12 research outputs found
Secondary constructions of vectorial -ary weakly regular bent functions
In \cite{Bapic, Tang, Zheng} a new method for the secondary construction of
vectorial/Boolean bent functions via the so-called property was
introduced. In 2018, Qi et al. generalized the methods in \cite{Tang} for the
construction of -ary weakly regular bent functions. The objective of this
paper is to further generalize these constructions, following the ideas in
\cite{Bapic, Zheng}, for secondary constructions of vectorial -ary weakly
regular bent and plateaued functions. We also present some infinite families of
such functions via the -ary Maiorana-McFarland class. Additionally, we give
another characterization of the property for the -ary case via
second-order derivatives, as it was done for the Boolean case in \cite{Zheng}
λ pμ§ λ°μλ©μ΄μ μμ΄ κ°μ μνΈμκ΄λ
νμλ
Όλ¬Έ (λ°μ¬)-- μμΈλνκ΅ λνμ : μ κΈ°Β·μ»΄ν¨ν°κ³΅νλΆ, 2017. 2. λ
Έμ’
μ .In this dissertation, the cross-correlation between two differently decimated sequences of a -ary m-sequence is considered. Two main contributions are as follows.
First, for an odd prime , , and a -ary m-sequence of period , the cross-correlation between two decimated sequences by and are investigated. Two cases of , with and with odd are considered. The value distribution of the cross-correlation function for each case is completely deterimined. Also, by using these decimated sequences, two new families of -ary sequences of period with good correlation property are constructed.
Second, an upper bound on the magnitude of the cross-correlation function between two decimated sequences of a -ary m-sequence is derived. The two decimation factors are and , where is an odd prime, , and . In fact, these two sequences corresponds to the sequences used for the construction of -ary Kasami sequences decimated by . The upper bound is given as .
Also, using this result, an upper bound of the cross-correlation magnitude between a -ary m-sequence and its decimated sequence with the decimation factor is derived.1 Introduction 1
1.1 Background 1
1.2 Overview of This Dissertation 7
2 Preliminaries 9
2.1 Finite Fields 9
2.2 Trace Functions and Sequences 11
2.3 Cross-Correlation Between Two Sequences 13
2.4 Characters and Weils Bound 15
2.5 Trace-Orthogonal Basis 16
2.6 Known Exponential Sums 17
2.7 Cross-Correlation of -ary Kasami Sequence Family 18
2.8 Previous Results on the Cross-Correlation for Decimations with 20
2.9 Cross-Correlation Between Two Decimated Sequences by and or 23
3 New -ary Sequence Families of Period with Good Correlation Property Using Two Decimated Sequences 26
3.1 Cross-Correlation for the Case of 27
3.2 Cross-Correlation for the Case of 37
3.3 Construction of New Sequence Families 43
4 Upper Bound on the Cross-Correlation Between Two Decimated -ary Sequences 52
4.1 Cross-Correlation Between and 53
4.2 Cross-Correlation Between and 66
5 Conclusions 69
Bibliography 72
Abstract (In Korean) 80Docto