On error linear complexity of new generalized cyclotomic binary sequences of period p2
Introduction
The theory of cyclotomy is widely adopted in cryptography. A typical application is the design of pseudorandom sequences. By defining the (generalized) cyclotomic classes modulo an integer, it can design a family of pseudorandom sequences with desired cryptographic features. The classical examples are the Legendre sequences that derived from cyclotomic classes modulo an odd prime and the Jacobi sequences that derived from generalized cyclotomic classes modulo a product of two odd distinct primes. Attention is also paid to the generalized cyclotomic classes modulo a general number (such as a prime-power) in the literature. Such sequences include the Hall' sequence of length p, the Ding–Helleseth–Lam sequence of length p, Ding–Helleseth generalized cyclotomic sequence of length pq, and generalized cyclotomic sequences of length or length N (a general number). The cryptographic measures-autocorrelation, crosscorrelation, linear complexity, trace representation are considered for such kind sequences. See the related works such as [2], [9], [10], [12], [15], [17], [20], [22], [23], [24] and the references therein.
Recently, a new family of binary sequences were introduced by Xiao, Zeng, Li and Helleseth [21] by defining the generalized cyclotomic classes modulo for odd prime p. Now we introduce the generalized cyclotomic classes modulo .
Let and g be a primitive root1 modulo . The generalized cyclotomic classes for is defined by and Then the authors of [21] chose even f and an integer to define a new -periodic binary sequence : where and The notation above means that . They determined the linear complexity (see below for the notion) of the proposed sequences for for some integer .
Theorem 1 ([21, Thm. 1]) Let be the binary sequence of period defined in Eq. (1) with (integer ) and any b for defining and . If , then the linear complexity of is
For a sequence to be cryptographically strong, its linear complexity should be large, but it's not significantly reduced by changing a few terms. This directs to the notion of the k-error linear complexity. For integers , the k-error linear complexity over of , denoted by , is the smallest linear complexity (over ) that can be obtained by changing at most k terms of the sequence per period, see [19], and see [11] for the related even earlier defined sphere complexity. Clearly and when w equals the number of nonzero terms of per period, i.e., the weight of .
The main contribution of this work is to determine the k-error linear complexity of in Eq. (1) for any even number f (including considered in [21]). The main results are presented in the following two theorems. The proof of Theorem 2 appears in Section 4. Some necessary lemmas are introduced in Section 3. A crucial tool for the proof is the Fermat quotients, which is introduced in Section 2. In Section 5, Theorem 3 gives a lower bound on the k-error linear complexity when 2 is not a primitive root modulo .
Theorem 2 (Main theorem) Let be the binary sequence of period defined in Eq. (1) with even f and any b for defining and . If 2 is a primitive root modulo , then the k-error linear complexity of satisfies if , and if .
Section snippets
Fermat quotients
In this section, we interpret that the construction of in Eq. (1) is related to Fermat quotients. Certain similar constructions can be found in [3], [4], [5], [6], [7], [13], [16].
For integers , the Fermat quotient is the value in at u defined by where , if we set , see [18].
Thanks to the facts that for and , we define
Auxiliary lemmas
In this section, we present some necessary lemmas needed in the proofs. In the sequel, the notation denotes the cardinality of the set Z.
Lemma 1 Let be defined for by Fermat quotients as in Sect. 2. Then we have for , Proof Since for , we get which completes the proof. □
Lemma 2 Let be defined for by Fermat quotients as in Sect. 2. Let and . Then for each
Proof of the main theorem
Proof of Theorem 2 From the construction (1), we see that the weight of is , i.e., there are many 1's in one period. Changing all terms of 0's of will lead to the constant 1-sequence, whose linear complexity is 1. And changing all terms of 1's will lead to the constant 0-sequence. So we always assume that . The generating polynomial of is of the form where is defined in Eq. (6). We
A lower bound
We have the following lower bound on the k-error linear complexity when 2 is not a primitive root modulo .
Theorem 3 Let be the binary sequence of period defined in Eq. (1) with even f and any b for defining and . If , then the k-error linear complexity of satisfies where is the order of 2 modulo p.
Proof First we show the order of 2 modulo is λp. Under the assumption on , we see that the order of 2 modulo is of the form mp
Acknowledgements
Parts of this work were written during a very pleasant visit of Chenhuang Wu and Zhixiong Chen to the Hong Kong University of Science and Technology and the Hong Kong Polytechnic University in 2018. They wish to thank the host for the hospitality and the Fujian Provincial Department of Human Resources and Social Security of P. R. China for financial support (File No. 2017-368). The authors also wish to thank the anonymous referees and the editor for their time and positive comments.
The work was
References (24)
- et al.
New generalized cyclotomy and its applications
Finite Fields Appl.
(1998) - et al.
Linear complexity of binary sequences derived from Euler quotients with prime-power modulus
Inf. Process. Lett.
(2012) - et al.
Linear complexity of pseudorandom sequences generated by Fermat quotients and their generalizations
Inf. Process. Lett.
(2012) - et al.
The linear complexity of binary sequences of length 2p with optimal three-level autocorrelation
Inf. Process. Lett.
(2016) - et al.
New classes of quaternary cyclotomic sequence of length with high linear complexity
Inf. Process. Lett.
(2012) - et al.
The largest known Wieferich numbers
Integers
(2018) - et al.
A new construction of zero-difference balanced functions and its applications
IEEE Trans. Inf. Theory
(2013) Trace representation and linear complexity of binary sequences derived from Fermat quotients
Sci. China Inf. Sci.
(2014)- et al.
On the linear complexity of binary threshold sequences derived from Fermat quotients
Des. Codes Cryptogr.
(2013) - et al.
Linear complexity of binary sequences derived from polynomial quotients