On the linear complexity of Sidel'nikov Sequences over Fd

We study the linear complexity of sequences over the prime field Fd introduced by Sidel’nikov. For several classes of period length we can show that these sequences have a large linear complexity. For the ternary case we present exact results on the linear complexity using well known results on cyclotomic numbers. Moreover, we prove a general lower bound on the linear complexity profile for all of these sequences. The obtained results extend known results on the binary case. Finally we present an upper bound on the aperiodic autocorrelation

Autocorrelation of a class of quaternary sequences of period 2pm2p^m

Sequences with good randomness properties are quite important for stream ciphers. In this paper, a new class of quaternary sequences is constructed by using generalized cyclotomic classes of $\mathbb{Z}_{2p^m}$ $(m\geq1)$. The exact values of autocorrelation of these sequences are determined based on cyclotomic numbers of order $2$ with respect to $p^m$. Results show that the presented sequences have the autocorrelations with at most $4$ values

About the Linear Complexity of Ding-Hellesth Generalized Cyclotomic Binary Sequences of Any Period

We defined sufficient conditions for designing Ding-Helleseth sequences with arbitrary period and high linear complexity for generalized cyclotomies. Also we discuss the method of computing the linear complexity of Ding-Helleseth sequences in the general case

The Computational Complexity of Quantum Determinants

In this work, we study the computational complexity of quantum determinants, a $q$-deformation of matrix permanents: Given a complex number $q$ on the unit circle in the complex plane and an $n\times n$ matrix $X$, the $q$-permanent of $X$ is defined as $$\mathrm{Per}_q(X) = \sum_{\sigma\in S_n} q^{\ell(\sigma)}X_{1,\sigma(1)}\ldots X_{n,\sigma(n)},$$ where $\ell(\sigma)$ is the inversion number of permutation $\sigma$ in the symmetric group $S_n$ on $n$ elements. The function family generalizes determinant and permanent, which correspond to the cases $q=-1$ and $q=1$ respectively. For worst-case hardness, by Liouville's approximation theorem and facts from algebraic number theory, we show that for primitive $m$-th root of unity $q$ for odd prime power $m=p^k$, exactly computing $q$-permanent is $\mathsf{Mod}_p\mathsf{P}$-hard. This implies that an efficient algorithm for computing $q$-permanent results in a collapse of the polynomial hierarchy. Next, we show that computing $q$-permanent can be achieved using an oracle that approximates to within a polynomial multiplicative error and a membership oracle for a finite set of algebraic integers. From this, an efficient approximation algorithm would also imply a collapse of the polynomial hierarchy. By random self-reducibility, computing $q$-permanent remains to be hard for a wide range of distributions satisfying a property called the strong autocorrelation property. Specifically, this is proved via a reduction from $1$-permanent to $q$-permanent for $O(1/n^2)$ points $z$ on the unit circle. Since the family of permanent functions shares common algebraic structure, various techniques developed for the hardness of permanent can be generalized to $q$-permanents

Random Number Generation: Types and Techniques

What does it mean to have random numbers? Without understanding where a group of numbers came from, it is impossible to know if they were randomly generated. However, common sense claims that if the process to generate these numbers is truly understood, then the numbers could not be random. Methods that are able to let their internal workings be known without sacrificing random results are what this paper sets out to describe. Beginning with a study of what it really means for something to be random, this paper dives into the topic of random number generators and summarizes the key areas. It covers the two main groups of generators, true-random and pseudo-random, and gives practical examples of both. To make the information more applicable, real life examples of currently used and currently available generators are provided as well. Knowing the how and why of a number sequence without knowing the values that will come is possible, and this thesis explains how it is accomplished

Cryptographic Analysis of Random Sequences

Cryptographically strong random sequences are essential in cryptography, digital signatures, challenge-response systems, and in Monte Carlo simulation. This thesis examines techniques for cryptographic hardening of random sequences that are not cryptographically strong. Specific random sequences that are considered include d-sequences, that is sequences that are reciprocals of primes, and a new sequence obtained by the use of a specific two-dimensional mesh array. It is shown that the use of many-to-one mapping on blocks of the raw sequence improves the quality of autocorrelation function. Various types of many-to-one mappings are used and their effect on the autocorrelation function is compared. Sequences are also compared using another measure of randomness.Computer Science Departmen

VLT/SPHERE robust astrometry of the HR8799 planets at milliarcsecond-level accuracy Orbital architecture analysis with PyAstrOFit

HR8799 is orbited by at least four giant planets, making it a prime target for the recently commissioned Spectro-Polarimetric High-contrast Exoplanet REsearch (VLT/SPHERE). As such, it was observed on five consecutive nights during the SPHERE science verification in December 2014. We aim to take full advantage of the SPHERE capabilities to derive accurate astrometric measurements based on H-band images acquired with the Infra-Red Dual-band Imaging and Spectroscopy (IRDIS) subsystem, and to explore the ultimate astrometric performance of SPHERE in this observing mode. We also aim to present a detailed analysis of the orbital parameters for the four planets. We report the astrometric positions for epoch 2014.93 with an accuracy down to 2.0 mas, mainly limited by the astrometric calibration of IRDIS. For each planet, we derive the posterior probability density functions for the six Keplerian elements and identify sets of highly probable orbits. For planet d, there is clear evidence for nonzero eccentricity ($e \simeq 0.35$), without completely excluding solutions with smaller eccentricities. The three other planets are consistent with circular orbits, although their probability distributions spread beyond $e = 0.2$, and show a peak at $e \simeq 0.1$ for planet e. The four planets have consistent inclinations of about $30\deg$ with respect to the sky plane, but the confidence intervals for the longitude of ascending node are disjoint for planets b and c, and we find tentative evidence for non-coplanarity between planets b and c at the $2 \sigma$ level.Comment: 23 pages, 14 figure