doi: 10.3934/amc.2021038
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On the pseudorandom properties of $ k $-ary Sidel'nikov sequences

Research Center for Number Theory and Its Applications, School of Mathematics, Northwest University, Xi'an 710127, China

* Corresponding author: Yixin Ren

Received  March 2021 Early access August 2021

In 2002 Mauduit and Sárközy started to study finite sequences of
$ k $
symbols
$ E_{N} = \left(e_{1},e_{2},\cdots,e_{N}\right)\in \mathcal{A}^{N}, $
where
$ \mathcal{A} = \left\{a_{1},a_{2},\cdots,a_{k}\right\}(k\in \mathbb{N},k\geq 2) $
is a finite set of
$ k $
symbols. Later many pseudorandom sequences of
$ k $
symbols have been given and studied by using number theoretic methods. In this paper we study the pseudorandom properties of the
$ k $
-ary Sidel'nikov sequences with length
$ q-1 $
by using the estimates for certain character sums with exponential function, where
$ q $
is a prime power. Our results show that Sidel'nikov sequences enjoy good well-distribution measure and correlation measure. Furthermore, we prove that the set of size
$ \phi(q-1) $
of
$ k $
-ary Sidel'nikov sequences is collision free and possesses the strict avalanche effect property provided that
$ k = o(q^{\frac{1}{4}}) $
, where
$ \phi $
denotes Euler's totient function.
Citation: Huaning Liu, Yixin Ren. On the pseudorandom properties of $ k $-ary Sidel'nikov sequences. Advances in Mathematics of Communications, doi: 10.3934/amc.2021038
References:
[1]

R. Ahlswede, C. Mauduit and A. Sárközy, Large families of pseudorandom sequences of $k$ symbols and their complexity-Part Ⅰ, in General Theory of Information Transfer and Combinatorics, Lecture Notes in Comput. Sci., Vol. 4123, Springer, Berlin, 2006,293–307. doi: 10.1007/11889342_16.  Google Scholar

[2]

R. Ahlswede, C. Mauduit and A. Sárközy, Large families of pseudorandom sequences of $k$ symbols and their complexity-Part Ⅱ, in General Theory of Information Transfer and Combinatorics, Lecture Notes in Comput. Sci., Vol. 4123, Springer, Berlin, 2006,308–325. doi: 10.1007/11889342_17.  Google Scholar

[3]

S. Alaca and G. Millar, Character values of the Sidelnikov-Lempel-Cohn-Eastman sequences, Cryptogr. Commun., 9 (2017), 665-682.  doi: 10.1007/s12095-016-0208-3.  Google Scholar

[4]

H. Aly and W. Meidl, On the linear complexity and $k$-error linear complexity over $\mathbb{F}_p$ of the $d$-ary Sidelnikov sequence, IEEE Trans. Inform. Theory, 53 (2007), 4755-4761.  doi: 10.1109/TIT.2007.909129.  Google Scholar

[5]

H. Aly and A. Winterhof, On the $k$-error linear complexity over $\mathbb{F}_p$ of Legendre and Sidelnikov sequences, Des. Codes Cryptogr., 40 (2006), 369-374.  doi: 10.1007/s10623-006-0023-5.  Google Scholar

[6]

H. Alzer and S. Koumandos, On a trigonometric sum of Vinogradov, J. Number Theory, 105 (2004), 251-261.  doi: 10.1016/j.jnt.2003.10.003.  Google Scholar

[7]

N. Brandstätter and W. Meidl, On the linear complexity of Sidelnikov sequences over $ \mathbb{F}_{d}$, in: Proceedings of SETA'06, Lecture Notes in Comput. Sci., Vol. 4086, Springer-Verlag, Berlin, Heidelberg, (2006), 47–60.  Google Scholar

[8]

N. Brandstätter and W. Meidl, On the linear complexity of Sidelnikov sequences over nonprime fields, J. Complexity, 24 (2008), 648-659.  doi: 10.1016/j.jco.2008.04.002.  Google Scholar

[9]

Z.-X. ChenX.-N. Du and C.-H. Wu, Pseudorandomness of certain sequences of $k$ symbols with length $pq$, J. Comput. Sci. Tech., 26 (2011), 276-282.  doi: 10.1007/s11390-011-9434-5.  Google Scholar

[10]

J.-S. ChungY.-S. KimT.-H. LimJ.-S. No and H. Chung, On some properties of $M$-ary Sidelnikov sequences, IEICE Trans. Fundamentals, E92-A (2009), 342-345.  doi: 10.1587/transfun.E92.A.342.  Google Scholar

[11]

J.-H. Chung and K. Yang, Bounds on the linear complexity and the $1$-error linear complexity over $\mathbb{F}_p$ of $M$-ary Sidelnikov sequences, in: Proceedings of SETA'06, Lecture Notes in Comput. Sci., Vol. 4086, Springer-Verlag, Berlin, Heidelberg, 2006, 74–87. doi: 10.1007/11863854_7.  Google Scholar

[12]

T. Cochrane, On a trigonometric inequality of Vinogradov, J. Number Theory, 27 (1987), 9-16.  doi: 10.1016/0022-314X(87)90045-X.  Google Scholar

[13]

C. Ding and J. Yin, Sets of optimal frequency-hopping sequences, IEEE Trans. Inform. Theory, 54 (2008), 3741-3745.  doi: 10.1109/TIT.2008.926410.  Google Scholar

[14]

E. Dobrowolski and K. S. Williams, An upper bound for the sum $\sum_{n = a+1}^{a+H}f(n)$ for a certain class of functions $f$, Proc. Amer. Math. Soc., 114 (1992), 29-35.  doi: 10.1090/s0002-9939-1992-1068118-6.  Google Scholar

[15]

Y.-C. Eun, H.-Y. Song and G. M. Kyureghyan, One-error linear complexity over $\mathbb{F}_p$ of Sidelnikov sequences, in Proc. Int. Conf. Sequ. Appl., Seoul 2004, Lecture Notes in Comput. Sci., Vol. 3486, Springer-Verlag, Berlin, Heidelberg, 2005,154–165. doi: 10.1007/11423461_9.  Google Scholar

[16]

M. Z. GaraevF. LucaI. E. Shparlinski and A. Winterhof, On the lower bound of the linear complexity over $\mathbb{F}_p$ of Sidelnikov sequences, IEEE Trans. Inform. Theory, 52 (2006), 3299-3304.  doi: 10.1109/TIT.2006.876352.  Google Scholar

[17]

D. Gomez and A. Winterhof, Multiplicative character sums of Fermat quotients and pseudorandom sequences, Period. Math. Hungar., 64 (2012), 161-168.  doi: 10.1007/s10998-012-3747-1.  Google Scholar

[18]

Y. K. Han and K. Yang, On the Sidelnikov sequences as frequency-hopping sequences, IEEE Trans. Inform. Theory, 55 (2009), 4279-4285.  doi: 10.1109/TIT.2009.2025569.  Google Scholar

[19]

T. HellesethS.-H. Kim and J.-S. No, Linear complexity over $\mathbb{F}_p$ and trace representation of Lempel-Cohn-Eastman sequences, IEEE Trans. Inform. Theory, 49 (2003), 1548-1552.  doi: 10.1109/TIT.2003.811924.  Google Scholar

[20]

T. HellesethM. MaasJ. E. Mathiassen and T. Segers, Linear complexity over $\mathbb{F}_p$ of Sidelnikov sequences, IEEE Trans. Inform. Theory, 50 (2004), 2468-2472.  doi: 10.1109/TIT.2004.834854.  Google Scholar

[21]

Y. Kim, D. Kim and H. Song, Some properties of $2$-dimensional array structure of Sidelnikov sequences of period $q^{d}-1$, Proceedings of IWSDA'13, (2013), 20–23. Google Scholar

[22]

Y.-J. Kim and H.-Y. Song, Cross correlation of Sidel'nikov sequences and their constant multiples, IEEE Trans. Inform. Theory, 53 (2007), 1220-1224.  doi: 10.1109/TIT.2006.890723.  Google Scholar

[23]

Y.-S. KimJ.-S. ChungJ.-S. No and H. Chung, On the autocorrelation distributions of Sidelnikov sequences, IEEE Trans. Inform. Theory, 51 (2005), 3303-3307.  doi: 10.1109/TIT.2005.853310.  Google Scholar

[24]

Y.-S. Kim, J.-S. Chung, J.-S. No and H. Chung, On the linear complexity over $\mathbb{F}_p$ of $M$-ary Sidelnikov sequences, in Proc. IEEE ISIT 2005, Sept. (2005), 2007–2011. doi: 10.1109/ISIT.2005.1523697.  Google Scholar

[25]

Y.-S. Kim, J.-S. Chung, J.-S. No and H. Chung, Linear complexity over $\mathbb{F}_p$ of ternary Sidelnikov sequences, in: Proceedings of SETA'06, Lecture Notes in Comput. Sci., Vol. 4086, Springer-Verlag, Berlin, Heidelberg, (2006), 61–73. doi: 10.1007/11863854_6.  Google Scholar

[26]

Y.-S. KimJ.-S. ChungJ.-S. No and H. Chung, New families of $M$-ary sequences with low correlation constructed from Sidelnikov sequences, IEEE Trans. Inform. Theory, 54 (2008), 3768-3774.  doi: 10.1109/TIT.2008.926428.  Google Scholar

[27]

Y.-T. KimD. S. Kim and H.-Y. Song, New $M$-ary sequence families with low correlation from the array structure of Sidelnikov sequences, IEEE Trans. Inform. Theory, 61 (2015), 655-670.  doi: 10.1109/TIT.2014.2371461.  Google Scholar

[28]

G. M. Kyureghyan and A. Pott, On the linear complexity of the Sidelnikov-Lempel-Cohn-Eastman sequences, Des. Codes Cryptogr., 29 (2003), 149-164.  doi: 10.1023/A:1024156525801.  Google Scholar

[29]

A. LempelM. Cohn and W. L. Eastman, A class of balanced binary sequences with optimal autocorrelation properties, IEEE Trans. Inform. Theory, 23 (1977), 38-42.  doi: 10.1109/TIT.1977.1055672.  Google Scholar

[30]

A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties, IEEE Trans. Inform. Theory, 20 (1974), 90-94.  doi: 10.1109/TIT.1974.1055169.  Google Scholar

[31]

H. Liu and B. Gao, A large family of pseudorandom sequences of $k$ symbols with length $pq$, Acta Arith., 181 (2017), 1-26.  doi: 10.4064/aa8452-5-2017.  Google Scholar

[32]

K.-H. Mak, More constructions of pseudorandom sequences of $k$ symbols, Finite Fields Appl., 25 (2014), 222-233.  doi: 10.1016/j.ffa.2013.09.006.  Google Scholar

[33]

C. Mauduit and A. Sárközy, On finite pseudorandom sequences of $k$ symbols, Indag. Math. (N.S.), 13 (2002), 89-101.  doi: 10.1016/S0019-3577(02)90008-X.  Google Scholar

[34]

W. Meidl and A. Winterhof, Some notes on the linear complexity of Sidelnikov-Lempel-Cohn-Eastman sequences, Des. Codes Cryptogr., 38 (2006), 159-178.  doi: 10.1007/s10623-005-6340-2.  Google Scholar

[35]

D. Peng and P. Fan, Lower bounds on the Hamming auto- and cross correlations of frequency-hopping sequences, IEEE Trans. Inform. Theory, 50 (2004), 2149-2154.  doi: 10.1109/TIT.2004.833362.  Google Scholar

[36]

V. M. Sidel'nikov, Some $k$-valued pseudo-random sequences and nearly equidistant codes, Problems Inform. Transmission, 5 (1969), 12-16.   Google Scholar

[37]

M. K. Song and H.-Y. Song, Hamming correlation properties of the array structure of Sidelnikov sequences, Des. Codes Cryptogr., 87 (2019), 2537-2551.  doi: 10.1007/s10623-019-00636-7.  Google Scholar

[38]

V. Tóth, Extension of the notion of collision and avalanche effect to sequences of $k$ symbols, Period. Math. Hungar., 65 (2012), 229-238.  doi: 10.1007/s10998-012-1005-1.  Google Scholar

[39]

I. M. Vinogradov, Elements of Number Theory, New York: Dover, 1954.  Google Scholar

[40]

H. B. Yu, Estimates of character sums with exponential function, Acta Arith., 97 (2001), 211-218.  doi: 10.4064/aa97-3-2.  Google Scholar

[41]

K. T. Yu, On a trigonometric inequality of Vinogradov, J. Number Theory, 49 (1994), 287-294.  doi: 10.1006/jnth.1994.1094.  Google Scholar

[42]

N. Y. Yu and G. Gong, New construction of $M$-ary sequence families with low correlation from the structure of Sidelnikov sequences, IEEE Trans. Inform. Theory, 56 (2010), 4061-4070.  doi: 10.1109/TIT.2010.2050793.  Google Scholar

show all references

References:
[1]

R. Ahlswede, C. Mauduit and A. Sárközy, Large families of pseudorandom sequences of $k$ symbols and their complexity-Part Ⅰ, in General Theory of Information Transfer and Combinatorics, Lecture Notes in Comput. Sci., Vol. 4123, Springer, Berlin, 2006,293–307. doi: 10.1007/11889342_16.  Google Scholar

[2]

R. Ahlswede, C. Mauduit and A. Sárközy, Large families of pseudorandom sequences of $k$ symbols and their complexity-Part Ⅱ, in General Theory of Information Transfer and Combinatorics, Lecture Notes in Comput. Sci., Vol. 4123, Springer, Berlin, 2006,308–325. doi: 10.1007/11889342_17.  Google Scholar

[3]

S. Alaca and G. Millar, Character values of the Sidelnikov-Lempel-Cohn-Eastman sequences, Cryptogr. Commun., 9 (2017), 665-682.  doi: 10.1007/s12095-016-0208-3.  Google Scholar

[4]

H. Aly and W. Meidl, On the linear complexity and $k$-error linear complexity over $\mathbb{F}_p$ of the $d$-ary Sidelnikov sequence, IEEE Trans. Inform. Theory, 53 (2007), 4755-4761.  doi: 10.1109/TIT.2007.909129.  Google Scholar

[5]

H. Aly and A. Winterhof, On the $k$-error linear complexity over $\mathbb{F}_p$ of Legendre and Sidelnikov sequences, Des. Codes Cryptogr., 40 (2006), 369-374.  doi: 10.1007/s10623-006-0023-5.  Google Scholar

[6]

H. Alzer and S. Koumandos, On a trigonometric sum of Vinogradov, J. Number Theory, 105 (2004), 251-261.  doi: 10.1016/j.jnt.2003.10.003.  Google Scholar

[7]

N. Brandstätter and W. Meidl, On the linear complexity of Sidelnikov sequences over $ \mathbb{F}_{d}$, in: Proceedings of SETA'06, Lecture Notes in Comput. Sci., Vol. 4086, Springer-Verlag, Berlin, Heidelberg, (2006), 47–60.  Google Scholar

[8]

N. Brandstätter and W. Meidl, On the linear complexity of Sidelnikov sequences over nonprime fields, J. Complexity, 24 (2008), 648-659.  doi: 10.1016/j.jco.2008.04.002.  Google Scholar

[9]

Z.-X. ChenX.-N. Du and C.-H. Wu, Pseudorandomness of certain sequences of $k$ symbols with length $pq$, J. Comput. Sci. Tech., 26 (2011), 276-282.  doi: 10.1007/s11390-011-9434-5.  Google Scholar

[10]

J.-S. ChungY.-S. KimT.-H. LimJ.-S. No and H. Chung, On some properties of $M$-ary Sidelnikov sequences, IEICE Trans. Fundamentals, E92-A (2009), 342-345.  doi: 10.1587/transfun.E92.A.342.  Google Scholar

[11]

J.-H. Chung and K. Yang, Bounds on the linear complexity and the $1$-error linear complexity over $\mathbb{F}_p$ of $M$-ary Sidelnikov sequences, in: Proceedings of SETA'06, Lecture Notes in Comput. Sci., Vol. 4086, Springer-Verlag, Berlin, Heidelberg, 2006, 74–87. doi: 10.1007/11863854_7.  Google Scholar

[12]

T. Cochrane, On a trigonometric inequality of Vinogradov, J. Number Theory, 27 (1987), 9-16.  doi: 10.1016/0022-314X(87)90045-X.  Google Scholar

[13]

C. Ding and J. Yin, Sets of optimal frequency-hopping sequences, IEEE Trans. Inform. Theory, 54 (2008), 3741-3745.  doi: 10.1109/TIT.2008.926410.  Google Scholar

[14]

E. Dobrowolski and K. S. Williams, An upper bound for the sum $\sum_{n = a+1}^{a+H}f(n)$ for a certain class of functions $f$, Proc. Amer. Math. Soc., 114 (1992), 29-35.  doi: 10.1090/s0002-9939-1992-1068118-6.  Google Scholar

[15]

Y.-C. Eun, H.-Y. Song and G. M. Kyureghyan, One-error linear complexity over $\mathbb{F}_p$ of Sidelnikov sequences, in Proc. Int. Conf. Sequ. Appl., Seoul 2004, Lecture Notes in Comput. Sci., Vol. 3486, Springer-Verlag, Berlin, Heidelberg, 2005,154–165. doi: 10.1007/11423461_9.  Google Scholar

[16]

M. Z. GaraevF. LucaI. E. Shparlinski and A. Winterhof, On the lower bound of the linear complexity over $\mathbb{F}_p$ of Sidelnikov sequences, IEEE Trans. Inform. Theory, 52 (2006), 3299-3304.  doi: 10.1109/TIT.2006.876352.  Google Scholar

[17]

D. Gomez and A. Winterhof, Multiplicative character sums of Fermat quotients and pseudorandom sequences, Period. Math. Hungar., 64 (2012), 161-168.  doi: 10.1007/s10998-012-3747-1.  Google Scholar

[18]

Y. K. Han and K. Yang, On the Sidelnikov sequences as frequency-hopping sequences, IEEE Trans. Inform. Theory, 55 (2009), 4279-4285.  doi: 10.1109/TIT.2009.2025569.  Google Scholar

[19]

T. HellesethS.-H. Kim and J.-S. No, Linear complexity over $\mathbb{F}_p$ and trace representation of Lempel-Cohn-Eastman sequences, IEEE Trans. Inform. Theory, 49 (2003), 1548-1552.  doi: 10.1109/TIT.2003.811924.  Google Scholar

[20]

T. HellesethM. MaasJ. E. Mathiassen and T. Segers, Linear complexity over $\mathbb{F}_p$ of Sidelnikov sequences, IEEE Trans. Inform. Theory, 50 (2004), 2468-2472.  doi: 10.1109/TIT.2004.834854.  Google Scholar

[21]

Y. Kim, D. Kim and H. Song, Some properties of $2$-dimensional array structure of Sidelnikov sequences of period $q^{d}-1$, Proceedings of IWSDA'13, (2013), 20–23. Google Scholar

[22]

Y.-J. Kim and H.-Y. Song, Cross correlation of Sidel'nikov sequences and their constant multiples, IEEE Trans. Inform. Theory, 53 (2007), 1220-1224.  doi: 10.1109/TIT.2006.890723.  Google Scholar

[23]

Y.-S. KimJ.-S. ChungJ.-S. No and H. Chung, On the autocorrelation distributions of Sidelnikov sequences, IEEE Trans. Inform. Theory, 51 (2005), 3303-3307.  doi: 10.1109/TIT.2005.853310.  Google Scholar

[24]

Y.-S. Kim, J.-S. Chung, J.-S. No and H. Chung, On the linear complexity over $\mathbb{F}_p$ of $M$-ary Sidelnikov sequences, in Proc. IEEE ISIT 2005, Sept. (2005), 2007–2011. doi: 10.1109/ISIT.2005.1523697.  Google Scholar

[25]

Y.-S. Kim, J.-S. Chung, J.-S. No and H. Chung, Linear complexity over $\mathbb{F}_p$ of ternary Sidelnikov sequences, in: Proceedings of SETA'06, Lecture Notes in Comput. Sci., Vol. 4086, Springer-Verlag, Berlin, Heidelberg, (2006), 61–73. doi: 10.1007/11863854_6.  Google Scholar

[26]

Y.-S. KimJ.-S. ChungJ.-S. No and H. Chung, New families of $M$-ary sequences with low correlation constructed from Sidelnikov sequences, IEEE Trans. Inform. Theory, 54 (2008), 3768-3774.  doi: 10.1109/TIT.2008.926428.  Google Scholar

[27]

Y.-T. KimD. S. Kim and H.-Y. Song, New $M$-ary sequence families with low correlation from the array structure of Sidelnikov sequences, IEEE Trans. Inform. Theory, 61 (2015), 655-670.  doi: 10.1109/TIT.2014.2371461.  Google Scholar

[28]

G. M. Kyureghyan and A. Pott, On the linear complexity of the Sidelnikov-Lempel-Cohn-Eastman sequences, Des. Codes Cryptogr., 29 (2003), 149-164.  doi: 10.1023/A:1024156525801.  Google Scholar

[29]

A. LempelM. Cohn and W. L. Eastman, A class of balanced binary sequences with optimal autocorrelation properties, IEEE Trans. Inform. Theory, 23 (1977), 38-42.  doi: 10.1109/TIT.1977.1055672.  Google Scholar

[30]

A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties, IEEE Trans. Inform. Theory, 20 (1974), 90-94.  doi: 10.1109/TIT.1974.1055169.  Google Scholar

[31]

H. Liu and B. Gao, A large family of pseudorandom sequences of $k$ symbols with length $pq$, Acta Arith., 181 (2017), 1-26.  doi: 10.4064/aa8452-5-2017.  Google Scholar

[32]

K.-H. Mak, More constructions of pseudorandom sequences of $k$ symbols, Finite Fields Appl., 25 (2014), 222-233.  doi: 10.1016/j.ffa.2013.09.006.  Google Scholar

[33]

C. Mauduit and A. Sárközy, On finite pseudorandom sequences of $k$ symbols, Indag. Math. (N.S.), 13 (2002), 89-101.  doi: 10.1016/S0019-3577(02)90008-X.  Google Scholar

[34]

W. Meidl and A. Winterhof, Some notes on the linear complexity of Sidelnikov-Lempel-Cohn-Eastman sequences, Des. Codes Cryptogr., 38 (2006), 159-178.  doi: 10.1007/s10623-005-6340-2.  Google Scholar

[35]

D. Peng and P. Fan, Lower bounds on the Hamming auto- and cross correlations of frequency-hopping sequences, IEEE Trans. Inform. Theory, 50 (2004), 2149-2154.  doi: 10.1109/TIT.2004.833362.  Google Scholar

[36]

V. M. Sidel'nikov, Some $k$-valued pseudo-random sequences and nearly equidistant codes, Problems Inform. Transmission, 5 (1969), 12-16.   Google Scholar

[37]

M. K. Song and H.-Y. Song, Hamming correlation properties of the array structure of Sidelnikov sequences, Des. Codes Cryptogr., 87 (2019), 2537-2551.  doi: 10.1007/s10623-019-00636-7.  Google Scholar

[38]

V. Tóth, Extension of the notion of collision and avalanche effect to sequences of $k$ symbols, Period. Math. Hungar., 65 (2012), 229-238.  doi: 10.1007/s10998-012-1005-1.  Google Scholar

[39]

I. M. Vinogradov, Elements of Number Theory, New York: Dover, 1954.  Google Scholar

[40]

H. B. Yu, Estimates of character sums with exponential function, Acta Arith., 97 (2001), 211-218.  doi: 10.4064/aa97-3-2.  Google Scholar

[41]

K. T. Yu, On a trigonometric inequality of Vinogradov, J. Number Theory, 49 (1994), 287-294.  doi: 10.1006/jnth.1994.1094.  Google Scholar

[42]

N. Y. Yu and G. Gong, New construction of $M$-ary sequence families with low correlation from the structure of Sidelnikov sequences, IEEE Trans. Inform. Theory, 56 (2010), 4061-4070.  doi: 10.1109/TIT.2010.2050793.  Google Scholar

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