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On the Planarity of Certain Dembowski-Ostrom Polynomials

Year 2022, Volume: 17 Issue: 2, 261 - 269, 25.11.2022
https://doi.org/10.29233/sdufeffd.1053097

Abstract

Planar mappings, defined by Dembowski and Ostrom, are identified as a means to construct projective planes. Then, many important applications of planar mappings appear in different fields such as cryptography and coding theory. In this paper, we provide sufficient and necessary conditions for the planarity of certain Dembowski-Ostrom polynomials over the finite field extension of degree three with odd characteristic. In particular, we completely determine the coefficients of the given Dembowski-Ostrom polynomials to be planar.

Supporting Institution

Scientific Research Fund of Suleyman Demirel University

Project Number

FYL-2020-7985

References

  • D. Bartoli and M. Bonini, “Planar polynomials arising from linearized polynomials”, Journal of Algebra and Its Applications, https://doi.org/10.1142/S0219498822500025, 2020.
  • E. R. Berlekamp, “Algebraic coding theory (revised edition) ”, World Scientific, 2015.
  • A. Blokhuis, R. S. Coulter, M. Henderson and C. M. O’Keefe, “Permutations amongst the Dembowski-Ostrom polynomials”, Finite Fields and Applications, Springer, Berlin, Heidelberg, 37-42, 2001.
  • C. Carlet, C. Ding and J. Yuan, “Linear codes from perfect nonlinear mappings and their secret sharing schemes”, IEEE Transactions on Information Theory, 51, 2089-2102, 2005.
  • R. S. Coulter and M. Henderson, “Commutative presemifields and semifields”, Advances in Mathematics, 217, 282-304, 2008.
  • R. S. Coulter and R. W. Matthews, “Planar functions and planes of Lenz-Barlotti class II”, Designs, Codes and Cryptography, 10, 167-184, 1997.
  • P. Dembowski and T. G. Ostrom, “Planes of order n with collineation groups of order n2”, Mathematische Zeitschrift, 103, 239-258, 1968.
  • U. Dempwolff, “More translation planes and semifields from Dembowski–Ostrom polynomials”, Designs, Codes and Cryptography, 68, 81-103, 2013.
  • C. Ding and H. Niederreiter, “Systematic authentication codes from highly nonlinear functions”, IEEE Transactions on Information Theory, 50, 2421-2428, 2004.
  • C. Ding and J. Yin, “Signal sets from functions with optimum nonlinearity”, IEEE transactions on communications, 55, 936-940, 2007.
  • C. Ding and J. Yuan, “A family of optimal constant-composition codes”, IEEE transactions on information theory, 51, 3668-3671, 2005.
  • E. M. Gabidulin, “Theory of codes with maximum rank distance”, Problemy peredachi informatsii, 21 3-16, 1985.
  • G. Kyureghyan and F. Özbudak, “Planarity of products of two linearized polynomials”, Finite Fields and Their Applications, 18, 1076-1088, 2012.
  • G. Kyureghyan, F. Özbudak, and A. Pott, “Some planar maps and related function fields”, Contemporary Mathematics, 574, 87-114, 2012.
  • R. Lidl and H. Niederreiter, Finite fields, Cambridge university press, 1997.
  • W. Meier and O. Staffelbach, “Nonlinearity criteria for cryptographic functions”, In Workshop on the Theory and Application of Cryptographic Techniques, Springer, Berlin, pp. 549-562, 1989.
  • K. Nyberg and L. R. Knudsen, “Provable security against a differential attack”, Journal of Cryptology, 8, 27-37, 1995.
  • K. Nyberg, “Perfect nonlinear S-boxes”, In Workshop on the Theory and Application of Cryptographic Techniques, Springer, Berlin, Heidelberg, pp. 378-386, 1991.
  • O. Ore, “Contributions to the theory of finite fields”, Transactions of the American Mathematical Society, 36, 243-274, 1934.
  • O. Ore, “On a special class of polynomials”, Transactions of the American Mathematical Society, 35, 559-584, 1933.
  • S. Puchinger and A. Wachter-Zeh, “Fast operations on linearized polynomials and their applications in coding theory”, Journal of Symbolic Computation, 89, 194-215, 2018.
  • L. Ronyai and T. Szőnyi, “Planar functions over finite fields”, Combinatorica, 9, 315-320, 1989.
  • S. Samardjiska and D. Gligoroski, “Quadratic permutations, complete mappings and mutually orthogonal latin squares”, Mathematica Slovaca, 67, 1129-1146, 2017.
  • K. U. Schmidt and Y. Zhou, “Planar functions over fields of characteristic two”, Journal of Algebraic Combinatorics, 40, 503-526, 2014.
  • D. Silva, F. R. Kschischang and R. Koetter, “A rank-metric approach to error control in random network coding”, IEEE Transactions on Information Theory, 54, 3951-3967, 2008.
  • R. P. Singh, “Counterexamples to Dembowski and Ostrom conjecture on Planar function”, arXiv preprint arXiv:2104.01942, 2021.
  • X. Zhang, B. Wu and Z. Liu, “Dembowski-Ostrom polynomials from reversed Dickson polynomials”, Journal of Systems Science and Complexity, 18, 259-271, 2016.
  • Y. Zhou, “(2n,2n,2n,1)‐Relative Difference Sets and Their Representations”, Journal of Combinatorial Designs, 21, 563-584, 2013.

Bazı Dembowski-Ostrom Polinomlarının Planaritesi Üzerine

Year 2022, Volume: 17 Issue: 2, 261 - 269, 25.11.2022
https://doi.org/10.29233/sdufeffd.1053097

Abstract

Dembowski ve Ostrom tarafından tanımlanan planar dönüşümler projektif düzlemler oluşturmanın bir yolu olarak ortaya çıkmıştır. Sonrasında, planar dönüşümlerin kriptografi ve kodlama teorisi gibi farklı alanlarda birçok önemli uygulaması yapılmıştır. Bu çalışmada, tek karakteristiğe sahip üçüncü dereceden sonlu cisim genişlemeleri üzerinde tanımlanan belirli bir formdaki Dembowski-Ostrom polinomlarının planaritesi için gerek ve yeter koşullar elde edilmiştir. Özel olarak, verilen Dembowski-Ostrom polinomlarının planar olmasını sağlayan katsayılar tamamıyla belirlenmiştir.

Project Number

FYL-2020-7985

References

  • D. Bartoli and M. Bonini, “Planar polynomials arising from linearized polynomials”, Journal of Algebra and Its Applications, https://doi.org/10.1142/S0219498822500025, 2020.
  • E. R. Berlekamp, “Algebraic coding theory (revised edition) ”, World Scientific, 2015.
  • A. Blokhuis, R. S. Coulter, M. Henderson and C. M. O’Keefe, “Permutations amongst the Dembowski-Ostrom polynomials”, Finite Fields and Applications, Springer, Berlin, Heidelberg, 37-42, 2001.
  • C. Carlet, C. Ding and J. Yuan, “Linear codes from perfect nonlinear mappings and their secret sharing schemes”, IEEE Transactions on Information Theory, 51, 2089-2102, 2005.
  • R. S. Coulter and M. Henderson, “Commutative presemifields and semifields”, Advances in Mathematics, 217, 282-304, 2008.
  • R. S. Coulter and R. W. Matthews, “Planar functions and planes of Lenz-Barlotti class II”, Designs, Codes and Cryptography, 10, 167-184, 1997.
  • P. Dembowski and T. G. Ostrom, “Planes of order n with collineation groups of order n2”, Mathematische Zeitschrift, 103, 239-258, 1968.
  • U. Dempwolff, “More translation planes and semifields from Dembowski–Ostrom polynomials”, Designs, Codes and Cryptography, 68, 81-103, 2013.
  • C. Ding and H. Niederreiter, “Systematic authentication codes from highly nonlinear functions”, IEEE Transactions on Information Theory, 50, 2421-2428, 2004.
  • C. Ding and J. Yin, “Signal sets from functions with optimum nonlinearity”, IEEE transactions on communications, 55, 936-940, 2007.
  • C. Ding and J. Yuan, “A family of optimal constant-composition codes”, IEEE transactions on information theory, 51, 3668-3671, 2005.
  • E. M. Gabidulin, “Theory of codes with maximum rank distance”, Problemy peredachi informatsii, 21 3-16, 1985.
  • G. Kyureghyan and F. Özbudak, “Planarity of products of two linearized polynomials”, Finite Fields and Their Applications, 18, 1076-1088, 2012.
  • G. Kyureghyan, F. Özbudak, and A. Pott, “Some planar maps and related function fields”, Contemporary Mathematics, 574, 87-114, 2012.
  • R. Lidl and H. Niederreiter, Finite fields, Cambridge university press, 1997.
  • W. Meier and O. Staffelbach, “Nonlinearity criteria for cryptographic functions”, In Workshop on the Theory and Application of Cryptographic Techniques, Springer, Berlin, pp. 549-562, 1989.
  • K. Nyberg and L. R. Knudsen, “Provable security against a differential attack”, Journal of Cryptology, 8, 27-37, 1995.
  • K. Nyberg, “Perfect nonlinear S-boxes”, In Workshop on the Theory and Application of Cryptographic Techniques, Springer, Berlin, Heidelberg, pp. 378-386, 1991.
  • O. Ore, “Contributions to the theory of finite fields”, Transactions of the American Mathematical Society, 36, 243-274, 1934.
  • O. Ore, “On a special class of polynomials”, Transactions of the American Mathematical Society, 35, 559-584, 1933.
  • S. Puchinger and A. Wachter-Zeh, “Fast operations on linearized polynomials and their applications in coding theory”, Journal of Symbolic Computation, 89, 194-215, 2018.
  • L. Ronyai and T. Szőnyi, “Planar functions over finite fields”, Combinatorica, 9, 315-320, 1989.
  • S. Samardjiska and D. Gligoroski, “Quadratic permutations, complete mappings and mutually orthogonal latin squares”, Mathematica Slovaca, 67, 1129-1146, 2017.
  • K. U. Schmidt and Y. Zhou, “Planar functions over fields of characteristic two”, Journal of Algebraic Combinatorics, 40, 503-526, 2014.
  • D. Silva, F. R. Kschischang and R. Koetter, “A rank-metric approach to error control in random network coding”, IEEE Transactions on Information Theory, 54, 3951-3967, 2008.
  • R. P. Singh, “Counterexamples to Dembowski and Ostrom conjecture on Planar function”, arXiv preprint arXiv:2104.01942, 2021.
  • X. Zhang, B. Wu and Z. Liu, “Dembowski-Ostrom polynomials from reversed Dickson polynomials”, Journal of Systems Science and Complexity, 18, 259-271, 2016.
  • Y. Zhou, “(2n,2n,2n,1)‐Relative Difference Sets and Their Representations”, Journal of Combinatorial Designs, 21, 563-584, 2013.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Makaleler
Authors

Zehra Aksoy 0000-0003-4534-3958

Barış Bülent Kırlar 0000-0003-0112-1974

Project Number FYL-2020-7985
Publication Date November 25, 2022
Published in Issue Year 2022 Volume: 17 Issue: 2

Cite

IEEE Z. Aksoy and B. B. Kırlar, “On the Planarity of Certain Dembowski-Ostrom Polynomials”, Süleyman Demirel University Faculty of Arts and Science Journal of Science, vol. 17, no. 2, pp. 261–269, 2022, doi: 10.29233/sdufeffd.1053097.