STATISTICAL PROPERTIES OF PSEUDORANDOM SEQUENCES

Random numbers (in one sense or another) have applications in computer simulation, Monte Carlo integration, cryptography, randomized computation, radar ranging, and other areas. It is impractical to generate random numbers in real life, instead sequences of numbers (or of bits) that appear to be ``random yet repeatable are used in real life applications. These sequences are called pseudorandom sequences. To determine the suitability of pseudorandom sequences for applications, we need to study their properties, in particular, their statistical properties. The simplest property is the minimal period of the sequence. That is, the shortest number of steps until the sequence repeats. One important type of pseudorandom sequences is the sequences generated by feedback with carry shift registers (FCSRs). In this dissertation, we study statistical properties of N-ary FCSR sequences with odd prime connection integer q and least period (q-1)/2. These are called half-ℓ-sequences. More precisely, our work includes: The number of occurrences of one symbol within one period of a half-ℓ-sequence; The number of pairs of symbols with a fixed distance between them within one period of a half-ℓ-sequence; The number of triples of consecutive symbols within one period of a half-ℓ-sequence. In particular we give a bound on the number of occurrences of one symbol within one period of a binary half-ℓ-sequence and also the autocorrelation value in binary case. The results show that the distributions of half-ℓ-sequences are fairly flat. However, these sequences in the binary case also have some undesirable features as high autocorrelation values. We give bounds on the number of occurrences of two symbols with a fixed distance between them in an ℓ-sequence, whose period reaches the maximum and obtain conditions on the connection integer that guarantee the distribution is highly uniform. In another study of a cryptographically important statistical property, we study a generalization of correlation immunity (CI). CI is a measure of resistance to Siegenthaler\u27s divide and conquer attack on nonlinear combiners. In this dissertation, we present results on correlation immune functions with regard to the q-transform, a generalization of the Walsh-Hadamard transform, to measure the proximity of two functions. We give two definitions of q-correlation immune functions and the relationship between them. Certain properties and constructions for q-correlation immune functions are discussed. We examine the connection between correlation immune functions and q-correlation immune functions

Constructing a Ternary FCSR with a Given Connection Integer

FCSRs have been proposed as an alternative to LFSRs for the design of stream ciphers. In 2009, a new ring representation of FCSRs was presented. This new representation preserves the statistical properties and circumvents the weaknesses of the Fibonacci and the Galois FCSRs. Moreover an extension of the ring FCSRs called ternary FCSRs has been proposed. They are suitable for hardware and software implementations of FCSRs. In this paper, we show a method of constructing a ternary FCSR with a given connection integer for hardware implementation. The construction is simple and convenient. And the ternary FCSRs we get are able to meet the hardware criteria

The relationship between personality characteristics and Expressed Emotion in significant relationships

The relationship between Expressed Emotion (EE) and relapse and clinical outcomes in individuals with mental and physical health difficulties is well established. This thesis explores the relationship between EE and individual psychological factors. Part One is a systematic review of the psychological factors associated with different EE domains in caregivers of individuals with First Episode Psychosis. Fourteen studies met inclusion criteria. The review found that EE-Criticism and emotional over-involvement (EOI) have mutual as well as distinct psychological correlates in caregivers of individuals with a first episode of psychosis, and that these may differ from the correlates of EE found amongst caregivers of individuals with chronic psychosis. Part Two is an online study investigating personality characteristics that are associated with EE in a non-clinical sample. Two-hundred-and-fifty participants completed measures of EE, self-criticism/ self-reassurance, locus of control, interpersonal functioning and personality. Higher scores on the self-criticism and interpersonal difficulties scales and lower scores on the ability to self-reassure scale were associated with both higher EE-Criticism and EOI. Having a more external locus of control was also associated with higher EOI, and higher EE-Criticism with being less open. This knowledge can be used to identify profiles of caregivers who may benefit from additional, targeted therapeutic interventions. The findings suggest that these personality characteristics warrant further investigation in longitudinal studies looking at the development of EE. Part Three is a critical appraisal of the research process. It considers issues pertaining to the construct of EE and how they impacted on the thesis. It also reflects on the methodological choices that were made and how things might have been done differently, and puts forward suggestions for future research