No abstract

This paper addresses learning stochastic rules especially on an inter-attribute relation based on a Minimum Description Length (MDL) principle with a finite number of examples, assuming an application to the design of in telligent relational database systems. The stochastic rule in this paper consists of a model giving the structure like the depen dencies of a Bayesian Belief Network (BBN) and. some stochastic parameters each indicat ing a conditional probability of an attribute value given the state determined by the other attributes' values in the same record. Espe cially, we propose the extended version of the algorithm of Chow and Liu in that our learn ing algorithm selects the model in the range where the dependencies among the attributes are represented by some general plural num ber of trees.

In Bayes score-based Bayesian network structure learning (BNSL), we are to specify two prior probabilities: over the structures and over the parameters. In this paper, we mainly consider the parameter priors, in particular for the BDeu (Bayesian Dirichlet equivalent uniform) and Jeffreys' prior. In model selection, given examples, we typically consider how well a model explains the examples and how simple the model is, and choose the best one for the criteria. In this sense, if a model A is better than another model B for both of the two criteria, it is reasonable to choose the model A. In this paper, we prove that the BDeu violates such a regularity, and that we will face a fatal situation in BNSL: the BDeu tends to add a variable to the current parent set of a variable X even when the conditional entropy reaches to zero. In general, priors should be reflected by the learner's belief, and should not be rejected from a general point of view. However, this paper suggests that the underlying belief of the BDeu contradicts with our intuition in some cases, which has not been known until this paper appears.

This paper shows a theoretical property on the Markov chain of genetic algorithms: the stationary distribution focuses on the uniform population with the optimal solution as mutation and crossover probabilities go to zero and some selective pressure defined in this paper goes to infinity. Moreover, as a result, a sufficient condition for ergodicity is derived when a simulated annealing-like strategy is considered. Additionally, the uniform crossover counterpart of the Vose-Liepins formula is derived using the Markov chain model.

Abstract. 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 miller's 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. IntroductionThe discrete logarithm problem for the multiplicative group * q of a finite field can be solved in subexponential time using the Index Calculus method, which appears to have been first discovered by Kraitchik [14, 15] in the 1920's and subsequently rediscovered and extended by many mathematicians. (See, for example, [1] and [43], and for a nice summary of the current state-of-the-art, see [29].) For this reason, it was proposed independently by Miller [23] and Koblitz [12] that for cryptographic purposes, one should replace * q by the group of rational points E( q ) on an elliptic curve, thus leading to the Elliptic Curve Discrete Logarithm Problem, which we abbreviate as the ECDL problem. Indeed, Victor Miller gives in his article [23, page 423] two reasons why "it is extremely unlikely that an 'index calculus' attack on elliptic curves will ever be able to work." Miller's reasons may be briefly summarized as follows:(1) It is difficult to find elliptic curves E/É with a large number of small rational points. This observation may be split into two pieces. (a) It is difficult to find elliptic curves E/É with high rank. (b) It is difficult to find elliptic curves E/É generated by points of small height.