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On the security of the WOTS-PRF signature scheme
Some two-weight and three-weight linear codes
1. | Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China |
2. | Department of Mathematics, KAIST, Daejeon, 305-701, Korea |
3. | School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China |
$\Bbb F_q$ |
$q = p^m$ |
$p$ |
$m$ |
$t$ |
$D \subset \Bbb F_q^t$ |
$\mbox{Tr}_m$ |
$\Bbb F_q$ |
$\Bbb F_p$ |
$p$ |
$\mathcal C_D$ |
$ \mathcal C_D = \{\textbf{c}(a_1,a_2, ..., a_t): a_1, a_2, ..., a_t ∈ \Bbb F_{p^m}\}, $ |
$\textbf{c}(a_1,a_2, ..., a_t) = \big(\mbox{Tr}_m(a_1x_1+a_2x_2+···+a_tx_t)\big)_{(x_1,x_2, ..., x_t)∈ D}.$ |
$\mathcal C_D$ |
$D = \{(x_1,x_2, ..., x_t) ∈ \Bbb F_q^t \setminus \{(0,0, ..., 0)\}: \mbox{Tr}_m(x_1^2+x_2^2+···+x_t^2) = 0\}$ |
$D = \{(x_1,x_2, ..., x_t) ∈ \Bbb F_q^t: \mbox{Tr}_m(x_1^2+x_2^2+···+x_t^2) = 1\}$ |
$\mathcal C_D$ |
$tm$ |
$tm$ |
$\mathcal C_D$ |
$\mathcal C_D$ |
References:
[1] |
L. D. Baumert and R. J. McEliece,
Weights of irreducible cyclic codes, Inf. Control, 20 (1972), 158-175.
doi: 10.1016/S0019-9958(72)90354-3. |
[2] |
B. Berndt, R. Evans and K. Williams, Gauss and Jacobi Sums, John Wiley & Sons company, New York, 1998.
![]() |
[3] |
A. R. Calderbank and J. M. Goethals,
Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152.
|
[4] |
A. R. Calderbank and W. M. Kantor,
The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122.
doi: 10.1112/blms/18.2.97. |
[5] |
C. Carlet, C. Ding and J. Yuan,
Linear codes from perfect nonlinear mappings and their
secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.
doi: 10.1109/TIT.2005.847722. |
[6] |
C. Ding,
Codes from Difference Sets, World Scientific, Singapore, 2015. |
[7] |
C. Ding,
Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275.
doi: 10.1109/TIT.2015.2420118. |
[8] |
C. Ding, T. Helleseth, T. Klove and X. Wang,
A general construction of authentication codes, IEEE Trans. Inf. Theory, 53 (2007), 2229-2235.
doi: 10.1109/TIT.2007.896872. |
[9] |
C. Ding, C. Li, N. Li and Z. Zhou,
Three-weight cyclic codes and their weight distributions, Discr. Math., 339 (2016), 415-427.
doi: 10.1016/j.disc.2015.09.001. |
[10] |
C. Ding, Y. Liu, C. Ma and L. Zeng,
The weight distributions of the duals of cyclic codes
with two zeros, IEEE Trans. Inf. Theory, 57 (2011), 8000-8006.
doi: 10.1109/TIT.2011.2165314. |
[11] |
C. Ding, J. Luo and H. Niederreiter, Two-weight codes punctured from irreducible cyclic
codes, in Proceedings of the First Worshop on Coding and Cryptography (eds. Y. Li, et al. ),
World Scientific, Singapore, 4 (2008), 119-124.
doi: 10.1142/9789812832245_0009. |
[12] |
C. Ding and H. Niederreiter,
Cyclotomic linear codes of order 3, IEEE Trans. Inf. Theory, 53 (2007), 2274-2277.
doi: 10.1109/TIT.2007.896886. |
[13] |
C. Ding and X. Wang,
A coding theory construction of new systematic authentication codes, Theor. Comp. Sci., 330 (2005), 81-99.
doi: 10.1016/j.tcs.2004.09.011. |
[14] |
C. Ding and J. Yang,
Hamming weights in irreducible cyclic codes, Discr. Math., 313 (2013), 434-446.
doi: 10.1016/j.disc.2012.11.009. |
[15] |
C. Ding and J. Yin,
Algebraic constructions of constant composition codes, IEEE Trans. Inf. Theory, 51 (2005), 1585-1589.
doi: 10.1109/TIT.2005.844087. |
[16] |
K. Ding and C. Ding, Binary linear codes with three weights, IEEE Comm. Letters, 18 (2014), 1879-1882. Google Scholar |
[17] |
K. Ding and C. Ding,
A class of two-weight and three-weight codes and their applications in
secret sharing, IEEE Trans. Inf. Theory, 61 (2015), 5835-5842.
doi: 10.1109/TIT.2015.2473861. |
[18] |
C. Li, Q. Yue and F. W. Fu,
A construction of several classes of two-weight and three-weight
linear codes, Appl. Alg. Eng. Comm. Comp., 28 (2017), 11-30.
doi: 10.1007/s00200-016-0297-4. |
[19] |
S. Li, T. Feng and G. Ge,
On the weight distribution of cyclic codes with Niho exponents, IEEE Trans. Inf. Theory, 60 (2014), 3903-3912.
doi: 10.1109/TIT.2014.2318297. |
[20] |
R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley Publishing Inc., 1983.
![]() |
[21] |
J. Luo and T. Helleseth,
Constant composition codes as subcodes of cyclic codes, IEEE Trans. Inf. Theory, 57 (2011), 7482-7488.
doi: 10.1109/TIT.2011.2161631. |
[22] |
C. Ma, L. Zeng, Y. Liu, D. Feng and C. Ding,
The weight enumerator of a class of cyclic
codes, IEEE Trans. Inf. Theory, 57 (2011), 397-402.
doi: 10.1109/TIT.2010.2090272. |
[23] |
F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane,
Generalizations of Gleason's theorem
on weight enumerators of self-dual codes, IEEE Trans. Inf. Theory, 18 (1972), 794-805.
doi: 10.1109/tit.1972.1054898. |
[24] |
F. J. MacWilliams and N. J. A. Sloane,
The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977. |
[25] |
C. Tang, N. Li, Y. Qi, Z. Zhou and T. Helleseth,
Linear codes with two or three weights from
weakly regular bent functions, IEEE Trans. Inf. Theory, 62 (2016), 1166-1176.
doi: 10.1109/TIT.2016.2518678. |
[26] |
G. Vega,
The weight distribution of an extended class of reducible cyclic codes, IEEE Trans. Inf. Theory, 58 (2012), 4862-4869.
doi: 10.1109/TIT.2012.2193376. |
[27] |
M. Xiong,
The weight distributions of a class of cyclic codes, Finite Fields Appl., 18 (2012), 933-945.
doi: 10.1016/j.ffa.2012.06.001. |
[28] |
J. Yang, M. Xiong, C. Ding and J. Luo,
Weight distribution of a class of cyclic codes with
arbitrary number of zeros, IEEE Trans. Inf. Theory, 59 (2013), 5985-5993.
doi: 10.1109/TIT.2013.2266731. |
[29] |
S. Yang and Z. Yao,
Complete weight enumerators of a class of linear codes, Discr. Math., 340 (2017), 729-739.
doi: 10.1016/j.disc.2016.11.029. |
[30] |
S. Yang, X. Kong and C. Tang,
A construction of linear codes and their complete weight
enumerators, Finite Fields Appl., 48 (2017), 196-226.
doi: 10.1016/j.ffa.2017.08.001. |
[31] |
J. Yuan and C. Ding,
Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212.
doi: 10.1109/TIT.2005.860412. |
[32] |
X. Zeng, L. Hu, W. Jiang, Q. Yue and X. Cao,
The weight distribution of a class of p-ary
cyclic codes, Finite Fields Appl., 16 (2010), 56-73.
doi: 10.1016/j.ffa.2009.12.001. |
[33] |
Z. Zhou, N. Li, C. Fan and T. Helleseth,
Linear codes with two or three weights from quadratic
Bent functions, Des. Codes Cryptogr., 81 (2016), 283-295.
doi: 10.1007/s10623-015-0144-9. |
show all references
References:
[1] |
L. D. Baumert and R. J. McEliece,
Weights of irreducible cyclic codes, Inf. Control, 20 (1972), 158-175.
doi: 10.1016/S0019-9958(72)90354-3. |
[2] |
B. Berndt, R. Evans and K. Williams, Gauss and Jacobi Sums, John Wiley & Sons company, New York, 1998.
![]() |
[3] |
A. R. Calderbank and J. M. Goethals,
Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152.
|
[4] |
A. R. Calderbank and W. M. Kantor,
The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122.
doi: 10.1112/blms/18.2.97. |
[5] |
C. Carlet, C. Ding and J. Yuan,
Linear codes from perfect nonlinear mappings and their
secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.
doi: 10.1109/TIT.2005.847722. |
[6] |
C. Ding,
Codes from Difference Sets, World Scientific, Singapore, 2015. |
[7] |
C. Ding,
Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275.
doi: 10.1109/TIT.2015.2420118. |
[8] |
C. Ding, T. Helleseth, T. Klove and X. Wang,
A general construction of authentication codes, IEEE Trans. Inf. Theory, 53 (2007), 2229-2235.
doi: 10.1109/TIT.2007.896872. |
[9] |
C. Ding, C. Li, N. Li and Z. Zhou,
Three-weight cyclic codes and their weight distributions, Discr. Math., 339 (2016), 415-427.
doi: 10.1016/j.disc.2015.09.001. |
[10] |
C. Ding, Y. Liu, C. Ma and L. Zeng,
The weight distributions of the duals of cyclic codes
with two zeros, IEEE Trans. Inf. Theory, 57 (2011), 8000-8006.
doi: 10.1109/TIT.2011.2165314. |
[11] |
C. Ding, J. Luo and H. Niederreiter, Two-weight codes punctured from irreducible cyclic
codes, in Proceedings of the First Worshop on Coding and Cryptography (eds. Y. Li, et al. ),
World Scientific, Singapore, 4 (2008), 119-124.
doi: 10.1142/9789812832245_0009. |
[12] |
C. Ding and H. Niederreiter,
Cyclotomic linear codes of order 3, IEEE Trans. Inf. Theory, 53 (2007), 2274-2277.
doi: 10.1109/TIT.2007.896886. |
[13] |
C. Ding and X. Wang,
A coding theory construction of new systematic authentication codes, Theor. Comp. Sci., 330 (2005), 81-99.
doi: 10.1016/j.tcs.2004.09.011. |
[14] |
C. Ding and J. Yang,
Hamming weights in irreducible cyclic codes, Discr. Math., 313 (2013), 434-446.
doi: 10.1016/j.disc.2012.11.009. |
[15] |
C. Ding and J. Yin,
Algebraic constructions of constant composition codes, IEEE Trans. Inf. Theory, 51 (2005), 1585-1589.
doi: 10.1109/TIT.2005.844087. |
[16] |
K. Ding and C. Ding, Binary linear codes with three weights, IEEE Comm. Letters, 18 (2014), 1879-1882. Google Scholar |
[17] |
K. Ding and C. Ding,
A class of two-weight and three-weight codes and their applications in
secret sharing, IEEE Trans. Inf. Theory, 61 (2015), 5835-5842.
doi: 10.1109/TIT.2015.2473861. |
[18] |
C. Li, Q. Yue and F. W. Fu,
A construction of several classes of two-weight and three-weight
linear codes, Appl. Alg. Eng. Comm. Comp., 28 (2017), 11-30.
doi: 10.1007/s00200-016-0297-4. |
[19] |
S. Li, T. Feng and G. Ge,
On the weight distribution of cyclic codes with Niho exponents, IEEE Trans. Inf. Theory, 60 (2014), 3903-3912.
doi: 10.1109/TIT.2014.2318297. |
[20] |
R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley Publishing Inc., 1983.
![]() |
[21] |
J. Luo and T. Helleseth,
Constant composition codes as subcodes of cyclic codes, IEEE Trans. Inf. Theory, 57 (2011), 7482-7488.
doi: 10.1109/TIT.2011.2161631. |
[22] |
C. Ma, L. Zeng, Y. Liu, D. Feng and C. Ding,
The weight enumerator of a class of cyclic
codes, IEEE Trans. Inf. Theory, 57 (2011), 397-402.
doi: 10.1109/TIT.2010.2090272. |
[23] |
F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane,
Generalizations of Gleason's theorem
on weight enumerators of self-dual codes, IEEE Trans. Inf. Theory, 18 (1972), 794-805.
doi: 10.1109/tit.1972.1054898. |
[24] |
F. J. MacWilliams and N. J. A. Sloane,
The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977. |
[25] |
C. Tang, N. Li, Y. Qi, Z. Zhou and T. Helleseth,
Linear codes with two or three weights from
weakly regular bent functions, IEEE Trans. Inf. Theory, 62 (2016), 1166-1176.
doi: 10.1109/TIT.2016.2518678. |
[26] |
G. Vega,
The weight distribution of an extended class of reducible cyclic codes, IEEE Trans. Inf. Theory, 58 (2012), 4862-4869.
doi: 10.1109/TIT.2012.2193376. |
[27] |
M. Xiong,
The weight distributions of a class of cyclic codes, Finite Fields Appl., 18 (2012), 933-945.
doi: 10.1016/j.ffa.2012.06.001. |
[28] |
J. Yang, M. Xiong, C. Ding and J. Luo,
Weight distribution of a class of cyclic codes with
arbitrary number of zeros, IEEE Trans. Inf. Theory, 59 (2013), 5985-5993.
doi: 10.1109/TIT.2013.2266731. |
[29] |
S. Yang and Z. Yao,
Complete weight enumerators of a class of linear codes, Discr. Math., 340 (2017), 729-739.
doi: 10.1016/j.disc.2016.11.029. |
[30] |
S. Yang, X. Kong and C. Tang,
A construction of linear codes and their complete weight
enumerators, Finite Fields Appl., 48 (2017), 196-226.
doi: 10.1016/j.ffa.2017.08.001. |
[31] |
J. Yuan and C. Ding,
Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212.
doi: 10.1109/TIT.2005.860412. |
[32] |
X. Zeng, L. Hu, W. Jiang, Q. Yue and X. Cao,
The weight distribution of a class of p-ary
cyclic codes, Finite Fields Appl., 16 (2010), 56-73.
doi: 10.1016/j.ffa.2009.12.001. |
[33] |
Z. Zhou, N. Li, C. Fan and T. Helleseth,
Linear codes with two or three weights from quadratic
Bent functions, Des. Codes Cryptogr., 81 (2016), 283-295.
doi: 10.1007/s10623-015-0144-9. |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
|
Frequency |
0 | 1 |
|
Frequency |
0 | 1 |
Frequency | |
0 | 1 |
Frequency | |
0 | 1 |
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