(1) THE NON PROBABILISTIC APPROACH TO LEARNING THE BEST PREDICTION
(2) THE SIMPLE IDEAL CIPHER SYSTEM
Professor Boris Ryabko, Department of Applied Maths and Cybernetics, Siberian State University of Telecommunication and Computer Science

(1) The nonprobabilistic approach to learning the best prediction

Abstract: The problem of predicting a sequence $x_1, x_2,.... $ where each $x_i$ belongs to a finite alphabet $A$ is considered. Each letter $x_{t+1}$ is predicted using information on the word $x_1, x_2, ...., x_t $ only. We use the game theoretical interpretation which can be traced to Laplace where there exists a gambler who tries to estimate probabilities for the letter $x_{t+1}$ in order to maximize his capital. The optimal method of prediction is described for the case when the sequence $x_1, x_2,.... $ is generated by a stationary and ergodic source. It turns out that the optimal method is based only on estimations of conditional probabilities. In particular, it means that if we work in the framework of the ergodic and stationary source model, we cannot consider pattern recognition and other complex and interesting tools, because even the optimal method does not need them. That is why we suggest a so-called nonprobabilistic approach which is not based on the stationary and ergodic source model and show that complex algorithms of prediction can be considered in the framework of this approach.

The new approach is to consider a set of all infinite sequences (over a given finite alphabet) and estimate the size of sets of predictable sequences with the help of the Hausdorff dimension. This approach enables us first, to show that there exist large sets of well predictable sequences which have zero measure for each stationary and ergodic measure. (In fact, it means that such sets are invisible in the framework of the ergodic and stationary source model and shows the necessity of the new approach.) Second, it is shown that there exist quite large sets of such sequences that can be predicted well by complex algorithms which use not only estimations of conditional probabilities.

(2) The simple ideal cipher system

Abstract: It is well known in cryptography that it is easy to construct an unbreakable secret-key cipher system if a plaintext source generates letters which are independent and have equal probabilities even in case the length of a key sequence is much less than length of message. We suggest a new secret-key cipher system in which a generated message is transformed into two parts in such a way that the biggest part consists of independent and equiprobable bits and only this part is encrypted. The complexity of the method is exponentially less than that for other known methods. (Complexity of a method is characterized by two numbers: The time of calculation (in bit operations) and the memory size (in bits) of the computer which is necessary in order to execute the program defining a coder and a decoder.

These seminars were held at the Department of Computer Science, Royal Holloway, University of London on 24 September 2001.

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