Pseudorandom Vectors Generation Using Elliptic Curves And Applications

In this paper we present, using the arithmetic of elliptic curves over finite fields, an algorithm for the efficient generation of sequence of uniform pseudorandom vectors in high dimension with long period, that simulates sample sequence of a sequence of independent identically distributed random variables, with values in the hypercube $[0,1]^d$ with uniform distribution. As an application, we obtain, in the discrete time simulation, an efficient algorithm to simulate, uniformly distributed sample path sequence of a sequence of independent standard Wiener processes. This could be employed for use, in the full history recursive multi-level Picard approximation method, for numerically solving the class of semilinear parabolic partial differential equations of the Kolmogorov type

Inferring Sequences Produced by a Linear Congruential Generator on Elliptic Curves Using Coppersmith's Methods

International audienceWe analyze the security of the Elliptic Curve Linear Con-gruential Generator (EC-LCG). We show that this generator is insecure if sufficiently many bits are output at each iteration. In 2007, Gutierrez and Ibeas showed that this generator is insecure given a certain amount of most significant bits of some consecutive values of the sequence. Using the Coppersmith's methods, we are able to improve their security bounds

GROUP-THEORETIC GENERATION OF NON-UNIFORM PSEUDO-RANDOM SEQUENCES FOR SIMULATION

Abstract: Many applications involving statistical simulation, such as Monte Carlo methods, require non-uniform random sequences. These are usually created by first generating a uniform sequence and then using techniques such as rejection sampling or transformation. In this paper we introduce a new method to directly generate, without transformation or rejection, some non-uniform pseudo-random sequences. This method is a group-theoretic analogue of linear congruential pseudo-random number generation. We provide examples of such sequences, involving computations in Jacobian groups of plane algebraic curves, that have both good theoretical and statistical properties

The quadratic extension extractor for (hyper)elliptic curves in odd characteristic

We propose a simple and efficient deterministic extractor for the (hyper)elliptic curve C, defined over Fq2, where q is some power of an odd prime. Our extractor, for a given point P on C, outputs the first Fq-coefficient of the abscissa of the point P. We show that if a point P is chosen uniformly at random in C, the element extracted from the point P is indistinguishable from a uniformly random variable in Fq

Developing a flexible and expressive realtime polyphonic wave terrain synthesis instrument based on a visual and multidimensional methodology

The Jitter extended library for Max/MSP is distributed with a gamut of tools for the generation, processing, storage, and visual display of multidimensional data structures. With additional support for a wide range of media types, and the interaction between these mediums, the environment presents a perfect working ground for Wave Terrain Synthesis. This research details the practical development of a realtime Wave Terrain Synthesis instrument within the Max/MSP programming environment utilizing the Jitter extended library. Various graphical processing routines are explored in relation to their potential use for Wave Terrain Synthesis

Perfect codes in the Lee and Chebyshev metrics and iterating Rédei functions

Orientadores: Sueli Irene Rodrigues Costa, Daniel Nelson Panario RodriguezTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: O conteúdo desta tese insere-se dentro de duas áreas de pesquisa muito ativas: a teoria de códigos corretores de erros e sistemas dinâmicos sobre corpos finitos. Para abordar problemas em ambos os tópicos introduzimos um tipo de sequência finita que chamamos v-séries. No conjunto destas definimos uma métrica que induz uma estrutura de poset usada no estudo das possíveis estruturas de grupo abeliano representadas por códigos perfeitos na métrica de Chebyshev. Por outro lado, cada v-série é associada a uma árvore com raiz, a qual terá um papel importante em resultados relacionados à estrutura dinâmica de iterações de funções de Rédei. Na teoria de códigos corretores de erros, estudamos códigos perfeitos na métrica de Lee e na métrica de Chebyshev (correspondentes à métrica lp para p=1 e p=infinito respetivamente). Os principais resultados aqui estão relacionados com a descrição dos códigos q-ários n-dimensionais com raio de empacotamento e que sejam perfeitos nestas métricas, a obtenção de suas matrizes geradoras e a classificação destes, a menos de isometrias e a menos de isomorfismos. Varias construções de códigos perfeitos e famílias interessantes destes códigos com respeito à métrica de Chebyshev são apresentadas. Em sistemas dinâmicos sobre corpos finitos centramos nossa atenção em iterações de funções de Rédei, sendo o principal resultado um teorema estrutural para estas funções, o qual permite estender vários resultados sobre funções de Rédei. Este teorema pode também ser aplicado para outras classes de funções permitindo obter provas alternativas mais simples de alguns resultados conhecidos como o número de componentes conexas, o número de pontos periódicos e o valor esperado para o período e preperíodo da aplicação exponencial sobre corpos finitosAbstract: The content of this thesis is inserted in two very active research areas: the theory of error correcting codes and dynamical systems over finite fields. To approach problems in both topics we introduce a type of finite sequence called v-series. A metric is introduced in the set of such sequences inducing a poset structure used to determine all possible abelian group structures represented by perfect codes in the Chebyshev metric. Moreover, each v-serie is associated with a rooted tree, which has an important role in results related to the cycle structure of iterating Rédei functions. Regarding the theory of error correcting codes, we study perfect codes in the Lee metric and Chebyshev metric (corresponding to the lp metric for p=1 and p=infinity, respectively). The main results here are related to the description of n-dimensional q-ary codes with packing radius e which are perfect in these metrics, obtaining their generator matrices and their classification up to isometry and up to isomorphism. Several constructions of perfect codes in the Chebyshev metric are given and interesting families of such codes are presented. Regarding dynamical system over finite fields we focus on iterating Rédei functions, where our main result is a structural theorem, which allows us to extend several results on Rédei functions. The above theorem can also be applied to other maps, allowing simpler proofs of some known results related to the number of components, the number of periodic points and the expected value for the period and preperiod for iterating exponentiations over finite fieldsDoutoradoMatematica AplicadaDoutor em Matemática Aplicada2012/10600-2FAPESPCAPE

Publications of the Jet Propulsion Laboratory, July 1969 - June 1970

JPL bibliography of technical reports released from July 1969 through June 197