Joe Suzuki

A Markov chain analysis on simple genetic algorithms

This paper addresses a Markov chain analysis of genetic algorithms (GAs), in particular for a variety called a modified elitist strategy. The modified elitist strategy generates the current population of M individuals by reserving the individual with the highest fitness value from the previous generation and generating M-1 individuals through a generation change. The authors analysis is based on a Markov chain: by assuming a simple GA in which the genetic operation in the generation changes is restricted to selection, crossover, and mutation, and by evaluating the eigenvalues of the transition matrix of the Markov chain, the convergence rate of the GAs is computed in terms of a mutation probability /spl mu/. In this way, the authors show the probability that the population includes the individual with the highest fitness value is lower-bounded by 1-O(|/spl lambda/*|/sup n/), |/spl lambda/*| >

Elliptic Curve Discrete Logarithms and the Index Calculus

The discrete logarithm problem forms the basis of numerous cryptographic systems. The most effective attack on the discrete logarithm problem in the multiplicative group of a finite field is via the index calculus, but no such method is known for elliptic curve discrete logarithms. Indeed, Miller [23] has given a brief heuristic argument as to why no such method can exist. IN this note we give a detailed analysis of the index calculus for elliptic curve discrete logarithms, amplifying and extending millers remarks. Our conclusions fully support his contention that the natural generalization of the index calculus to the elliptic curve discrete logarithm problem yields an algorithm with is less efficient than a brute-force search algorithm.

Comparing the MOV and FR reductions in elliptic curve cryptography

This paper addresses the discrete logarithm problem in elliptic curve cryptography. In particular, we generalize the Menezes, Okamoto, and Vanstone (MOV) reduction so that it can be applied to some non-supersingular elliptic curves (ECs); decrypt Frey and Ruck (FR)s idea to describe the detail of the FR reduction and to implement it for actual elliptic curves with finite fields on a practical scale; and based on them compare the (extended) MOV and FR reductions from an algorithmic point of view. (This paper has primarily an expository role.)

A CTW scheme for some FSM models

The paper addresses a modified version of the CTW (context tree weighting) which deals with some FSM (finite state machine) models as well as the FSMX (FSM X) models at little expense of computing in encoding/decoding.

Universal coding and universal prediction

We consider predicting the next outcome x/sub n+1/ from the past sequence x/sub 1//sup n/, n=0,1...,1 for a given binary sequence x/sub 1//sup /spl infin//. We establish a rigorous framework of universal prediction based on another universal coding which was set up by Davisson.

Universal prediction and universal coding

Although prediction schemes called “universal” are now abundant, very little attention has been devoted to the definition of universal prediction. This paper addresses, for α-nary (α ≥ 2) sequences, the criteria of successful universal prediction and the prediction schemes that achieve the goals. We propose the following criteria: for any probability measures in a given measure class, the error probability of prediction (in the problem of predicting a value for the next outcome) and the conditional probability of the next outcome given the past sequence (in the problem of predicting a probability distribution for the next outcome, i.e., the probability assignment problem) should converge to the optimal values in probability (weakly universal) and almost surely (strongly universal). We present a small review of various results on universal prediction, and give several results relating the developed criteria to each other and to various prediction submeasures. Furthermore, we explore the connection between universal prediction and universal coding. The proposed criteria are applied to any measure class as well as stationary ergodic measures.

Comparing the multilevel pattern matching code and the Lempel-Ziv codes

The asymptotic performance of the multilevel pattern matching (MPM) code and a Lempel-Ziv77 (LZ) code are compared. It is known that both codes have the Shannon entropy as the asymptotic performance if they are employed for stationary ergodic sources, but the redundancy of MPM is less than that of LZ. It is shown that there exists a large set of sequences that can be compressed well by LZ but cannot be compressed by MPM.

On the error probability of model selection for classification

We estimate a conditional probability P(y|x) of class y/spl isin/Y given attribute x/spl isin/X from training examples, where X and Y are respectively infinite and finite sets. The estimated conditional probability is used for classification in which a class y is guessed from an attribute x based on the conditional probability P(y|x). The procedure can be also applied to order identification of Markov models. We derive the asymptotically exact error probability in model selection for an arbitrary function d(/spl middot/) which determines the selection procedure as well as the information criterion.

A CTW scheme for non-tree sources

This paper addresses a modified version of the context tree weighting (CTW) scheme for FV noiseless universal coding. The CTW assumes that the source is some tree source. Although it is known that the computation of the CTW in coding/decoding is O(Dn), the redundancy gets worse in the case where the source is outside the tree sources. The proposed scheme deals with a more wider source class.

An extension on learning Bayesian belief networks based on MDL principle

Bayesian belief network (BBN) is a framework for representation/inference of some knowledge with uncertainty. Since the process of constructing a BBN manually by experts is time-consuming in general, some method supporting the task is needed. We proposed an algorithm for acquiring some BBN automatically from finite examples based on minimum description length (MDL) principle. This paper addresses an improvement which relaxes a constraint that the original scheme held on the representation. In BBNs, attributes and stochastic dependencies between them are expressed as nodes and directed links connecting them, respectively, where each attribute may be a predicate, a numerical data, etc., and each dependence is numerically expressed as the conditional probability of one attribute given other attributes if their dependence exists. Therefore, in general, BBNs are represented in terms of the network structure and the conditional probabilities.