Abstract
In this paper, we discuss the properties of the derivative of a special function, and propose a general approach to estimating a class of trigonometric sums based on the derivative of the special function. Then we apply the approach to three trigonometric sums and get three new estimates. Using the estimate of the first trigonometric sum, we deduce new upper and lower bounds of the arithmetic mean value for a trigonometric sum of Vinogradov. Using the estimate of the second trigonometric sum, we derive a new upper bound on the imbalance properties of Linear Feedback Shift Register subsequences. We also deduce a new lower bound on the nonlinearity of the Carlet–Feng vectorial Boolean function with the estimate of the third trigonometric sum.
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Acknowledgements
The authors would like to thank Dr. Ziran Tu for his valuable suggestions. The authors also thank the anonymous referees whose comments significantly improved the quality of the paper. This work was supported in part by the Open Foundation of Hubei Key Laboratory of Applied Mathematics (Hubei University) (Grant No. HBAM202101), in part by the National Natural Science Foundation of China (Grant No. 32061123007, 61902285), in part by the Natural Science Foundation of Hubei Province (Grant No. 2019CFB099), in part by and the Fundamental Research Funds for the Central Universities (Program No. 2662018QD043).
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Appendices
Appendix A
Proof of Lemma 2
Clearly, we have \(\varphi _{k}^{(1)}(t)=\frac{1}{2} h\left( \frac{\pi (k+1)}{M}\right) -\frac{1}{2} h\left( \frac{\pi (k+1)}{M}+t\right) +\frac{t}{2} h^{(1)}\left( \frac{\pi (k+1)}{M}+t\right) \) where \( 0\le t\le \frac{\pi }{M} \). Moreover,
Case 1: M is odd.
When \( -1\le k\le \frac{M-1}{2}-2 \), since \(0\le \frac{\pi (k+1)}{M} \le \frac{\pi (k+1)}{M}+t \le \frac{\pi (k+1)}{M}+\frac{\pi }{M}\le \frac{(M-1)\pi }{2 M} < \frac{\pi }{2}\), we have
from Theorem 1(3). Integrating twice and considering that \( \varphi _{k}(0)=\varphi _{k}^{(1)}(0)=0 \), we have
Therefore,
When \( k= \frac{M-1}{2} -1\), since \(\frac{(M-1)\pi }{2 M} = \frac{\pi (k+1)}{M} \le \frac{\pi (k+1)}{M}+t \le \frac{\pi (k+1)}{M}+\frac{\pi }{M} = \frac{(M+1)\pi }{2 M} \), we have
from Theorem 1(3). Integrating twice and considering that \( \varphi _{k}(0)=\varphi _{k}^{(1)}(0)=0 \), we have
Therefore,
When \( \frac{M-1}{2}\le k\le {M-2} \), since \(\frac{\pi }{2} \le \frac{(M+1)\pi }{2M}\le \frac{\pi (k+1)}{M} \le \frac{\pi (k+1)}{M}+t \le \frac{\pi (k+1)}{M}+\frac{\pi }{M} \le \pi \), we have
from Theorem 1(3). Integrating twice and considering that \( \varphi _{k}(0)=\varphi _{k}^{(1)}(0)=0 \), we have
Therefore,
From Theorem 1(1), we have
Using the comparison of a series and the Riemann integral, we have
then
So, we have
from (33) and Theorem 1(3). Similarly, we have
Therefore, we can get
from (32), (34) and (35), where \(M\ge 2\) is odd.
Case 2: M is even.
In this case, similarly as above, we have
where \(M\ge 2\) is even. The proof is completed. \(\square \)
Appendix B
Proof of Lemma 6
Let \(F(t)=t h(t)-\int _{0}^{t} h(x) d x\left( 0 \le t \le \frac{\pi }{M}\right) \). It is easy to see that \( F(0)=0 \) and \( F^{(1)}(t)=th^{(1)}(t) \), and we know that \( h^{(1)}(t) \) is strictly monotonically decreasing in \( [0,\pi ] \) from Theorem 1(3), so
Moreover,
Clearly,
Using the comparison of a series and the Riemann integral, we have
Then
Moreover,
Appendix C
Proof of Lemma 9
let \(F(t)=t h\left( \frac{\pi }{2}+t\right) -\int _{\frac{\pi }{2}}^{\frac{\pi }{2}+t} h(x) d x\left( 0 \le t \le \frac{\pi }{ 2M}\right) \). It is easy to see that \( F(0)=0 \) and \( F^{(1)}(t)=th^{(1)}(\frac{\pi }{2}+t) \), and we know that \( h^{(1)}(t) \) is strictly monotonically decreasing in \( [0,\pi ] \) from Theorem 1(3), so
Moreover,
Clearly,
Using the comparison of a series and the Riemann integral, we have
From Theorem 1(3), then
Clearly, \(h^{(1)}\left( \frac{\pi }{2}\right) =0\) and \(h^{(2)}\left( \frac{2\pi }{3}\right) =\frac{10 \sqrt{3}}{9}-\frac{243}{4 \pi ^{3}}\). Then we get
\(\square \)
Proof of Proposition 1
According to Theorem 1(3), we get
where \( 0\le t\le \frac{\pi }{M} \). Moreover,
Integral together, we have
Then
\(\square \)
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Tong, Y., Zeng, X., Zhang, S. et al. The estimates of trigonometric sums and new bounds on a mean value, a sequence and a cryptographic function. Des. Codes Cryptogr. 91, 921–949 (2023). https://doi.org/10.1007/s10623-022-01140-1
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DOI: https://doi.org/10.1007/s10623-022-01140-1
Keywords
- Trigonometric sum
- Arithmetic mean value
- Linear feedback shift register
- Carlet–Feng (vectorial )Boolean function
- Nonlinearity