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 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

On the kk-error linear complexity of binary sequences derived from the discrete logarithm in finite fields

Let $q=p^r$ be a power of an odd prime $p$. We study binary sequences $\sigma=(\sigma_0,\sigma_1,\ldots)$ with entries in $\{0,1\}$ defined by using the quadratic character $\chi$ of the finite field $\mathbb{F}_q$: $$\sigma_n=\left\{ \begin{array}{ll} 0,& \mathrm{if}\quad n= 0,\\ (1-\chi(\xi_n))/2,&\mathrm{if}\quad 1\leq n< q, \end{array} \right.$$ for the ordered elements $\xi_0,\xi_1,\ldots,\xi_{q-1}\in \mathbb{F}_q$. The $\sigma$ is Legendre sequence if $r=1$. Our first contribution is to prove a lower bound on the linear complexity of $\sigma$ for $r\geq 2$. The bound improves some results of Meidl and Winterhof. Our second contribution is to study the $k$-error linear complexity of $\sigma$ for $r=2$. It seems that we cannot settle the case when $r>2$ and leave it open

On qq-nearly bent Boolean functions

For each non-constant Boolean function $q$, Klapper introduced the notion of $q$-transforms of Boolean functions. The {\em $q$-transform} of a Boolean function $f$ is related to the Hamming distances from $f$ to the functions obtainable from $q$ by nonsingular linear change of basis. In this work we discuss the existence of $q$-nearly bent functions, a new family of Boolean functions characterized by the $q$-transform. Let $q$ be a non-affine Boolean function. We prove that any balanced Boolean functions (linear or non-linear) are $q$-nearly bent if $q$ has weight one, which gives a positive answer to an open question (whether there exist non-affine $q$-nearly bent functions) proposed by Klapper. We also prove a necessary condition for checking when a function isn't $q$-nearly bent

Linear complexity of quaternary sequences over Z_4 derived from generalized cyclotomic classes modulo 2p

We determine the exact values of the linear complexity of 2p-periodic quaternary sequences over Z_4 (the residue class ring modulo 4) defined from the generalized cyclotomic classes modulo 2p in terms of the theory of of Galois rings of characteristic 4, where p is an odd prime. Compared to the case of quaternary sequences over the finite field of order 4, it is more dificult and complicated to consider the roots of polynomials in Z_4[X] due to the zero divisors in Z_4 and hence brings some interesting twists. We answer an open problem proposed by Kim, Hong and Song

A General Construction of Binary Sequences with Optimal Autocorrelation

A general construction of binary sequences with low autocorrelation are considered in the paper. Based on recent progresses about this topic and this construction, several classes of binary sequences with optimal autocorrelation and other low autocorrelation are presented

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

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