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Title of Journal: Des Codes Cryptogr

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Abbravation: Designs, Codes and Cryptography

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Springer US

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DOI

10.1007/s11829-012-9203-6

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1573-7586

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Primitive normal bases for quartic and cubic exten

Authors: Dirk Hachenberger
Publish Date: 2015/03/14
Volume: 77, Issue: 2-3, Pages: 335-350
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Abstract

For the Galois field extension mathbb F qn over mathbb F q we let PN nq denote the number of primitive elements of mathbb F qn which are normal over mathbb F q We derive lower bounds for PN 3q and PN 4q the number of primitive normal elements in cubic and quartic extensions Our reasoning relies on basic projective geometry A comparision with the exact numbers for PN 3q and PN 4q where qle 32 altogether 36 instances indicates that these bounds are very good we even achieve equality in 14 casesThis paper is dedicated to the memory of Scott Vanstone I thank Scott for many discussions on finite fields and finite geometries which once have raised my interest in studying specific normal bases I am also very grateful to Scott for once supporting my Habilitation at the University of Augsburg published as 7 Finally I wish to thank Thomas Gruber for supporting me with the computational results included in Tables 1 and 2

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  1. Composition of recursions and nonlinear complexity of periodic binary sequences
  2. Practical-time attacks against reduced variants of MISTY1
  3. On the largest affine sub-families of a family of NFSR sequences
  4. The dimension of subcode-subfields of shortened generalized Reed–Solomon codes
  5. On explicit factors of cyclotomic polynomials over finite fields
  6. Two classes of optimal two-dimensional OOCs
  7. Sequences with small correlation
  8. Improved algorithms for finding low-weight polynomial multiples in $$\mathbb {F}_{2}^{}[x]$$ and some cryptographic applications
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  10. Distinguisher-based attacks on public-key cryptosystems using Reed–Solomon codes
  11. A new table of permutation codes
  12. Bent functions embedded into the recursive framework of $${\mathbb{Z}}$$ -bent functions
  13. Nonexistence of CW (110, 100)
  14. Point compression for the trace zero subgroup over a small degree extension field
  15. The Diffie–Hellman problem and generalization of Verheul’s theorem
  16. Modular independence and generator matrices for codes over $${\mathbb {Z}_m}$$
  17. Improved lower bounds on sizes of single-error correcting codes
  18. A combinatorial problem related to sparse systems of equations
  19. Some results concerning cryptographically significant mappings over GF(2 n )
  20. A note on the reducibility of binary affine polynomials
  21. Computing isogenies between supersingular elliptic curves over $${\mathbb {F}}_p$$
  22. Some cyclic codes of length 2 p n
  23. On the correlation distribution of Delsarte–Goethals sequences
  24. Algebraic decoding of folded Gabidulin codes
  25. Inner balance of symmetric designs
  26. On the construction of Griesmer codes of dimension 5
  27. Applications of representation theory to wireless communications
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