350 rub
Journal Highly available systems №3 for 2015 г.
Article in number:
About one class of skew linear recurrences of maximal period over Galois rings
Authors:
M.A. Goltvanitsa - Researcher, Center of Special Development of Ministry of Defense of RF (Moscow). E-mail: goltv91@mail.ru
Abstract:
Here a class of skew LRS oriented toward fast implementation is investigated. For the sequences from this class necessary and sufficient conditions for achieving maximal period τ are obtained. For the skew MP LRS from this class the ranks (as LRS over the module sS) are determined and the uniqueness of the minimal polynomials over S is proved.
Pages: 28-49
References

 

  1. Alferov A.P., Zubov A.JU., Kuzmin A.S., CHeremushkin A.V. Osnovy kriptografii M.: Gelios ARV. 2001.
  2. Preneel B. Introduction to the Proceedings of the Fast Software Encryption 1994 Workshop // Lectures Notes in Comput. Sci. 1995. V. 1008. P. 1−5.
  3. Tsaban B., Vishne U. Efficient Linear Feedback Shift Registers with Maximal Period // Finite Fields and Their Applications. 2002. V. 8. № 2. P. 256−267.
  4. Zeng G., Han W., He K. Word-oriented feedback shift register: σ-LFSR // URL: http://eprint.iacr.org/2007/114 (dataobrashhenija 29.06.2015).
  5. Zeng G., He K.C., Han W. A trinomial type of σ-LFSR oriented toward software implementation // Science in China Series F - Information Sciences. 2007. V. 50. № 3. P. 359−372.
  6. Guang Zeng, Yang Yang, Wenbao Han and Shuqin Fan Word Oriented Cascade Jump σ-LFSR // Lectures Notes in Comput. Sci. 2009. V. 5527. P. 127−136.
  7. Nechaev A.A. Konechnye kolca glavnykh idealov // Matematicheskijj sbornik. 1973. T. 9. № 3. S. 350−366.
  8. Kurakin V.L., Kuzmin A.S., Mikhalev A.V., Nechaev A.A. Linear recurring sequences over rings and modules // J. of Math. Sciences. 1995. V. 76 № 6. P. 2793−2915.
  9. Nechaev A.A. Finite Rings with Applications. Handbook of Algebra / Edited by M. Hazewinkel. 2008. № 5. P. 213−320.
  10. Nechaev A.A. Kod Kerdoka v ciklicheskojj forme // Diskretnaja matematika. 1989. T. 1. № 4. S. 123−139.
  11. Goltvanica M.A., Zajjcev S.N., Nechaev A.A. Skruchennye linejjnye rekurrenty maksimalnogo perioda nad kolcami Galua // Fundamentalnaja i prikladnaja matematika 2012. T. 17. № 3. S. 5−23.
  12. Kurakin V.L., Mikhalev A.V., Nechaev A.A.andTsypyschev V.N. Linear and polylinear recurring sequences over abelian groups and modules // Journal of Mathematical Sciences. 2000. V. 102. № 6. P. 4598−4626.
  13. Nechaev A.A. Linejjnye rekurrentnye posledovatelnosti nad kommutativnymi kolcami // Diskretnaja matematika. 1991. T. 3. № 4. S. 107−121.
  14. Nechaev A.A. Ciklovye tipy linejjnykh podstanovok nad konechnymi kommutativnymi kolcami // Matematicheskijj sbornik. 1993. T. 184. № 3. S. 21−56.
  15. McDonald B.R. Finite Rings with Identity. NewYork: MarcelDekker. 1974.
  16. Glukhov M.M., Elizarov V.P., Nechaev A.A. Algebra. T. II. M.: Gelios ARV. 2003.
  17. Virgilo Sison Bases of the Galois Ring GR(pr,m) over the Integer Ring Zpr // URL: http://arxiv.org/abs/1410.0289 (dataobrashhenija 29.06.2015).
  18. Ghorpade S.R., Hasan S.U., Kumari M. Primitive polynomials, Singer cycles, and word-oriented linear feedback shift registers // Des. CodesCryptogr. 2011. № 58. P. 123−134.
  19. Sudhir R. Ghorpade, Samrith Ram Block companion Singer cycles, primitive recursive vector sequences, and coprime polynomial pairs over finite fields // Finite Fields Appl. 2011. V. 17. № 5. P. 461−472.
  20. Goltvanitsa M.A., Nechaev A.A., Zaitsev S.N. Skew LRS of maximal period over Galois rings // Mat. Vopr. Kriptogr. 2014. V. 5. № 2. P. 37−46.
  21. Goltvanitsa M.A. A construction of skew LRS of maximal period over finite fields based on the defining tuples of factors // Mat. Vopr. Kriptogr. 2013. V. 4. № 2. P. 59−72.
  22. Chen E.,Tseng D. The splitting Subspace Conjecture // Finite Fields Appl. 2013. № 24. P. 15−28.
  23. Goltvanitsa M.A. Digit sequences of skew linear recurrences of maximal period over Galois rings // Mat. Vopr. Kriptogr. 2015. V. 6. № 2. P. 189−198.
  24. Lidl R.,Niderrajjter G. Konechnye polja. M.: Mir. 1988.