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Cryptography and Communications
Published in: Cryptography and Communications 2/2015

01-06-2015

The Walsh transform of a class of monomial functions and cyclic codes

Authors: Chengju Li, Qin Yue

Published in: Cryptography and Communications | Issue 2/2015

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Abstract

Let 𝔽 p be a finite field with p elements, where p is a prime. Let N ≥ 2 be an integer and f the least positive integer satisfying p f ≡ −1 (mod N). Then we let q = p 2f and r = q m . In this paper, we study the Walsh transform of the monomial function
\(f(x)=\text {Tr}_{r/p}(ax^{\frac {r-1} N})\)
for \(a \in \Bbb F_{r}^{*}\). We shall present the value distribution of the Walsh transform of f(x) and show that it takes at most \(\min \{p, N\}+1\) distinct values. In particular, we can obtain binary functions with three-valued Walsh transform and ternary functions with three-valued or four-valued Walsh transform. Furthermore, we present two classes of four-weight binary cyclic codes and six-weight ternary cyclic codes.
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Literature
1.
Berndt, B., Evans, R., Williams, K.: Gauss and Jacobi Sums. Wiley, New York (1997)
2.
Canteaut, A., Charpin, P., Dobbertin, H.: Binary m-sequences with three-valued crosscorrelation: A proof of Welch’s conjecture. IEEE Trans. Inf. Theory 46(1), 4–8 (2000)CrossRefMATHMathSciNet
3.
4.
5.
Dobbertin, H., Helleseth, T., Kumar, P. V., Martinsen, H.: Ternary m-sequences with three-valued cross-correlation function: New decimations of Welch and Niho type. IEEE Trans. Inf. Theory 47(4), 1473–1481 (2001)CrossRefMATHMathSciNet
6.
Feng, K., Luo, J.: Values distributions of exponential sums from perfect nonlinear functions and their applications. IEEE Trans. Inf. Theory 53(9), 3035–3041 (2007)CrossRefMathSciNet
7.
Helleseth, T.: Some results about the cross-correlation function between two maximal linear sequences. Discrete Math. 16(3), 209–232 (1976)CrossRefMATHMathSciNet
8.
Helleseth, T., Hu, L., Kholosha, A., Zeng, X., Li, N., Jiang, W.: Period-different m-sequences with at most four-valued cross correlation. IEEE Trans. Inf. Theory 55(7), 3305–3311 (2009)CrossRefMathSciNet
9.
Helleseth, T., Kholosha, A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 52(5), 2018–2032 (2006)CrossRefMATHMathSciNet
10.
Helleseth, T., Kholosha, A.: Crosscorrelation of m-sequences, exponential sums, bent functions and Jacobsthal sums. Cryptogr. Commun. 3(4), 281–291 (2011)CrossRefMATHMathSciNet
11.
Helleseth, T., Kholosha, A., Ness, G. J.: Characterization of m-sequences of lengths 22k −1 and 2 k −1 with three-valued cross correlation. IEEE Trans. Inf. Theory 53(6), 2236–2245 (2007)CrossRefMATHMathSciNet
12.
13.
Hou, X.D.: p-ary and q-ary versions of certain results about bent functions and resilient functions. Finite Fields Appl. 10, 566–582 (2004)CrossRefMATHMathSciNet
14.
15.
Jia, W., Zeng, X., Helleseth, T., Li, C.: A class of binomial bent functions over the finite fields of odd characterisitic. IEEE Trans. Inf. Theory 58(9), 6054–6063 (2006)CrossRefMathSciNet
16.
17.
18.
Li, N., Helleseth, T., Kholosha, A., Tang, X.: On the Walsh transform of a class of functions from Niho exponents. IEEE Trans. Inf. Theory 59(7), 4662–4667 (2013)CrossRefMathSciNet
19.
Li, N., Helleseth, T., Tang, X., Kholosha, A.: Several new classes of bent functions from Dillon exponents. IEEE Trans. Inf. Theory 59(3), 1818–1831 (2013)CrossRefMathSciNet
20.
Lidl, R., Niederreiter, H.: Finite fields. Addison-Wesley Publishing Inc. (1983)
21.
Luo, J., Feng, K.: On the weight distribution of two classes of cyclic codes. IEEE Trans. Inf. Theory 54(12), 5332–5344 (2008)CrossRefMathSciNet
22.
Luo, J., Feng, K.: Cyclic codes and sequences from generalized Coulter-Matthews funtion. IEEE Trans. Inf. Theory 54(12), 5345–5353 (2008)CrossRefMathSciNet
23.
Luo, J., Tang, Y., Wang, H.: Cyclic codes and sequences: The generalized Kasami case. IEEE Trans. Inf. Theory 56(5), 2130–2142 (2010)CrossRefMathSciNet
24.
25.
Ness, G.J., Helleseth, T.: Cross correlation of m-sequences of different lengths. IEEE Trans. Inf. Theory 52(4), 1637–1648 (2006)CrossRefMATHMathSciNet
26.
Ness, G.J., Helleseth, T.: A new three-valued cross correlation between m-sequences of different lengths. IEEE Trans. Inf. Theory 52(10), 4695–4701 (2006)CrossRefMathSciNet
27.
Ness, G.J., Helleseth, T.: A new family of four-valued cross correlation between m-sequences of different lengths. IEEE Trans. Inf. Theory 53(11), 4308–4313 (2007)CrossRefMathSciNet
28.
Ness, G.J., Helleseth, T., Kholosha, A.: On the correlation distribution of the Coulter-Matthews decimation. IEEE Trans. Inf. Theory 52(5), 2241–2247 (2006)CrossRefMATHMathSciNet
29.
Yuan, J., Carlet, C., Ding, C.: The weight distribution of a class of linear codes from perfect nonlinear functions. IEEE Trans. Inf. Theory 52(2), 712–717 (2006)CrossRefMATHMathSciNet
30.
Zeng, X., Hu, L., Jiang, W., Yue, Q., Cao, X.: The weight distribution of a class of p-ary cyclic codes. Finite Fields Appl. 16, 56–73 (2010)CrossRefMATHMathSciNet
31.
Metadata
Title
The Walsh transform of a class of monomial functions and cyclic codes
Authors
Chengju Li
Qin Yue
Publication date
01-06-2015
Publisher
Springer US
Published in
Cryptography and Communications / Issue 2/2015
Print ISSN: 1936-2447
Electronic ISSN: 1936-2455
DOI
https://doi.org/10.1007/s12095-014-0109-2

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