On kk-error linear complexity of pseudorandom binary sequences derived from Euler quotients

We investigate the $k$-error linear complexity of pseudorandom binary sequences of period $p^{\mathfrak{r}}$ derived from the Euler quotients modulo $p^{\mathfrak{r}-1}$, a power of an odd prime $p$ for $\mathfrak{r}\geq 2$. When $\mathfrak{r}=2$, this is just the case of polynomial quotients (including Fermat quotients) modulo $p$, which has been studied in an earlier work of Chen, Niu and Wu. In this work, we establish a recursive relation on the $k$-error linear complexity of the sequences for the case of $\mathfrak{r}\geq 3$. We also state the exact values of the $k$-error linear complexity for the case of $\mathfrak{r}=3$. From the results, we can find that the $k$-error linear complexity of the sequences (of period $p^{\mathfrak{r}}$) does not decrease dramatically for $k<p^{\mathfrak{r}-2}(p-1)^2/2$

On the kk-error linear complexity of binary sequences derived from polynomial quotients

We investigate the $k$-error linear complexity of $p^2$-periodic binary sequences defined from the polynomial quotients (including the well-studied Fermat quotients), which is defined by $$q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p ~ \mathrm{with} 0 \le q_{p,w}(u) \le p-1, ~u\ge 0,$$ where $p$ is an odd prime and $1\le w<p$. Indeed, first for all integers $k$, we determine exact values of the $k$-error linear complexity over the finite field \F_2 for these binary sequences under the assumption of f2 being a primitive root modulo $p^2$, and then we determine their $k$-error linear complexity over the finite field \F_p for either $0\le k<p$ when $w=1$ or $0\le k<p-1$ when $2\le w<p$. Theoretical results obtained indicate that such sequences possess `good' error linear complexity.Comment: 2 figure

Linear complexity problems of level sequences of Euler quotients and their related binary sequences

The Euler quotient modulo an odd-prime power $p^r~(r>1)$ can be uniquely decomposed as a $p$-adic number of the form $$\frac{u^{(p-1)p^{r-1}} -1}{p^r}\equiv a_0(u)+a_1(u)p+\ldots+a_{r-1}(u)p^{r-1} \pmod {p^r},~ \gcd(u,p)=1,$$ where $0\le a_j(u)<p$ for $0\le j\le r-1$ and we set all $a_j(u)=0$ if $\gcd(u,p)>1$. We firstly study certain arithmetic properties of the level sequences $(a_j(u))_{u\ge 0}$ over $\mathbb{F}_p$ via introducing a new quotient. Then we determine the exact values of linear complexity of $(a_j(u))_{u\ge 0}$ and values of $k$-error linear complexity for binary sequences defined by $(a_j(u))_{u\ge 0}$.Comment: 16 page

On error linear complexity of new generalized cyclotomic binary sequences of period p2p^2

We consider the $k$-error linear complexity of a new binary sequence of period $p^2$, proposed in the recent paper "New generalized cyclotomic binary sequences of period $p^2$", by Z. Xiao et al., who calculated the linear complexity of the sequences (Designs, Codes and Cryptography, 2017, https://doi.org/10.1007/s10623-017-0408-7). More exactly, we determine the values of $k$-error linear complexity over $\mathbb{F}_2$ for almost $k>0$ in terms of the theory of Fermat quotients. Results indicate that such sequences have good stability

Trace representation of pseudorandom binary sequences derived from Euler quotients

We give the trace representation of a family of binary sequences derived from Euler quotients by determining the corresponding defining polynomials. Trace representation can help us producing the sequences efficiently and analyzing their cryptographic properties, such as linear complexity.Comment: 16 page

Linear complexity of Legendre-polynomial quotients

We continue to investigate binary sequence $(f_u)$ over $\{0,1\}$ defined by $(-1)^{f_u}=\left(\frac{(u^w-u^{wp})/p}{p}\right)$ for integers $u\ge 0$, where $\left(\frac{\cdot}{p}\right)$ is the Legendre symbol and we restrict $\left(\frac{0}{p}\right)=1$. In an earlier work, the linear complexity of $(f_u)$ was determined for $w=p-1$ under the assumption of $2^{p-1}\not\equiv 1 \pmod {p^2}$. In this work, we give possible values on the linear complexity of $(f_u)$ for all $1\le w<p-1$ under the same conditions. We also state that the case of larger $w(\geq p)$ can be reduced to that of $0\leq w\leq p-1$.Comment: 11 page

Polynomial quotients: Interpolation, value sets and Waring's problem

For an odd prime $p$ and an integer $w\ge 1$, polynomial quotients $q_{p,w}(u)$ are defined by $$q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p ~~ \mathrm{with}~~ 0 \le q_{p,w}(u) \le p-1, ~~u\ge 0,$$ which are generalizations of Fermat quotients $q_{p,p-1}(u)$. First, we estimate the number of elements $1\le u<N\le p$ for which $f(u)\equiv q_{p,w}(u) \bmod p$ for a given polynomial $f(x)$ over the finite field $\mathbb{F}_p$. In particular, for the case $f(x)=x$ we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each element of $\mathbb{F}_p$ as sum of values of polynomial quotients, we prove some lower bounds on the size of their value sets, and then we apply these lower bounds to prove some bounds on the Waring number using results from bounds on additive character sums and additive number theory

A further study on the linear complexity of new binary cyclotomic sequence of length prp^r

Recently, a conjecture on the linear complexity of a new class of generalized cyclotomic binary sequences of period $p^r$ was proposed by Z. Xiao et al. (Des. Codes Cryptogr., DOI 10.1007/s10623-017-0408-7). Later, for the case $f$ being the form $2^r$ with $r\ge 1$, Vladimir Edemskiy proved the conjecture (arXiv:1712.03947). In this paper, under the assumption of $2^{p-1} \not\equiv 1 \bmod p^2$ and $\gcd(\frac{p-1}{{\rm {ord}}_{p}(2)},f)=1$, the conjecture proposed by Z. Xiao et al. is proved for a general $f$ by using the Euler quotient. Actually, a generic construction of $p^r$-periodic binary sequence based on the generalized cyclotomy is introduced in this paper, which admits a flexible support set and includes Xiao's construction as a special case, and then an efficient method to compute the linear complexity of the sequence by the generic construction is presented, based on which the conjecture proposed by Z. Xiao et al. could be easily proved under the aforementioned assumption

Chapter 10: Algebraic Algorithms

Our Chapter in the upcoming Volume I: Computer Science and Software Engineering of Computing Handbook (Third edition), Allen Tucker, Teo Gonzales and Jorge L. Diaz-Herrera, editors, covers Algebraic Algorithms, both symbolic and numerical, for matrix computations and root-finding for polynomials and systems of polynomials equations. We cover part of these large subjects and include basic bibliography for further study. To meet space limitation we cite books, surveys, and comprehensive articles with pointers to further references, rather than including all the original technical papers.Comment: 41.1 page

On Sequences with a Perfect Linear Complexity Profile

We derive B\'ezout identities for the minimal polynomials of a finite sequence and use them to prove a theorem of Wang and Massey on binary sequences with a perfect linear complexity profile. We give a new proof of Rueppel's conjecture and simplify Dai's original proof. We obtain short proofs of results of Niederreiter relating the linear complexity of a sequence s and K(s), which was defined using continued fractions. We give an upper bound for the sum of the linear complexities of any sequence. This bound is tight for sequences with a perfect linear complexity profile and we apply it to characterise these sequences in two new ways.Comment: 19 pages, 3 table