Sequences of irreducible polynomials without prescribed coefficients over odd prime fields

详细信息    查看全文

• 作者：S. Ugolini (1)
  
  1. Dipartimento di Matematica ; Via Sommarive 14 ; 38123聽 ; Povo ; TN ; Italy
  
• 关键词：Finite fields ; Polynomials ; Sequences ; 11R09 ; 11T55 ; 12E05
• 刊名：Designs, Codes and Cryptography
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：75
• 期：1
• 页码：145-155
• 全文大小：170 KB
• 参考文献：1. Chou W., Shparlinski I.: On the cycle structure of repeated exponentiation modulo a prime. J. Number Theory 107(2), 345鈥?56 (2004).
  2. Chu W.M.: Construction of irreducible polynomials using cubic transformation. Appl. Algebra Eng. Commun. Comput. 7(1), 15鈥?9 (1996).
  3. Cohen S.D.: The explicit construction of irreducible polynomials over finite fields. Des. Codes Cryptogr. 2(2), 169鈥?74 (1992).
  4. Garefalakis T.: On the action of inline-equation id-i-eq651"> ion-source format-t-e-x">\(GL_2 ({ F}_q)\) on irreducible polynomials over inline-equation id-i-eq652"> ion-source format-t-e-x">\({ F}_q\) . J. Pure Appl. Algebra 215(8), 1835鈥?843 (2011).
  5. Kyuregyan M.K.: Recurrent methods for constructing irreducible polynomials over inline-equation id-i-eq653"> ion-source format-t-e-x">\({ F}_q\) of odd characteristics. Finite Fields Appl. 9(1), 39鈥?8 (2003).
  6. Kyuregyan M.K.: Recurrent methods for constructing irreducible polynomials over inline-equation id-i-eq654"> ion-source format-t-e-x">\({ F}_q\) of odd characteristics. II. Finite Fields Appl. 12(3), 357鈥?78 (2006).
  7. Meyn H.: On the construction of irreducible self-reciprocal polynomials over finite fields. Appl. Algebra Eng. Commun. Comput. 1(1), 43鈥?3 (1990).
  8. Rogers T.G.: The graph of the square mapping on the prime fields. Discrete Math. 148(1鈥?), 317鈥?24 (1996).
  9. Sha M., Hu S.: Monomial dynamical systems of dimension over finite fields. Acta Arith. 148(4), 309鈥?31 (2011).
  10. Stichtenoth H., Topuzo臒lu A.: Factorization of a class of polynomials over finite fields. Finite Fields Appl. 18(1), 108鈥?22 (2012).
  11. Ugolini S.: Graphs associated with the map inline-equation id-i-eq655"> ion-source format-t-e-x">\({X} \mapsto X + {X}^{-1}\) in finite fields of characteristic three and five. J. Number Theory 133(4), 1207鈥?228 (2013).
  12. Ugolini S.: On the iterations of certain maps inline-equation id-i-eq656"> ion-source format-t-e-x">\(x \mapsto k \cdot (x+x^{-1})\) over finite fields of odd characteristic. Preprint arXiv (2013).
  13. Ugolini S.: Sequences of binary irreducible polynomials. Discrete Math. 313(22), 2656鈥?662 (2013).
  14. van de Woestijne C.: Deterministic equation solving over finite fields. Doctoral thesis, Leiden University. http://hdl.handle.net/1887/4392 (2006). Accessed 4 Nov 2013.
  15. Vasiga T., Shallit J.: On the iteration of certain quadratic maps over GF(p). Discrete Math. 277(1鈥?), 219鈥?40 (2004).
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Coding and Information Theory
  Data Structures, Cryptology and Information Theory
  Data Encryption
  Discrete Mathematics in Computer Science
  Information, Communication and Circuits
  
• 出版者：Springer Netherlands
• ISSN：1573-7586

文摘

In this paper we construct infinite sequences of monic irreducible polynomials with coefficients in odd prime fields by means of a transformation introduced by Cohen in 1992. We make no assumptions on the coefficients of the first polynomial \(f_0\) of the sequence, which belongs to \(\mathbf{F}_p [x]\) , for some odd prime \(p\) , and has positive degree \(n\) . If \(p^{2n}-1 = 2^{e_1} \cdot m\) for some odd integer \(m\) and non-negative integer \(e_1\) , then, after an initial segment \(f_0, \dots , f_s\) with \(s \le e_1\) , the degree of the polynomial \(f_{i+1}\) is twice the degree of \(f_i\) for any \(i \ge s\) .