A generalization of the Hall’s sextic residue sequences
Introduction
Let p be an odd prime and integer m ⩾ 1. We identify , the residue ring modulo pm, with the set {0, 1, … , pm − 1}. We also denote by the set of unit elements of . Since is a cyclic group under the multiplicative operation, let g be one of its generators (or primitive elements). Then we havewhere φ(−) is the Euler totient function. By defining different partitions of , the ring is extensively applied to constructing pseudorandom sequences in the literature. A typical partition is the (generalized) cyclotomic classes, see below for the definition.
In this paper, we always suppose that g is a primitive element modulo p2. We note that in this case g is a primitive element modulo pn for any n ⩾ 1 [16]. That is, the order of g in is . Let d be an even integer satisfying d∣(p − 1). Now for each n, we get the (generalized) cyclotomic classes of by definingwhich give a partition of . Letwe haveThus we represent the ring as followsAssume thatthen we havewhere ∅ denotes the empty set.
So one can construct a pm-periodic binary sequence {st}t⩾0 by definingwhere is called the characteristic set of {st}t⩾0. We note that {st}t⩾0 is called a cyclotomic sequence for m = 1 or a generalized cyclotomic sequence for m ⩾ 2. Some special cases have been investigated in the literature, see Table 1, in which the symbol “√” means that the trace function representation of the corresponding sequences has been investigated.
In this paper, we will consider {st}t⩾0 defined in (2) and its related sequences when m ⩾ 1 and d = 6. Definition 1 Let distinct integers , which is the residue class ring modulo 6. The triple subset {u,v,w} is admissible over if there exists an such that
One can easily verify that all admissible triples over are Proposition 1 Each admissible triple {u, v, w} over satisfies Proof Clearly. □
Now we introduce the new sequences. Let be the set of all admissible triples over . Definition 2 Let p = 6f + 1 be a prime and m ⩾ 1. Using Cl in (1) with d = 6, we construct a pm-periodic binary sequence {et}t⩾0 by definingwhere .
We remark that if {u, v, w} = {0, 2, 4} or {1, 3, 5}, {et}t⩾0 was considered in [18], [12]. So we exclude them here. Clearly {et}t⩾0 is a generalization and modification of the Hall’s sequences[9], [10]. We will present the linear complexity and trace function representation for {et}t⩾0.
Section snippets
Linear complexity
Let be the finite field of q elements. We recall that the linear complexity L(s) of an N-periodic binary sequence {st}t⩾0 is the least order L of a linear recurrence relation over which is satisfied by {st}t⩾0. And the polynomialis called the minimal polynomial of {st}t⩾0. The generating polynomial of {st}t⩾0 is defined byIt is easy to seehence
Trace representation (m = 2)
For applications, pseudorandom sequences should be implemented efficiently. Trace function is extensively applied to producing pseudorandom sequences and analyzing their pseudorandom properties. The trace function from to is defined byWe refer the readers to[15] for details.
In this section, we present a trace function representation for {et}t⩾0 defined as in Definition 2 when m = 2. The case of m > 2 seems more complicated, although it might be extended directly from
Final remarks
Cryptographically strong sequences should have an ideal balance property, long periods, large linear complexity, low correlation, etc. [4]. In this paper, we have constructed families of binary sequences of period pm over . The difference between the number of 0’s and the number of 1’s is 1 in such sequences, i.e., the sequences have optimum balance among 0’s and 1’s. The linear complexity is large enough since it can resist attacks using the Berlekamp–Massey algorithm [2], which tells us
Acknowledgments
The authors thank the anonymous referees for their valuable comments. The third referee analyzed the paper in details and pointed out Lemma 6 to clarify and simplify our proofs, we are very grateful for that too.
X.N.D. was partially supported by the National Natural Science Foundation of China under Grant 61063041, 61202395 and 61163038, the Program for New Century Excellent Talents in University, the Scientific Research Foundation of the State Human Resource Ministry and the Education Ministry
References (19)
- et al.
Sequences related to Legendre/Jacobi sequences
Inform. Sci.
(2007) - et al.
New generalized cyclotomy and its application
Finite Fields Appl.
(1998) - et al.
Trace representation of some generalized cyclotomic sequences of length pq
Inform. Sci.
(2008) - et al.
Some notes of prime-square sequences
J. Comput. Sci. Technol.
(2007) Algebraic Coding Theory
(1968)- et al.
Stream Ciphers and Number Theory
(1998) - et al.
On the linear complexity of Legendre sequence
IEEE Trans. Inform. Theory
(1998) - et al.
Trace representation of a new class of sextic sequences of period p ≡ 3 (mod 8)
IEICE Trans. Fundam. Electron. Commun. Comput. Sci.
(2009) - et al.
On the linear complexity of Hall’s sextic residue sequences
IEEE Trans. Inform. Theory
(2001)
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