In 1969, Riesel proposed a primality test for natural numbers of the form $h \cdot 2^n-1$, which includes the case $h=n$, the Woodall numbers $W_ {n} = n \cdot 2 ^ {n} -1$. In this paper, we utilize Riesel's primality test for Woodall numbers, and propose a primality testing algorithm for Woodall numbers. The implementations of the algorithm and its optimization are discussed.

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In 1948, Ward defined elliptic divisibility sequences satisfying a certain recurrence relation. An elliptic divisibility sequence arises from any choice of elliptic curve and initial point on that curve. In this paper, we define a hard problem in the theory of elliptic divisibility sequences (EDS-DH problem), which is computationally equivalent to the elliptic curve Diffie-Hellman problem.

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The tanh rule and the double-exponential (DE) formula are known empirically and theoretically as quite efficient quadrature formulas, especially for integrals with endpoint singularity, including algebraic singularity and logarithmic singularity. Furthermore, in the case of integrals with algebraic singularity, explicit error bounds have been given for those formulas, which enables us to guarantee their approximation accuracy. In the case of integrals with logarithmic singularity, however, such explicit error bounds have not ever given thus far, although those formulas should work accurately in this case as well. This paper presents the desired theoretical explicit error bounds, with numerical experiments.

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Boneh and Franklin proposed a practical Identity-Based Encryption (IBE) in 2001 [1]. In order to embed an identity of users in the IBE, we need a hash function, called HashToPoint. The dominant computation of HashToPoint is the scalar multiplication by a large cofactor $c$, which is relatively expensive compared with other cryptographic functions in the IBE. In this paper, we present a list of cofactor $c$ with Hamming weight two, which can accelerate the computation of HashToPoint. Indeed the timing of our implementation of HashToPoint using the proposed cofactor is reduced by about 30% on a desktop PC.

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In the real market, the term structure of forward rates exhibits some humps. The quadratic Gaussian term structure models or affine term structure models well explain this phenomena. In this research, we give a new insight, where we understand the humps as multi-soliton that are related to KdV solitons.

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We describe a new transform that generates a dictionary of bases for handling data on a graph by combining recursive partitioning of the graph and the Laplacian eigenvectors of each subgraph. Similar to the wavelet packet and local cosine dictionaries for regularly sampled signals, this dictionary of bases on the graph allows one to select an orthonormal basis that is most suitable to one's task at hand using a best-basis type algorithm. We also describe a few related transforms including a version of the Haar wavelet transform on a graph, each of which may be useful in its own right.

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In a preceding paper, we proved the discrete compactness properties of Rellich type for some 2D discontinuous Galerkin finite element methods (DGFEM), that is, the strong $L^2$ convergence of some subfamily of finite element functions bounded in an $H^1$-like mesh-dependent norm. In this note, we will show the strong $L^p$ convergence of the above subfamily for $1 \le p < \infty$. To this end, we will utilize the duality mappings and special auxiliary problems. The results are applicable to numerical analysis of various semi-linear problems.

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This paper presents analytic solutions of the shape derivatives (Fréchet derivatives with respect to domain variation) for singular points of cost functions in shape-optimization problems for the domain in which the boundary value problem of a partial differential equation is defined. A design variable is given by a domain mapping. Cost functions are defined as functionals of the design variable and the solution to the boundary value problem. The analytic solutions for singular points such as crack tips and boundary points of the mixed boundary conditions on a smooth boundary are obtained by using the generalized $J$ integral.

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We give an algorithm which computes $r$, defined by K. Kato in the paper [1], which is an important invariant for Artin-Schreier extensions of surfaces $X$ over fields of positive characteristic. The Swan conductor gives the invariant of ramifications concerning codimension 1 subvarieties of $X$. This $r$ gives the invariant of ramifications concerning codimension 2 subvarieties of $X$. The invariant $r$ is important to calculate the Euler Poincaré characteristic of some smooth $l$-adic sheaf of rank 1 on an open dense subscheme $U$ of $X$.

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This paper analyzes the problem of speeding up single-scalar multiplication of a recently introduced type of elliptic curve, so-called ``twisted Edwards curve", and also presents a new construction of addition chains using the extended double-base number system. Our method uses the Fibonacci sequence. It was found through numerical investigation that our double-base chains can save time, compared with other methods in previous work.

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In the studies of mathematical statistics, we often consider discrete distributions and their corresponding stochastic processes. Especially, probabilistic limit theorems of them may give us some progress in mathematical finance. There exist not so many properties of discrete distributions on $\mathbb{R}^d$. In this paper, we treat multiple zeta functions as to define several forms of discrete distributions on $\mathbb{R}^d$ including those with infinitely many mass points. Our purpose is to obtain new methods in the relations between multiple infinite series and high dimensional integral calculus, which can provide us more opportunities to handle high dimensional phenomenon.

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There exist not many treatable multivariable functions with respect to multidimensional discrete distributions. In this paper, we pick up multidimensional finite Euler products and show when they can generate characteristic functions. Moreover, the infinite divisibility of them are studied as well as non-infinite divisibility which are rarely seen in multidimensional discrete case with infinitely many mass points. The relation between series representations of zeta functions is also studied by adjusting to the Shintani zeta type.

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In this paper we propose a practical cost-effective model to estimate the credit risk of a large portfolio of real estate non-recourse loans. It uses information that is as easy to get and update as possible, such as real estate investment indices and macroeconomic indices. Empirical characteristics of real estates can be taken into account, such as serial correlations, cross-sectional correlations within individual properties, lagged effects of macroeconomic factors.

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The security of the optimal Ate pairing using the BN curves is based on the hardness of the DLP over $\text{GF}(p^{12})$. At CRYPTO 2006, Joux et al. proposed the number field sieve over $\text{GF}(p^n)$, but the number field sieve needs multi-dimensional sieving. In this paper, we deal with the multi-dimensional sieving, and discuss its parameter sizes such as the dimension of sieving and the size of the sieving region from some experiments of the multi-dimensional sieving. Using efficient parameters, we have solved the DLP over $\text{GF}(p^{12})$ of 203 bits in about 43 hours using a PC of 16 CPU cores.

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We analyze convergence properties of the parallel classical block Jacobi method for the symmetric eigenvalue problem using dynamic ordering strategy of Bečka et al. It is shown that the method is globally convergent. It is also shown that the order of convergence is ultimately quadratic if there are no multiple eigenvalues.

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Block Krylov subspace methods are efficient solvers for linear systems with multiple right-hand sides in terms of the number of iterations and computational time. As one of Block Krylov subspace methods, the Block BiCR method has been proposed by Zhang et al. in 2013. This method often shows a smooth convergence behavior compared with the Block BiCG method. However, the accuracy of the approximate solution generated by the Block BiCR method often deteriorates. In this paper we propose a modified Block BiCR method in order to improve the accuracy of the approximate solutions.

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In this paper, we derive the Block BiCGSTAB($\ell$) method which is developed by extending the BiCGSTAB($\ell$) method. We also propose some techniques to improve convergence properties by applying orthogonalization and to improve the accuracy of the approximate solutions by additional matrix multiplications. Some numerical experiments indicate that the performance of the Block BiCGSTAB($\ell$) method with those stabilization techniques can be higher than that of the Block BiCGSTAB method.

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The pairing-based cryptosystem was proposed in 2001, and it provides efficient implementations of identity-based encryption (IBE) and attribute-based encryption (ABE). In 2010, Costello and Stebila introduced the concept of fixed argument pairing, which can be applied to many applications of pairings, and, to compute these pairings, they proposed an efficient algorithm based on the Miller algorithm. In this paper, we propose a method for computing fixed argument pairings, based on the elliptic net method proposed by Stange.

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Estimating the number of pairing-friendly elliptic curves is important for obtaining such a curve with a suitable security level and high efficiency. For 128-bit security level, M. Naehrig and J. Boxall estimated the number of Barreto-Naehrig (BN) curves. For future use, we extend their results to higher security levels, that is, to count Kachisa-Schaefer-Scott (KSS) curves with 192- and 224-bit security levels. Our efficient counting is based on a number-theoretic conjecture, called the Bateman-Horn conjecture. We verify the validity of using the conjecture and confirm that an enough amount of KSS curves can be obtained for practical use.

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In the present paper, we introduce a framework of a discrete stochastic calculus based on Parisian walk, a special kind of symmetric random walk in the complex plane, listing some results analogue to those for complex Brownian motion. We also discuss, as an application to mathematical finance, a Parisian-walk analogue of Heston's stochastic volatility model.

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The Strassen algorithm and Winograd's variant accelerate matrix multiplication by using fewer arithmetic operations than standard matrix multiplication. Although many papers have been published to accelerate single- as well as double-precision matrix multiplication by using these algorithms, no research to date has been undertaken to accelerate multiple precision matrix multiplication. In this paper, we propose a multiple precision matrix multiplication program for matrices of any size and test its performance. We also reveal special properties of our program through its application to LU decomposition.

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The aim of this article is to present a cryptosystem with a new key exchange protocol based on Diophantine equations of polynomial type. Our protocol is inspired by that of H. Yosh whose security comes from a translation of Diophantine equations. We suggest here a key exchange protocol relying on the hardness of solving Diophantine equations in the ring of $S$-integers.

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The present paper describes a solution to a non-parametric shape optimization problem of a rubber bushing in order to adjust a function of the reaction force with respect to static displacement to a desired function. The main problem is defined as a static hyperelastic problem considering a large deformation and a non-linear constitutive equation. The squared error norm of the work done by compulsory displacement and the volume are chosen as cost functions. The shape derivatives of the cost functions are derived theoretically. An iterative algorithm based on the $H^1$ gradient method is used to solve the shape optimization problem.

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