In 1989, Tsujii, Fujioka, and Hirayama proposed a family of multivariate public key cryptosystems, where the public key is given as a set of multivariate rational functions of degree 4. These cryptosystems are constructed via composition of two quadratic rational maps. In this paper, we present the cryptanalysis of this family of cryptosystems. The key point of our attack is to transform a problem of decomposition of two rational maps into a problem of decomposition of two polynomial maps. We develop a new improved 2R decomposition method and other new techniques, which allows us to find an equivalent decomposition of the rational maps to break the system completely. For the example suggested for practical applications, it is very fast to derive an equivalent private key, and it requires only a few seconds on a standard PC.

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Let <i>A</i> and <i>B</i> be non-negative self-adjoint operators in a separable Hilbert space such that their form sum <i>C</i> is densely defined. It is shown that the Trotter product formula holds for imaginary parameter values in the <i>L</i> ²-norm, that is, one has <div class="gsh_dspfr"> \lim_{no+\infty} \int\limits^T_{-T} \left\|\left(e^{-itA/n}e^{-itB/n} ight)^nh - e^{-itC}hight\|^2dt = 0

We consider a two-dimensional particle of charge e interacting with a homogeneous magnetic field perpendicular to the plane and a potential well which is transported along a closed loop in the plane. We show that a bound state corresponding to a simple isolated eigenvalue acquires at that Berry's phase equal to 2 sgn e times the number of flux quanta through the oriented area encircled by the loop. We also argue that this is a purely quantum effect since the corresponding Hannay angle is zero.

We consider the Laplacian in a straight planar strip with Dirichlet boundary which has two Neumann'windows' of the same length, the centres of which are 2l apart, and study the asymptotic behaviour of the discrete spectrum as l. It is shown that there are pairs of eigenvalues around each isolated eigenvalue of a single-window strip and their distances vanish exponentially in the limit l. We derive an asymptotic expansion also in the case where a single window gives rise to a threshold resonance which the presence of the other window turns into a single isolated eigenvalue.