Article

On the cycle structure of permutation polynomials

Sabancı University, MDBF, Orhanlı, 34956 Tuzla, İstanbul, Turkey
Finite Fields and Their Applications (Impact Factor: 0.46). 07/2008; 14(3):593-614. DOI: 10.1016/j.ffa.2007.08.003
Source: DBLP

ABSTRACT Any permutation of a finite field Fq can be represented by a polynomial Pn(x)=(⋯+((a0x+a1)q−2+a2)q−2+⋯+an)q−2+an+1, for some n⩾0. P0 is linear and the cycle structure of P1 is known. In this work we present the cycle structure of the polynomials P2(x) and P3(x) completely and give methods for constructing Pn(x) with full cycle, for arbitrary n⩾1.

Download full-text

Full-text

Available from: Ayca Cesmelioglu, Aug 03, 2015
1 Follower
 · 
98 Views
  • Source
    • "But this is not the case for interleavers derived from permutations like T . The next theorem cited from [4] "
    [Show abstract] [Hide abstract]
    ABSTRACT: In this work we establish some new interleavers based on permutation functions. The inverses of these interleavers are known over a finite field $\mathbb{F}_q$. For the first time M\"{o}bius and R\'edei functions are used to give new deterministic interleavers. Furthermore we employ Skolem sequences in order to find new interleavers with known cycle structure. In the case of R\'edei functions an exact formula for the inverse function is derived. The cycle structure of R\'edei functions is also investigated. The self-inverse and non-self-inverse versions of these permutation functions can be used to construct new interleavers.
    Advances in Mathematics of Communications 11/2010; 6(3). DOI:10.3934/amc.2012.6.347 · 0.65 Impact Factor
  • Source
    • "for some n ≥ 1 and a 0 , a 1 , · · · , a n+1 ∈ F q with a 0 = 0 or a linear polynomial P 0 (x) = cx + d ∈ F q [x]. The cycle structure of PPs given as in (1) was studied in [2] and also some conditions for obtaining PPs with full cycle were determined. "
    [Show abstract] [Hide abstract]
    ABSTRACT: For q > 2, Carlitz proved that the group of permutation polynomials (PPs) over F_q is generated by linear polynomials and x^{q-2}. Based on this result, this note points out a simple method for representing all PPs with full cycle over the prime field F_p, where p is an odd prime. We use the isomorphism between the symmetric group S_p of p elements and the group of PPs over F_p, and the well-known fact that permutations in S_p have the same cycle structure if and only if they are conjugate. Comment: 5 pages
  • Source
    • "Permutation monomials x n with all cycles of the same length are characterized in [24]. The cycle structure of Dickson permutation polynomials D n (x, a) where a ∈ {0, ±1} has been studied in [17]. The cycle structure of Möbius transformation has been described in [7]. "
    [Show abstract] [Hide abstract]
    ABSTRACT: In this work we introduce and study a set of new interleavers based on permutation polynomials and functions with known inverses over a finite field $\mathbb{F}_q$ for using in turbo code structures. We use Monomial, Dickson, M\"{o}bius and R\'edei functions in order to get new interleavers. In addition we employ Skolem sequences in order to find new interleavers with known cycle structure. As a byproduct we give an exact formula for the inverse of every R\'edei function. The cycle structure of R\'edei functions are also investigated. Finally, self-inverse versions of permutation functions are used to construct interleavers. These interleavers are their own de-interleavers and are useful for turbo coding and turbo decoding. Experiments carried out for self-inverse interleavers constructed using these kind of permutation polynomials and functions show excellent agreement with our theoretical results.
Show more