A New Method to Compute the 2-adic Complexity of Binary Sequences

In this paper, a new method is presented to compute the 2-adic complexity of pseudo-random sequences. With this method, the 2-adic complexities of all the known sequences with ideal 2-level autocorrelation are uniformly determined. Results show that their 2-adic complexities equal their periods. In other words, their 2-adic complexities attain the maximum. Moreover, 2-adic complexities of two classes of optimal autocorrelation sequences with period $N\equiv1\mod4$, namely Legendre sequences and Ding-Helleseth-Lam sequences, are investigated. Besides, this method also can be used to compute the linear complexity of binary sequences regarded as sequences over other finite fields.Comment: 16 page

Comments on "A New Method to Compute the 2-Adic Complexity of Binary Sequences"

We show that there is a very simple approach to determine the 2-adic complexity of periodic binary sequences with ideal two-level autocorrelation. This is the first main result by H. Xiong, L. Qu, and C. Li, IEEE Transactions on Information Theory, vol. 60, no. 4, pp. 2399-2406, Apr. 2014, and the main result by T. Tian and W. Qi, IEEE Transactions on Information Theory, vol. 56, no. 1, pp. 450-454, Jan. 2010

Further results on the 2-adic complexity of a class of balanced generalized cyclotomic sequences

In this paper, the 2-adic complexity of a class of balanced Whiteman generalized cyclotomic sequences of period $pq$ is considered. Through calculating the determinant of the circulant matrix constructed by one of these sequences, we derive a lower bound on the 2-adic complexity of the corresponding sequence, which further expands our previous work (Zhao C, Sun Y and Yan T. Study on 2-adic complexity of a class of balanced generalized cyclotomic sequences. Journal of Cryptologic Research,6(4):455-462, 2019). The result shows that the 2-adic complexity of this class of sequences is large enough to resist the attack of the rational approximation algorithm(RAA) for feedback with carry shift registers(FCSRs), i.e., it is in fact lower bounded by $pq-p-q-1$, which is far larger than one half of the period of the sequences. Particularly, the 2-adic complexity is maximal if suitable parameters are chosen.Comment: 1

A lower bound on the 2-adic complexity of Ding-Helleseth generalized cyclotomic sequences of period pnp^n

Let $p$ be an odd prime, $n$ a positive integer and $g$ a primitive root of $p^n$. Suppose $D_i^{(p^n)}=\{g^{2s+i}|s=0,1,2,\cdots,\frac{(p-1)p^{n-1}}{2}\}$, $i=0,1$, is the generalized cyclotomic classes with $Z_{p^n}^{\ast}=D_0\cup D_1$. In this paper, we prove that Gauss periods based on $D_0$ and $D_1$ are both equal to 0 for $n\geq2$. As an application, we determine a lower bound on the 2-adic complexity of a class of Ding-Helleseth generalized cyclotomic sequences of period $p^n$. The result shows that the 2-adic complexity is at least $p^n-p^{n-1}-1$, which is larger than $\frac{N+1}{2}$, where $N=p^n$ is the period of the sequence.Comment: 1

A lower bound on the 2-adic complexity of modified Jacobi sequence

Let $p,q$ be distinct primes satisfying $\mathrm{gcd}(p-1,q-1)=d$ and let $D_i$, $i=0,1,\cdots,d-1$, be Whiteman's generalized cyclotomic classes with $Z_{pq}^{\ast}=\cup_{i=0}^{d-1}D_i$. In this paper, we give the values of Gauss periods based on the generalized cyclotomic sets $D_0^{\ast}=\sum_{i=0}^{\frac{d}{2}-1}D_{2i}$ and $D_1^{\ast}=\sum_{i=0}^{\frac{d}{2}-1}D_{2i+1}$. As an application, we determine a lower bound on the 2-adic complexity of modified Jacobi sequence. Our result shows that the 2-adic complexity of modified Jacobi sequence is at least $pq-p-q-1$ with period $N=pq$. This indicates that the 2-adic complexity of modified Jacobi sequence is large enough to resist the attack of the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).Comment: 13 pages. arXiv admin note: text overlap with arXiv:1702.00822, arXiv:1701.0376

Determination of 2-Adic Complexity of Generalized Binary Sequences of Order 2

The generalized binary sequences of order 2 have been used to construct good binary cyclic codes [4]. The linear complexity of these sequences has been computed in [2]. The autocorrelation values of such sequences have been determined in [1] and [3]. Some lower bounds of 2-adic complexity for such sequences have been presented in [5] and [7]. In this paper we determine the exact value of 2-adic complexity for such sequences. Particularly, we improve the lower bounds presented in [5] and [7] and the condition for the 2-adic complexity reaching the maximum value.Comment: 5 page

The 4-Adic Complexity of A Class of Quaternary Cyclotomic Sequences with Period 2p

In cryptography, we hope a sequence over $\mathbb{Z}_m$ with period $N$ having larger $m$-adic complexity. Compared with the binary case, the computation of 4-adic complexity of knowing quaternary sequences has not been well developed. In this paper, we determine the 4-adic complexity of the quaternary cyclotomic sequences with period 2$p$ defined in [6]. The main method we utilized is a quadratic Gauss sum $G_{p}$ valued in $\mathbb{Z}_{4^N-1}$ which can be seen as a version of classical quadratic Gauss sum. Our results show that the 4-adic complexity of this class of quaternary cyclotomic sequences reaches the maximum if $5\nmid p-2$ and close to the maximum otherwise.Comment: 7 page

A note on Hall's sextic residue sequence: correlation measure of order kk and related measures of pseudorandomness

It is known that Hall's sextic residue sequence has some desirable features of pseudorandomness: an ideal two-level autocorrelation and linear complexity of the order of magnitude of its period $p$. Here we study its correlation measure of order $k$ and show that it is, up to a constant depending on $k$ and some logarithmic factor, of order of magnitude $p^{1/2}$, which is close to the expected value for a random sequence of length $p$. Moreover, we derive from this bound a lower bound on the $N$th maximum order complexity of order of magnitude $\log p$, which is the expected order of magnitude for a random sequence of length $p$

On the 2-Adic Complexity of the Ding-Helleseth-Martinsen Binary Sequences

We determine the 2-adic complexity of the Ding-Helleseth-Martinsen (DHM) binary sequences by using cyclotomic numbers of order four, "Gauss periods" and "quadratic Gauss sum" on finite field $\mathbb{F}_q$ and valued in $\mathbb{Z}_{2^N-1}$ where $q \equiv 5\pmod 8$ is a prime number and $N=2q$ is the period of the DHM sequences.Comment: 16 page

The 2-adic complexity of a class of binary sequences with optimal autocorrelation magnitude

Recently, a class of binary sequences with optimal autocorrelation magnitude has been presented by Su et al. based on interleaving technique and Ding-Helleseth-Lam sequences (Des. Codes Cryptogr., https://doi.org/10.1007/s10623-017-0398-5). And its linear complexity has been proved to be large enough to resist the B-M Algorighm (BMA) by Fan (Des. Codes Cryptogr., https://doi.org/10.1007/s10623-018-0456-7). In this paper, we study the 2-adic complexity of this class of binary sequences. Our result shows that the 2-adic complexity of this class of sequence is no less than one half of its period, i.e., its 2-adic complexity is large enough to resist the Rational Aproximation Algorithm (RAA).Comment: 9page