Corrigendum to New Generalized Cyclotomic Binary Sequences of Period p2p^2

New generalized cyclotomic binary sequences of period $p^2$ are proposed in this paper, where $p$ is an odd prime. The sequences are almost balanced and their linear complexity is determined. The result shows that the proposed sequences have very large linear complexity if $p$ is a non-Wieferich prime.Comment: In the appended corrigendum, we pointed out that the proof of Lemma 6 in the paper only holds for $f=2$ and gave a proof for any $f=2^r$ when $p$ is a non-Wieferich prim

Some New Balanced and Almost Balanced Quaternary Sequences with Low Autocorrelation

Quaternary sequences of both even and odd period having low autocorrelation are studied. We construct new families of balanced quaternary sequences of odd period and low autocorrelation using cyclotomic classes of order eight, as well as investigate the linear complexity of some known quaternary sequences of odd period. We discuss a construction given by Chung et al. in "New Quaternary Sequences with Even Period and Three-Valued Autocorrelation" [IEICE Trans. Fundamentals Vol. E93-A, No. 1 (2010)] first by pointing out a slight modification (thereby obtaining new families of balanced and almost balanced quaternary sequences of even period and low autocorrelation), then by showing that, in certain cases, this slight modification greatly simplifies the construction given by Shen et al. in "New Families of Balanced Quaternary Sequences of Even Period with Three-level Optimal Autocorrelation" [IEEE Comm. Letters DOI10.1109/LCOMM.2017.26611750 (2017)]. We investigate the linear complexity of these sequences as well

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

Linear complexity and trace representation of quaternary sequences over Z4\mathbb{Z}_4 based on generalized cyclotomic classes modulo pqpq

We define a family of quaternary sequences over the residue class ring modulo $4$ of length $pq$, a product of two distinct odd primes, using the generalized cyclotomic classes modulo $pq$ and calculate the discrete Fourier transform (DFT) of the sequences. The DFT helps us to determine the exact values of linear complexity and the trace representation of the sequences.Comment: 16 page

Linear complexity of generalized cyclotomic sequences of period 2pm2p^{m}

In this paper, we construct two generalized cyclotomic binary sequences of period $2p^{m}$ based on the generalized cyclotomy and compute their linear complexity, showing that they are of high linear complexity when $m\geq 2$

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

Linear Complexity of Geometric Sequences Defined by Cyclotomic Classes and Balanced Binary Sequences Constructed by the Geometric Sequences

Pseudorandom number generators are required to generate pseudorandom numbers which have good statistical properties as well as unpredictability in cryptography. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol, and showed the period, periodic autocorrelation and linear complexity of the sequence. Furthermore, Nogami et al. proposed a generalization of the sequence, and showed the period and periodic autocorrelation. In this paper, we first investigate linear complexity of the geometric sequences. In the case that the Chan--Games formula which describes linear complexity of geometric sequences does not hold, we show the new formula by considering the sequence of complement numbers, Hasse derivative and cyclotomic classes. Under some conditions, we can ensure that the geometric sequences have a large linear complexity from the results on linear complexity of Sidel'nikov sequences. The geometric sequences have a long period and large linear complexity under some conditions, however they do not have the balance property. In order to construct sequences that have the balance property, we propose interleaved sequences of the geometric sequence and its complement. Furthermore, we show the periodic autocorrelation and linear complexity of the proposed sequences. The proposed sequences have the balance property, and have a large linear complexity if the geometric sequences have a large one.Comment: 20 pages, 3 figures. arXiv admin note: text overlap with arXiv:1709.0516

On the Use of Cellular Automata in Symmetric Cryptography

In this work, pseudorandom sequence generators based on finite fields have been analyzed from the point of view of their cryptographic application. In fact, a class of nonlinear sequence generators has been modelled in terms of linear cellular automata. The algorithm that converts the given generator into a linear model based on automata is very simple and is based on the concatenation of a basic structure. Once the generator has been linearized, a cryptanalytic attack that exploits the weaknesses of such a model has been developed. Linear cellular structures easily model sequence generators with application in stream cipher cryptography.Comment: 25 pages, 0 figure

Cellular Automata in Stream Ciphers

A wide family of nonlinear sequence generators, the so-called clock-controlled shrinking generators, has been analyzed and identified with a subset of linear cellular automata. The algorithm that converts the given generator into a linear model based on automata is very simple and can be applied in a range of practical interest. Due to the linearity of these automata as well as the characteristics of this class of generators, a cryptanalytic approach can be proposed. Linear cellular structures easily model keystream generators with application in stream cipher cryptography.Comment: 26 pages, 1 figur

Autocorrelation and Linear Complexity of Quaternary Sequences of Period 2p Based on Cyclotomic Classes of Order Four

We examine the linear complexity and the autocorrelation properties of new quaternary cyclotomic sequences of period 2p. The sequences are constructed via the cyclotomic classes of order four