Our research group is composed of scholars working in the areas of discrete mathematics, nonlinear differential equations, information theory and numerical computation. We have organized "Seminar on Digital Analysis" so that members can hold common understanding and insight on the fundamental theories and ideas of digital mathematics. As speakers of this seminar, we have invited 16 researchers who are highly active in the areas of discrete mathematics, mathematical modeling, information theory and numerical computation. We have succeeded in getting common understanding on digital analysis through exciting discussions in each lecture of the seminar.

Stream ciphers provide probably the most important method of modern encipherment. The central problem in stream cipher cryptography is the difficulty of efficiently generating long unpredictable sequencers of binary signals form a short and random key. Although such an unpredictable sequence can be generated in various ways, linear feedback shift register (LFSR) sequences are employed in nearly all the methods. In this study, not using such sequences, we give a simple method to obtain a running key sequence of stream cipher system in which chaotic orbits of nonlinear maps are used. Two type of sequences of balanced I. I. D. (dependent and identically distributed) binary random variables have been defined, referred to as a chaotic binary sequence and a chaotic bit sequence, each of which is obtained from chaotic real-valued orbits generated by nonlinear maps. The stream cipher system proposed in this study has the following characteristics : 1) There are several parameters as short secret key : 2) The linear complexity of the running-key sequence of period N is nearly equal to N/2 : 3) The correlation property of ciphertexts is as good as those of the DES and FEAL

(1) We summarized the state of the art of this research subject and classified systematically various analysis methods so far known.(2) We pointed out that both the characteristics impedance function ZETA_0(s) and a bilinear, functionof the phase characteristics rheta(s) of distributed elements have to bc approximated by rational positive real functions. We proposed the method for approximating these quantities by positive real functions. This resulted in better property of convergence than the conventional Pade approximation method.(3) A new transient analysis method for multi-conductor transmission line terminated with lumped-constant elements was proposed and the validity by numerical simulation was shown. The method utilizes multi-dimensional WDF (Wave Digital Filter), which Fettweis proposed for partial differential equation. Our method solved the difficulty arising from the treatment of mixed lumped- and distributed systems.(4) fi new equivalent circuit for a dielectric Alter was proposed, where the equivalent circuit consists.i multiconductor transmission line and lumped constant elements and its element values are determined from experimental results. The total characteristics of the circuit are much coincident with physical results over wide frequency than usual model.(5) The properties of lossless bandpass ladder network composed of only two kinds of resonant sections consisting of lumped- and distributed-elements were investigated and the necessary and sufficient conditions for realization were given.(6) A theoretical lower bound for the ratio of the maximum error of the minimal approximation to that of the the least pth approximation by a rational function is given. Numerical examples on various kind of functions verified that the above lower bound is a good estimation for the actual values

We had the purpose to establish fundamental theories and to constitute the elements of computer aided mathematical analyzing software. We developed our research as we planned as the following :
(1) We proposed the fixed point theorem for fuzzy map which is obtained by modeling the system with uncertain property.
(2) We make the theory to prove numerically the existence of solution for nonlinear operator equations.
(3) Based on C++ and an object oriented language in which rational number arithmetic is implemented, we constituted 3 prototypes of object oriented software which can deal with various objects corresponding to interval analysis, automatic differentiation, function representation and so on.
(4) By extending the methods to prove numerically the existence of bifurcation point, we developed the theory to cancel singular points. We also applied our theory to various types of bifurcation phenomena and indicated that we can prove the existence of actual bifurcation phenomena.
(5) We proposed the theory to prove the existence of homoclinic orbits or heteroclinic orbits and prove their existence for actual examples.
(6) We proposed an algorithm to prove the existence of all solutions in a bounded region for finite dimensional nonlinear equations and proved that this algorithm stops within finite steps under the certain conditions.
(7) We proposed a method to prove the existence of all solutions with high speed in a bounded region for finite dimensional nonlinear equations with separability, whose example is VLSI circuit. (8) We could change the speed of calculation by the accuracy. Concretely, We proposed the method in which we can obtain the calculated results with super high speed when we demand its low accuracy and in which we can obtain the results with high speed even when we demand its high accuracy.
(9) We realized the obtained techniques of numerical method with guaranteed accuracy on our prototypes of the software. We applied our software to various nonlinear functional equations and indicated its usefulness.
(10) We combined the numerical method in the case that we demand its low accuracy and the numerical one in the case that we demand its high accuracy, by which we proposed the numerical method with high speed at our request of accuracy. We realized this method on our software and indicated its usefulness by the example of nonlinear circuit systems.
(11) We integrated the above organized investigations and remade a prototype of the software for computer aided nonlinear analysis which can correspond to the changes of the problem or the accuracy. We also indicated its usefulness by applying it to nonlinear circuit problems.

Recently, the study of nonlinear systems and nonlinear technologes has made a great advance. For example, the studies of optical fiber soliton communications, neural networks, fuzzy systems, analogue VLSI and so on have achieved much interests. Since such systems essentially use nonlinearity , new modelling techniques and reliable simulation techniques are required. Moreover, in the field of computer assisted design of nonlinear systems, it is very important to guarantee the accuracy of the result of calculation. For example, in the VLSI design, if we can validate the accuracy of modeling and numerical simulation, it enables to reduce the cost and period of design.
In this study, taking the accuracy of modeling into consideration, and guarantteng the accuracy of numerical simulation of the modeling, we have develpoed a numerical validation method through the total simulation process of nonlinear system.
In this year, we have improved the theory, algorithm and system developed until last year, and have made them more practical.
1.Effectiveness of our fuzzy modeling theory is confirmed by numerical simulation.
2.Automatic numerical validation method for general ordinary differential equations is developed. It provides a rigorous method for transient analysis.
3.Extending the technique used in all solution method for nonlinear equations, we have developed a inclusion method for solution sets of set-valued functions. It gives a more rigorous system analysing method together with modeling method with guaranteed accuracy .