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On the correlation measures of orders $ 3 $ and $ 4 $ of binary sequence of period $ p^2 $ derived from Fermat quotients
Research Center for Number Theory and Its Applications, School of Mathematics, Northwest University, Xi'an 710127, China |
$ p $ |
$ n $ |
$ (n, p) = 1 $ |
$ q_p(n) $ |
$ q_p(n)\equiv \frac{n^{p-1}-1}{p} \ (\bmod\ p), \quad 0\leq q_p(n)\leq p-1. $ |
$ q_p(kp) = 0 $ |
$ k\in \mathbb{Z} $ |
$ E_{p^2} = \left(e_0, e_1, \cdots, e_{p^2-1}\right)\in \{0, 1\}^{p^2} $ |
$ \begin{equation*} \begin{split} e_{n} = \left\{\begin{array}{ll} 0, & \hbox{if }\ 0\leq \frac{q_p(n)}{p}<\frac{1}{2}, \\ 1, & \hbox{if }\ \frac{1}{2}\leq \frac{q_p(n)}{p}<1, \end{array} \right. \end{split} \end{equation*} $ |
$ 2 $ |
$ 3 $ |
$ 4 $ |
References:
[1] |
H. Aly and A. Winterhof,
Boolean functions derived from Fermat quotients, Cryptogr. Commun., 3 (2011), 165-174.
doi: 10.1007/s12095-011-0043-5. |
[2] |
M.-C. Chang,
Short character sums with Fermat quotients, Acta Arith., 152 (2012), 23-38.
doi: 10.4064/aa152-1-3. |
[3] |
Z. Chen, Trace representation and linear complexity of binary sequences derived from Fermat quotients, Sci. China Inf. Sci., 57 (2014), 112109, 10 pp.
doi: 10.1007/s11432-014-5092-x. |
[4] |
Z. Chen and X. Du,
On the linear complexity of binary threshold sequences derived from Fermat quotients, Des. Codes Cryptogr., 67 (2013), 317-323.
doi: 10.1007/s10623-012-9608-3. |
[5] |
Z. Chen, L. Hu and X. Du, Linear complexity of some binary sequences derived from Fermat quotients, China Commun., 9 (2012), 105-108. Google Scholar |
[6] |
Z. Chen, A. Ostafe and A. Winterhof, Structure of pseudorandom numbers derived from Fermat quotients, In Arithmetic of Finite Fields, Lecture Notes in Computer Science 6087, Springer, Berlin, (2010), 73-85.
doi: 10.1007/978-3-642-13797-6_6. |
[7] |
Z. Chen and A. Winterhof,
Interpolation of Fermat quotients, SIAM J. Discr. Math., 28 (2014), 1-7.
doi: 10.1137/130907951. |
[8] |
X. Du, A. Klapper and Z. Chen,
Linear complexity of pseudorandom sequences generated by Fermat quotients and their generalizations, Inform. Process. Lett., 112 (2012), 233-237.
doi: 10.1016/j.ipl.2011.11.017. |
[9] |
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. |
[10] |
D. R. Heath-Brown, An estimate for Heilbronn's exponential sum, In Analytic Number Theory, Proceedings of a Conference in Honor of Heini Halberstam, Progr. Math., Birkhäuser, Boston, 139 (1996), 451-463. |
[11] |
C. Mauduit and A. Sárközy,
On finite pseudorandom binary sequencs I: Measure of pseudorandomness, the Legendre symbol, Acta Arith., 82 (1997), 365-377.
doi: 10.4064/aa-82-4-365-377. |
[12] |
A. Ostafe and I. E. Shparlinski,
Pseudorandomness and dynamics of Fermat quotients, SIAM J. Discr. Math., 25 (2011), 50-71.
doi: 10.1137/100798466. |
[13] |
I. E. Shparlinskii,
Fermat quotients: Exponential sums, value set and primitive roots, Bull. Lond. Math. Soc., 43 (2011), 1228-1238.
doi: 10.1112/blms/bdr058. |
[14] |
I. E. Shparlinski,
Character sums with Fermat quotients, Quart. J. Math., 62 (2011), 1031-1043.
doi: 10.1093/qmath/haq028. |
[15] |
I. E. Shparlinski,
Bounds of multiplicative character sums with Fermat quotients of primes, Bull. Aust. Math. Soc., 83 (2011), 456-462.
doi: 10.1017/S000497271000198X. |
show all references
References:
[1] |
H. Aly and A. Winterhof,
Boolean functions derived from Fermat quotients, Cryptogr. Commun., 3 (2011), 165-174.
doi: 10.1007/s12095-011-0043-5. |
[2] |
M.-C. Chang,
Short character sums with Fermat quotients, Acta Arith., 152 (2012), 23-38.
doi: 10.4064/aa152-1-3. |
[3] |
Z. Chen, Trace representation and linear complexity of binary sequences derived from Fermat quotients, Sci. China Inf. Sci., 57 (2014), 112109, 10 pp.
doi: 10.1007/s11432-014-5092-x. |
[4] |
Z. Chen and X. Du,
On the linear complexity of binary threshold sequences derived from Fermat quotients, Des. Codes Cryptogr., 67 (2013), 317-323.
doi: 10.1007/s10623-012-9608-3. |
[5] |
Z. Chen, L. Hu and X. Du, Linear complexity of some binary sequences derived from Fermat quotients, China Commun., 9 (2012), 105-108. Google Scholar |
[6] |
Z. Chen, A. Ostafe and A. Winterhof, Structure of pseudorandom numbers derived from Fermat quotients, In Arithmetic of Finite Fields, Lecture Notes in Computer Science 6087, Springer, Berlin, (2010), 73-85.
doi: 10.1007/978-3-642-13797-6_6. |
[7] |
Z. Chen and A. Winterhof,
Interpolation of Fermat quotients, SIAM J. Discr. Math., 28 (2014), 1-7.
doi: 10.1137/130907951. |
[8] |
X. Du, A. Klapper and Z. Chen,
Linear complexity of pseudorandom sequences generated by Fermat quotients and their generalizations, Inform. Process. Lett., 112 (2012), 233-237.
doi: 10.1016/j.ipl.2011.11.017. |
[9] |
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. |
[10] |
D. R. Heath-Brown, An estimate for Heilbronn's exponential sum, In Analytic Number Theory, Proceedings of a Conference in Honor of Heini Halberstam, Progr. Math., Birkhäuser, Boston, 139 (1996), 451-463. |
[11] |
C. Mauduit and A. Sárközy,
On finite pseudorandom binary sequencs I: Measure of pseudorandomness, the Legendre symbol, Acta Arith., 82 (1997), 365-377.
doi: 10.4064/aa-82-4-365-377. |
[12] |
A. Ostafe and I. E. Shparlinski,
Pseudorandomness and dynamics of Fermat quotients, SIAM J. Discr. Math., 25 (2011), 50-71.
doi: 10.1137/100798466. |
[13] |
I. E. Shparlinskii,
Fermat quotients: Exponential sums, value set and primitive roots, Bull. Lond. Math. Soc., 43 (2011), 1228-1238.
doi: 10.1112/blms/bdr058. |
[14] |
I. E. Shparlinski,
Character sums with Fermat quotients, Quart. J. Math., 62 (2011), 1031-1043.
doi: 10.1093/qmath/haq028. |
[15] |
I. E. Shparlinski,
Bounds of multiplicative character sums with Fermat quotients of primes, Bull. Aust. Math. Soc., 83 (2011), 456-462.
doi: 10.1017/S000497271000198X. |
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