (σ-LFSR) was considered by Zeng, Han and He [20]. For LFSRs as well as σ-LFSRs, those that are primitive (i.e., for which the corresponding infinite sequence is of maximal possible period) are of particular interest. The following conjecture was proposed in the binary case in [20] and was extended to the q-ary case in [8]:Conjecture 1.1. For positive integers m and n, the number of primitive σ-LFSRs of order n over F q m is given byThe notion of σ-LFSR is essentially equivalent to that of a splitting subspace previously defined by Niederreiter [16]: Given positive integers m, n and α ∈ F q mn , an m-dimensional F q -linear subspace of F q mn is said to be α-splitting if

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“…From the definition of a reciprocal polynomial, it is clear that if f (x) is irreducible over F q , then so is f * (x). Several authors have surveyed self-reciprocal irreducible monic (srim) polynomials and obtained many results; see [8,12,13,20,16].…”

Section: Introductionmentioning

confidence: 99%