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

Topics on Register Synthesis Problems

Pseudo-random sequences are ubiquitous in modern electronics and information technology. High speed generators of such sequences play essential roles in various engineering applications, such as stream ciphers, radar systems, multiple access systems, and quasi-Monte-Carlo simulation. Given a short prefix of a sequence, it is undesirable to have an efficient algorithm that can synthesize a generator which can predict the whole sequence. Otherwise, a cryptanalytic attack can be launched against the system based on that given sequence. Linear feedback shift registers (LFSRs) are the most widely studied pseudorandom sequence generators. The LFSR synthesis problem can be solved by the Berlekamp-Massey algorithm, by constructing a system of linear equations, by the extended Euclidean algorithm, or by the continued fraction algorithm. It is shown that the linear complexity is an important security measure for pseudorandom sequences design. So we investigate lower bounds of the linear complexity of different kinds of pseudorandom sequences. Feedback with carry shift registers (FCSRs) were first described by Goresky and Klapper. They have many good algebraic properties similar to those of LFSRs. FCSRs are good candidates as building blocks of stream ciphers. The FCSR synthesis problem has been studied in many literatures but there are no FCSR synthesis algorithms for multi-sequences. Thus one of the main contributions of this dissertation is to adapt an interleaving technique to develop two algorithms to solve the FCSR synthesis problem for multi-sequences. Algebraic feedback shift registers (AFSRs) are generalizations of LFSRs and FCSRs. Based on a choice of an integral domain R and π ∈ R, an AFSR can produce sequences whose elements can be thought of elements of the quotient ring R/(π). A modification of the Berlekamp-Massey algorithm, Xu\u27s algorithm solves the synthesis problem for AFSRs over a pair (R, π) with certain algebraic properties. We propose two register synthesis algorithms for AFSR synthesis problem. One is an extension of lattice approximation approach but based on lattice basis reduction and the other one is based on the extended Euclidean algorithm

ANALYSIS OF SECURITY MEASURES FOR SEQUENCES

Stream ciphers are private key cryptosystems used for security in communication and data transmission systems. Because they are used to encrypt streams of data, it is necessary for stream ciphers to use primitives that are easy to implement and fast to operate. LFSRs and the recently invented FCSRs are two such primitives, which give rise to certain security measures for the cryptographic strength of sequences, which we refer to as complexity measures henceforth following the convention. The linear (resp. N-adic) complexity of a sequence is the length of the shortest LFSR (resp. FCSR) that can generate the sequence. Due to the availability of shift register synthesis algorithms, sequences used for cryptographic purposes should have high values for these complexity measures. It is also essential that the complexity of these sequences does not decrease when a few symbols are changed. The k-error complexity of a sequence is the smallest value of the complexity of a sequence obtained by altering k or fewer symbols in the given sequence. For a sequence to be considered cryptographically ‘strong’ it should have both high complexity and high error complexity values. An important problem regarding sequence complexity measures is to determine good bounds on a specific complexity measure for a given sequence. In this thesis we derive new nontrivial lower bounds on the k-operation complexity of periodic sequences in both the linear and N-adic cases. Here the operations considered are combinations of insertions, deletions, and substitutions. We show that our bounds are tight and also derive several auxiliary results based on them. A second problem on sequence complexity measures useful in the design and analysis of stream ciphers is to determine the number of sequences with a given fixed (error) complexity value. In this thesis we address this problem for the k-error linear complexity of 2n-periodic binary sequences. More specifically: 1. We characterize 2n-periodic binary sequences with fixed 2- or 3-error linear complexity and obtain the counting function for the number of such sequences with fixed k-error linear complexity for k = 2 or 3. 2. We obtain partial results on the number of 2n-periodic binary sequences with fixed k-error linear complexity when k is the minimum number of changes required to lower the linear complexity

Squirals and beyond: Substitution tilings with singular continuous spectrum

The squiral inflation rule is equivalent to a bijective block substitution rule and leads to an interesting lattice dynamical system under the action of $\mathbb{Z}^{2}$. In particular, its balanced version has purely singular continuous diffraction. The dynamical spectrum is of mixed type, with pure point and singular continuous components. We present a constructive proof that admits a generalisation to bijective block substitutions of trivial height on $\mathbb{Z}^{d}$.Comment: 23 pages, 7 figure

Dynamical Directions in Numeration

International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper