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

Explicit Methods in Number Theory

These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes asymptotics for field extensions and class numbers, random matrices and L-functions, rational points on curves and higher-dimensional varieties, and aspects of lattice basis reduction

Minimum distance of error correcting codes versus encoding complexity, symmetry, and pseudorandomness

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2003.Includes bibliographical references (leaves 207-214).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.We study the minimum distance of binary error correcting codes from the following perspectives: * The problem of deriving bounds on the minimum distance of a code given constraints on the computational complexity of its encoder. * The minimum distance of linear codes that are symmetric in the sense of being invariant under the action of a group on the bits of the codewords. * The derandomization capabilities of probability measures on the Hamming cube based on binary linear codes with good distance properties, and their variations. Highlights of our results include: * A general theorem that asserts that if the encoder uses linear time and sub-linear memory in the general binary branching program model, then the minimum distance of the code cannot grow linearly with the block length when the rate is nonvanishing. * New upper bounds on the minimum distance of various types of Turbo-like codes. * The first ensemble of asymptotically good Turbo like codes. We prove that depth-three serially concatenated Turbo codes can be asymptotically good. * The first ensemble of asymptotically good codes that are ideals in the group algebra of a group. We argue that, for infinitely many block lengths, a random ideal in the group algebra of the dihedral group is an asymptotically good rate half code with a high probability. * An explicit rate-half code whose codewords are in one-to-one correspondence with special hyperelliptic curves over a finite field of prime order where the number of zeros of a codeword corresponds to the number of rational points.(cont.) * A sharp O(k-1/2) upper bound on the probability that a random binary string generated according to a k-wise independent probability measure has any given weight. * An assertion saying that any sufficiently log-wise independent probability measure looks random to all polynomially small read-once DNF formulas. * An elaborate study of the problem of derandomizability of AC₀ by any sufficiently polylog-wise independent probability measure. * An elaborate study of the problem of approximability of high-degree parity functions on binary linear codes by low-degree polynomials with coefficients in fields of odd characteristics.by Louay M.J. Bazzi.Ph.D

Quantum algorithms for algebraic problems

Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation, and in particular, on problems with an algebraic flavor.Comment: 52 pages, 3 figures, to appear in Reviews of Modern Physic

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics