JSIAM Letters Vol. 9(2017)

In this paper, we provide some numerical results for Malliavin sensitivity analysis in finance. We consider payoff functions with polynomial growth, and with and without discontinuous points and compare the results to that of the finite-difference and pathwise methods.

Permutation polynomials over a ring of modulo $2^w$ are well adopted to digital computers and digital signal processors, and so they are in particular expected to be useful for cryptography and pseudo random number generator{s}. For a longer period of the polynomial is demanded in general, we derive a necessary and sufficient condition that polynomials are permutating and their periods are the longest over the ring. We call polynomials which satisfy the condition ``one-stroke polynomials over the ring''.

Estimating the Frobenius norm of a matrix product C=XY without computing C explicitly is required in applications such as the one-sided block Jacobi method. In this paper, we analyze Bečka et al.'s estimator for this problem within a probabilistic framework. Specifically, we consider the set of matrices with the Frobenius norm $\|C\|_F^2$ and introduce some natural probability measure into it. Then, we show that if we choose a matrix randomly from this set and apply the estimator, the expected value of the square of this estimator is exactly $\|C\|_F^2$.

This paper presents a new methodology to compute first-order Greeks for barrier options under the framework of path-dependent payoff functions with European, Lookback, or Asian type and with time-dependent trigger levels. In particular, we develop chain rules for Wiener path integrals between two curves that arise in the computation of first-order Greeks for barrier options. We also illustrate the effectiveness of our method through numerical examples.

The Sakurai-Sugiura method with Rayleigh-Ritz projection (SS-RR method) finds the eigenvalues in a certain domain of the complex plane of large quadratic eigenvalue problems (QEPs). The SS-RR method can suffer from numerical instability when the coefficient matrices of the projected QEP vary widely in norm. To improve the numerical stability of the SS-RR method, we combine it with a numerically stable eigensolver for the small projected QEP. We analyze the backward stability of the proposed method and show, through numerical experiments, that it computes eigenpairs with backward errors that are smaller than those computed by the SS-RR method.

HFE (Patarin, Eurocrypt'96) is one of the most famous multivariate public key cryptosystems. Unfortunately, HFE has a serious trade-off between security and efficiency, which lacks HFE's practicality. Recently, Porras et al. proposed a new encryption scheme ZHFE at PQCrypto 2014. While its construction is similar to HFE, the security seems more than HFE. The present paper proposes a chosen ciphertext attack (CCA) on ZHFE. The CCA reduces the problem of recovering the univariate polynomial for decryption to the min-rank problem on HFE. Thus the CCA security of ZHFE is almost the same as the security of HFE against the min-rank attack.

We propose a mapping operator construction technique for error correction methods applied to a series of linear systems. The mapping operator is constructed based on information obtained in the previous solution step. The additional memory requirement for the construction is limited. Our numerical experiments based on an electromagnetic field analysis confirmed that the correction method with the proposed mapping operator improves the convergence of the iterative solver.

Charles et al. proposed hash functions based on the difficulty of computing isogenies between supersingular elliptic curves. Yoshida and Takashima then improved the 2-isogeny hash function computation by using some specific properties of 2-torsion points. In this paper, we extend the technique to 3-isogenies and give the efficient 3-isogeny hash computation based on a simple representation of the (backtracking) 3-torsion point. Moreover, we implement the 2- and 3-isogeny hash functions using Magma and show our 3-isogeny proposal has a comparable efficiency with the 2-isogeny one.

Hamiltonian Monte Carlo is a Markov chain Monte Carlo method that uses Hamiltonian dynamics to efficiently produce distant samples. It employs geometric numerical integration to simulate Hamiltonian dynamics, which is a key of its high performance. We present a Hamiltonian Monte Carlo method with adaptive step size control to further enhance the efficiency. We propose a new explicit, reversible, and volume-preserving integration method to adaptively set the step sizes, which does not violate the detailed balance condition or require a large increase in computational time.

We carry out a linear stability analysis for a free boundary between two fluids in a Hele-Shaw cell, which is driven by the Darcy's law and an injection or suction at the center. In general, stability of an interface is determined by the well-known Saffman-Taylor instability condition. The linear growth rate of the perturbation depends on two parameters; the rate of the injection/suction and the viscosity contrast of the two fluids. In this paper, we numerically find a parameter region, in which the interface can be stable (resp. unstable) even though it has been considered to be unstable (resp. stable) due to the Saffman-Taylor instability. For the case of the injection, it is suggested that such parameter region vanishes as time increases to infinity. However, the destabilization can be retarded for a sufficiently long time, as one tunes the viscosity and the injection rate of the injecting fluid.

The aim of this paper is to develop the method of fundamental solutions using weighted average condition and dummy points. We accomplish mathematical analysis, a unique existence of an approximate solution and an exponential decay of the approximation error, for a potential problem in disk, and show some numerical experiments, which exemplify our error estimate.

An efficient quadrature formula, known as the Single-Exponential (SE) formula, was proposed by Stenger for the definite integral of an exponentially decaying function over the semi-infinite interval. The formula was derived by combining the trapezoidal formula with a SE transformation. An error bound of the formula was already given. In this study, we investigate another SE formula obtained by replacing the transformation with Muhammad-Mori's SE transformation. Its error bound was determined by theoretical analysis. Numerical comparisons of Stenger's SE formula with that of Muhammad-Mori's are given as well.