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The cross-correlation distribution of a $p$-ary $m$-sequence of period $p^{2k}-1$ and its decimated sequence by $\frac{(p^{k}+1)^{2}}{2(p^{e}+1)}$
On the dual of (non)-weakly regular bent functions and self-dual bent functions
1. | Faculty of Mathematics, Otto-von-Guericke University, Universitätsplatz 2, 39106, Magdeburg, Germany, Germany |
2. | MDBF, Sabanci University, Orhanlı, Tuzla 34956, İstanbul, Turkey |
References:
[1] |
C. Carlet, On the secondary constructions of resilient and bent functions, in Coding, Cryptography and Combinatorics, Birkhäuser Basel, 2004, 3-28. |
[2] |
C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156.
doi: 10.1023/A:1008344232130. |
[3] |
C. Carlet, L. E. Danielsen, M. G. Parker and P. Solé, Self-dual bent functions, Int. J. Inform. Coding Theory, 1 (2010), 384-399.
doi: 10.1504/IJICOT.2010.032864. |
[4] |
C. Carlet, H. Dobbertin and G. Leander, Normal extensions of bent functions, IEEE Trans. Inform. Theory, 50 (2004), 2880-2885.
doi: 10.1109/TIT.2004.836681. |
[5] |
A. Çeşmelioǧlu, G. McGuire and W. Meidl, A construction of weakly and non-weakly regular bent functions, J. Comb. Theory Ser. A, 119 (2012), 420-429.
doi: 10.1016/j.jcta.2011.10.002. |
[6] |
A. Çeşmelioǧlu and W. Meidl, A construction of bent functions from plateaued functions, Des. Codes Cryptogr., 66 (2013), 231-242.
doi: 10.1007/s10623-012-9686-2. |
[7] |
Y. M. Chee, Y. Tan and X. D. Zhang, Strongly regular graphs constructed from $p$-ary bent functions, J. Algebr. Comb., 34 (2011), 251-266.
doi: 10.1007/s10801-010-0270-4. |
[8] |
Y. Edel and A. Pott, On the equivalence of nonlinear functions, in NATO Advanced Research Workshop on Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes, Amsterdam, 2009, 87-103. |
[9] |
K. Garaschuk and P. Lisoněk, On ternary Kloosterman sums modulo 12, Finite Fields Appl., 14 (2008), 1083-1090.
doi: 10.1016/j.ffa.2008.07.002. |
[10] |
F. Göloǧlu, G. McGuire and R. Moloney, Ternary Kloosterman sums modulo $18$ using Stickelberger's theorem, in Proceedings of SETA 2010 (eds. C. Carlet and A. Pott), Springer-Verlag, Berlin, 2010, 196-203.
doi: 10.1007/978-3-642-15874-2_16. |
[11] |
T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inform. Theory, 52 (2006), 2018-2032.
doi: 10.1109/TIT.2006.872854. |
[12] |
T. Helleseth and A. Kholosha, New binomial bent functions over the finite fields of odd characteristic, IEEE Trans. Inform. Theory, 56 (2010), 4646-4652.
doi: 10.1109/TIT.2010.2055130. |
[13] |
T. Helleseth and A. Kholosha, Crosscorrelation of m-sequences, exponential sums, bent functions and Jacobsthal sums, Cryptogr. Commun., 3 (2011), 281-291.
doi: 10.1007/s12095-011-0048-0. |
[14] |
X. D. Hou, Classification of self dual quadratic bent functions, Des. Codes Cryptogr., 63 (2012), 183-198.
doi: 10.1007/s10623-011-9544-7. |
[15] |
K. P. Kononen, M. J. Rinta-aho and K. O. Väänänen, On integer values of Kloosterman sums, IEEE Trans. Inform. Theory, 56 (2010), 4011-4013.
doi: 10.1109/TIT.2010.2050806. |
[16] |
N. G. Leander, Monomial bent functions, IEEE Trans. Inform. Theory, 52 (2006), 738-743.
doi: 10.1109/TIT.2005.862121. |
[17] |
R. Lidl and H. Niederreiter, Finite Fields, Second edition, Cambridge Univ. Press, Cambridge, 1997. |
[18] |
Y. Tan, A. Pott and T. Feng, Strongly regular graphs associated with ternary bent functions, J. Comb. Theory Ser. A, 117 (2010), 668-682.
doi: 10.1016/j.jcta.2009.05.003. |
[19] |
Y. Tan, J. Yang and X. Zhang, A recursive approach to construct $p$-ary bent functions which are not weakly regular, in Proceedings of IEEE International Conference on Information Theory and Information Security, Beijing, 2010, 156-159. |
show all references
References:
[1] |
C. Carlet, On the secondary constructions of resilient and bent functions, in Coding, Cryptography and Combinatorics, Birkhäuser Basel, 2004, 3-28. |
[2] |
C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156.
doi: 10.1023/A:1008344232130. |
[3] |
C. Carlet, L. E. Danielsen, M. G. Parker and P. Solé, Self-dual bent functions, Int. J. Inform. Coding Theory, 1 (2010), 384-399.
doi: 10.1504/IJICOT.2010.032864. |
[4] |
C. Carlet, H. Dobbertin and G. Leander, Normal extensions of bent functions, IEEE Trans. Inform. Theory, 50 (2004), 2880-2885.
doi: 10.1109/TIT.2004.836681. |
[5] |
A. Çeşmelioǧlu, G. McGuire and W. Meidl, A construction of weakly and non-weakly regular bent functions, J. Comb. Theory Ser. A, 119 (2012), 420-429.
doi: 10.1016/j.jcta.2011.10.002. |
[6] |
A. Çeşmelioǧlu and W. Meidl, A construction of bent functions from plateaued functions, Des. Codes Cryptogr., 66 (2013), 231-242.
doi: 10.1007/s10623-012-9686-2. |
[7] |
Y. M. Chee, Y. Tan and X. D. Zhang, Strongly regular graphs constructed from $p$-ary bent functions, J. Algebr. Comb., 34 (2011), 251-266.
doi: 10.1007/s10801-010-0270-4. |
[8] |
Y. Edel and A. Pott, On the equivalence of nonlinear functions, in NATO Advanced Research Workshop on Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes, Amsterdam, 2009, 87-103. |
[9] |
K. Garaschuk and P. Lisoněk, On ternary Kloosterman sums modulo 12, Finite Fields Appl., 14 (2008), 1083-1090.
doi: 10.1016/j.ffa.2008.07.002. |
[10] |
F. Göloǧlu, G. McGuire and R. Moloney, Ternary Kloosterman sums modulo $18$ using Stickelberger's theorem, in Proceedings of SETA 2010 (eds. C. Carlet and A. Pott), Springer-Verlag, Berlin, 2010, 196-203.
doi: 10.1007/978-3-642-15874-2_16. |
[11] |
T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inform. Theory, 52 (2006), 2018-2032.
doi: 10.1109/TIT.2006.872854. |
[12] |
T. Helleseth and A. Kholosha, New binomial bent functions over the finite fields of odd characteristic, IEEE Trans. Inform. Theory, 56 (2010), 4646-4652.
doi: 10.1109/TIT.2010.2055130. |
[13] |
T. Helleseth and A. Kholosha, Crosscorrelation of m-sequences, exponential sums, bent functions and Jacobsthal sums, Cryptogr. Commun., 3 (2011), 281-291.
doi: 10.1007/s12095-011-0048-0. |
[14] |
X. D. Hou, Classification of self dual quadratic bent functions, Des. Codes Cryptogr., 63 (2012), 183-198.
doi: 10.1007/s10623-011-9544-7. |
[15] |
K. P. Kononen, M. J. Rinta-aho and K. O. Väänänen, On integer values of Kloosterman sums, IEEE Trans. Inform. Theory, 56 (2010), 4011-4013.
doi: 10.1109/TIT.2010.2050806. |
[16] |
N. G. Leander, Monomial bent functions, IEEE Trans. Inform. Theory, 52 (2006), 738-743.
doi: 10.1109/TIT.2005.862121. |
[17] |
R. Lidl and H. Niederreiter, Finite Fields, Second edition, Cambridge Univ. Press, Cambridge, 1997. |
[18] |
Y. Tan, A. Pott and T. Feng, Strongly regular graphs associated with ternary bent functions, J. Comb. Theory Ser. A, 117 (2010), 668-682.
doi: 10.1016/j.jcta.2009.05.003. |
[19] |
Y. Tan, J. Yang and X. Zhang, A recursive approach to construct $p$-ary bent functions which are not weakly regular, in Proceedings of IEEE International Conference on Information Theory and Information Security, Beijing, 2010, 156-159. |
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