May  2016, 10(2): 291-306. doi: 10.3934/amc.2016006

Constructing commutative semifields of square order

1. 

Department of Algebra, Charles University, Sokolovská 83, 186 75 Prague, Czech Republic

2. 

School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, Australia

Received  April 2014 Revised  November 2015 Published  April 2016

The projection construction has been used to construct semifields of odd characteristic using a field and a twisted semifield [Commutative semifields from projection mappings, Designs, Codes and Cryptography, 61 (2011), 187--196]. We generalize this idea to a projection construction using two twisted semifields to construct semifields of odd characteristic. Planar functions and semifields have a strong connection so this also constructs new planar functions.
Citation: Stephen M. Gagola III, Joanne L. Hall. Constructing commutative semifields of square order. Advances in Mathematics of Communications, 2016, 10 (2) : 291-306. doi: 10.3934/amc.2016006
References:
[1]

A. A. Albert, Quasigroups I,, Trans. Amer. Math. Soc., 54 (1943), 507. doi: 10.2307/1990259. Google Scholar

[2]

J. Bierbrauer, Commutative semifields from projection mappings,, Des. Codes Crypt., 61 (2011), 187. doi: 10.1007/s10623-010-9447-z. Google Scholar

[3]

A. Blokhuis, R. S. Coulter, M. Henderson and C. M. O'Keefe, Permutations amongst the Dembowski-Ostrom polynomials,, in Finite Fields and Applications, (2001), 37. doi: 10.1007/978-3-642-56755-1_4. Google Scholar

[4]

L. Budaghyan and T. Helleseth, New commutative semifields defined by new PN multinomials,, Crypt. Commun., 3 (2011), 1. doi: 10.1007/s12095-010-0022-2. Google Scholar

[5]

L. Budaghyan and T. Helleseth, On isotopisms of commutative presemifields and CCZ-equivalence of functions,, Int. J. Found. Comput. Sci., 22 (2011), 1243. doi: 10.1142/S0129054111008684. Google Scholar

[6]

C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems,, Des. Codes Crypt., 15 (1998), 125. doi: 10.1023/A:1008344232130. Google Scholar

[7]

C. Carlet and C. Ding, Highly nonlinear mappings,, J. Complexity, 20 (2004), 205. doi: 10.1016/j.jco.2003.08.008. Google Scholar

[8]

R. S. Coulter and M. Henderson, Commutative presemifields and semifields,, Adv. Math., 217 (2008), 282. doi: 10.1016/j.aim.2007.07.007. Google Scholar

[9]

R. S. Coulter and R. W. Matthews, Planar functions and planes of Lenz-Barlotti class II,, Des. Codes Crypt., 10 (1997), 167. doi: 10.1023/A:1008292303803. Google Scholar

[10]

T. W. Cusick, C. Ding and A. R. Renvall, Stream Ciphers and Number Theory,, Elsevier, (2004). Google Scholar

[11]

P. Dembowski and T. G. Ostrom, Planes of order $n$ with collineation groups of order $n^2$,, Math. Z., 103 (1968), 239. doi: 10.1007/BF01111042. Google Scholar

[12]

C. Ding and J. Yin, Signal sets from functions with optimum nonlinearity,, IEEE Trans. Commun., 55 (2007), 936. doi: 10.1109/TCOMM.2007.894113. Google Scholar

[13]

The GAP Group, GAP - Groups, Algorithms and Programming,, Version 4.5.5, (2012). Google Scholar

[14]

K. J. Horadam, Hadamard Matrices and their Applications,, Princeton Univ. Press, (2007). Google Scholar

[15]

W. Jia, X. Zeng, T. Helleseth and C. Li, A class of binomial bent functions over the finite fields of odd characteristic,, IEEE Trans. Inf. Theory, 58 (2012), 6054. doi: 10.1109/TIT.2012.2199736. Google Scholar

[16]

D. E. Knuth, Finite semifields and projective planes,, J. Algebra, 2 (1965), 182. doi: 10.1016/0021-8693(65)90018-9. Google Scholar

[17]

G. Kyureghyan and A. Pott, Some theorems on planar mappings,, in Arithmetic of Finite Fields, (2008), 117. doi: 10.1007/978-3-540-69499-1_10. Google Scholar

[18]

G. Marino and O. Polverino, On the nuclei of a finite semifield,, Theory Appl. Finite Fields. Contemp. Math., 579 (2012), 123. doi: 10.1090/conm/579/11525. Google Scholar

[19]

A. Roy and A. J. Scott, Weighted complex projective 2-designs from bases: Optimal state determination by orthogonal measurements,, J. Math. Phys., 48 (2007), 1. doi: 10.1063/1.2748617. Google Scholar

[20]

J. Wedderburn, A theorem on finite algebras,, Trans. Amer. Math. Soc., 6 (1903), 349. doi: 10.2307/1986226. Google Scholar

[21]

Y. Zhou and A. Pott, A new family of semifields with 2 parameters,, Adv. Math., 234 (2013), 43. doi: 10.1016/j.aim.2012.10.014. Google Scholar

show all references

References:
[1]

A. A. Albert, Quasigroups I,, Trans. Amer. Math. Soc., 54 (1943), 507. doi: 10.2307/1990259. Google Scholar

[2]

J. Bierbrauer, Commutative semifields from projection mappings,, Des. Codes Crypt., 61 (2011), 187. doi: 10.1007/s10623-010-9447-z. Google Scholar

[3]

A. Blokhuis, R. S. Coulter, M. Henderson and C. M. O'Keefe, Permutations amongst the Dembowski-Ostrom polynomials,, in Finite Fields and Applications, (2001), 37. doi: 10.1007/978-3-642-56755-1_4. Google Scholar

[4]

L. Budaghyan and T. Helleseth, New commutative semifields defined by new PN multinomials,, Crypt. Commun., 3 (2011), 1. doi: 10.1007/s12095-010-0022-2. Google Scholar

[5]

L. Budaghyan and T. Helleseth, On isotopisms of commutative presemifields and CCZ-equivalence of functions,, Int. J. Found. Comput. Sci., 22 (2011), 1243. doi: 10.1142/S0129054111008684. Google Scholar

[6]

C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems,, Des. Codes Crypt., 15 (1998), 125. doi: 10.1023/A:1008344232130. Google Scholar

[7]

C. Carlet and C. Ding, Highly nonlinear mappings,, J. Complexity, 20 (2004), 205. doi: 10.1016/j.jco.2003.08.008. Google Scholar

[8]

R. S. Coulter and M. Henderson, Commutative presemifields and semifields,, Adv. Math., 217 (2008), 282. doi: 10.1016/j.aim.2007.07.007. Google Scholar

[9]

R. S. Coulter and R. W. Matthews, Planar functions and planes of Lenz-Barlotti class II,, Des. Codes Crypt., 10 (1997), 167. doi: 10.1023/A:1008292303803. Google Scholar

[10]

T. W. Cusick, C. Ding and A. R. Renvall, Stream Ciphers and Number Theory,, Elsevier, (2004). Google Scholar

[11]

P. Dembowski and T. G. Ostrom, Planes of order $n$ with collineation groups of order $n^2$,, Math. Z., 103 (1968), 239. doi: 10.1007/BF01111042. Google Scholar

[12]

C. Ding and J. Yin, Signal sets from functions with optimum nonlinearity,, IEEE Trans. Commun., 55 (2007), 936. doi: 10.1109/TCOMM.2007.894113. Google Scholar

[13]

The GAP Group, GAP - Groups, Algorithms and Programming,, Version 4.5.5, (2012). Google Scholar

[14]

K. J. Horadam, Hadamard Matrices and their Applications,, Princeton Univ. Press, (2007). Google Scholar

[15]

W. Jia, X. Zeng, T. Helleseth and C. Li, A class of binomial bent functions over the finite fields of odd characteristic,, IEEE Trans. Inf. Theory, 58 (2012), 6054. doi: 10.1109/TIT.2012.2199736. Google Scholar

[16]

D. E. Knuth, Finite semifields and projective planes,, J. Algebra, 2 (1965), 182. doi: 10.1016/0021-8693(65)90018-9. Google Scholar

[17]

G. Kyureghyan and A. Pott, Some theorems on planar mappings,, in Arithmetic of Finite Fields, (2008), 117. doi: 10.1007/978-3-540-69499-1_10. Google Scholar

[18]

G. Marino and O. Polverino, On the nuclei of a finite semifield,, Theory Appl. Finite Fields. Contemp. Math., 579 (2012), 123. doi: 10.1090/conm/579/11525. Google Scholar

[19]

A. Roy and A. J. Scott, Weighted complex projective 2-designs from bases: Optimal state determination by orthogonal measurements,, J. Math. Phys., 48 (2007), 1. doi: 10.1063/1.2748617. Google Scholar

[20]

J. Wedderburn, A theorem on finite algebras,, Trans. Amer. Math. Soc., 6 (1903), 349. doi: 10.2307/1986226. Google Scholar

[21]

Y. Zhou and A. Pott, A new family of semifields with 2 parameters,, Adv. Math., 234 (2013), 43. doi: 10.1016/j.aim.2012.10.014. Google Scholar

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