This paper presents a graph-based approach to designing arithmetic circuits over Galois fields (GFs) using normal basis representations. The proposed method is based on a graph-based circuit description called Galois-field Arithmetic Circuit Graph (GF-ACG). First, we extend GF-ACG representation to describe GFs defined by normal basis in addition to polynomial basis. We then apply the extended design method to Massey-Omura parallel multipliers which are well known as typical multipliers based on normal basis. We present the formal description of the multipliers in a hierarchical manner and show that the verification time can be greatly reduced in comparison with those of the conventional techniques. In addition, we design GF exponentiation circuits consisting of the Massey-Omura parallel multipliers and an inversion circuit over composite field GF(((22)2)2) in order to demonstrate the advantages of normal-basis circuits over polynomial-basis ones.

We try to use a computer algebra system Mathematica as a test case generation system. In test case generation, we generally need to solve equations and inequalities. The main reason why we take Mathematica is because it has a built-in function to solve equations and inequalities. In this paper, we deal with both black-box testing and white-box testing. First, we show two black-box test case generation procedures described in Mathematica. The first one is based on equivalence partitioning. Mathematica explicitly shows a case that test cases do no exist. This is an advantage in using Mathematica. The second procedure is a modification of the first one adopting boundary value analysis. For implementation of boundary value analysis, we give a formalization for it. Next, we show a white-box test case generation procedure. For this purpose, we also give a model for source programs. It is like a control flow graph model. The proposed procedure analyzes a model description of a program.

Let G(s)=C(sI - A)-1B+D be a given system where entries of A,B,C,D are polynomials in a parameter k. Then H∞ norm || G(s) ||∞ of G(s) is a function of k, and [9] presents an algorithm to express 1/(||G(s) ||∞)2 as a root of a bivariate polynomial, assuming feedthrough term D to be zero. This paper extends the algorithm in two ways: The first extension is the form of the function to be expressed. The extended algorithm can treat, not only H∞ norm, but also functions that appear in the celebrated KYP Lemma. The other extension is the range of the frequency. While H∞ norm considers the supremum of the maximum singular value of G(i ω) for the infinite range 0 ≤ω ≤ ∞ of ω, the extended algorithm treats the norm for the finite frequency range ω ≤ ω ≤ ω- (ω, ω- ∈ R ∪ ∞). Those two extensions allow the algorithm to be applied to wider area of control problems. We give illustrative numerical examples where we apply the extended algorithm to the computation of the frequency-restricted norm, i.e., the supremum of the maximum singular value of G(i ω) (ω- ≤ ω ≤ ω-).

This paper presents a formal approach to verify arithmetic circuits using symbolic computer algebra. Our method describes arithmetic circuits directly with high-level mathematical objects based on weighted number systems and arithmetic formulae. Such circuit description can be effectively verified by polynomial reduction techniques using Grobner Bases. In this paper, we describe how the symbolic computer algebra can be used to describe and verify arithmetic circuits. The advantageous effects of the proposed approach are demonstrated through experimental verification of some arithmetic circuits such as multiply-accumulator and FIR filter. The result shows that the proposed approach has a definite possibility of verifying practical arithmetic circuits.

H∞ optimal control is one of the most successful achievements in the post modern control theory. In the H∞ optimal control, we design a controller that minimizes the H∞ norm of a given system. Although the algorithms to solve the problem have already been reported, they focus on numerical systems (systems without any unknown parameters) and, can not be applied for parametric systems (systems with unknown parameters). Given a parametric system, this paper presents an algorithm to compute the optimal H∞ norm of the system achieved by an output feedback controller. The optimal H∞ norm is expressed as , where φ(k) denotes a root of a bivariate polynomial. A numerical example is given to show the effectiveness of the algorithm.

In recent years, algorithms based on Computer Algebra ([1]-[3]) have been introduced into a range of control design problems because of the capacity to handle unknown parameters as indeterminates. This feature of algorithms in Computer Algebra reduces the costs of computer simulation and the trial and error process involved, enabling us to design and analyze systems more theoretically with the behavior of given parameters. In this paper, we apply Computer Algebra algorithms to H∞ control theory, representing one of the most successful achievements in post-modern control theory. More specifically, we consider the H∞ norm minimization problem using a state feedback controller. This problem can be formulated as follows: Suppose that we are given a plant described by the linear differential equation = Ax + B1w + B2u, z = Cx + Du, where A,B1,B2,C,D are matrices whose entries are polynomial in an unknown parameter k. We apply a state feedback controller u = -F x to the plant, where F is a design parameter, and obtain the system = (A - B2F)x + B1w, z =(C - DF)x. Our task is to compute the minimum H∞ norm of the transfer function G(s)(=(C - DF)(sI - A + B2F)-1B1) from w to z achieved using a static feedback controller u = -Fx, where F is a constant matrix. In the H∞ control theory, it is only possible to check if there is a controller such that ||G(s)||∞ < γ is satisfied for a given number γ, where ||G(s)||∞ denotes the H∞ norm of the transfer function G(s). Thus, a typical procedure to solve the H∞ optimal problem would involve a bisection method, which cannot be applied to plants with parameters. In this paper, we present a new method of solving the H∞ norm minimization problem that can be applied to plants with parameters. This method utilizes QE (Quantifier Elimination) and a variable elimination technique in Computer Algebra, and expresses the minimum of the H∞ norm as a root of a bivariate polynomial. We also present a numerical example to illustrate each step of the algorithm.

We introduce an efficient interpolation attack which gives the tighter upper bound of the complexity and the number of pairs of plaintexts and ciphertexts required for the attack. In the previously known interpolation attack there is a problem in that the required complexity for the attack can be overestimated. We solve this problem by first, finding the actual number of coefficients in the polynomial used in the attack by using a computer algebra system, and second, by finding the polynomial with fewer coefficients by choosing the plaintexts. We apply this interpolation attack to the block cipher SNAKE and succeeded in attacking many ciphers in the SNAKE family. When we evaluate the resistance of a block cipher to interpolation attack, it is necessary to apply the interpolation attack described in this paper.