Introduction to algebraic approaches for solving isogeny path-finding problems (Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties)

The isogeny path-finding is a computational problem that finds an isogeny connecting two given isogenous elliptic curves. The hardness of the isogeny path-finding problem supports the fundamental security of isogeny-based cryptosystems. In this paper, we introduce an algebraic approach for solving the isogeny path-finding problem. The basic idea is to reduce the isogeny problem to a system of algebraic equations using modular polynomials, and to solve the system by Gröbner basis computation. We report running time of the algebraic approach for solving the isogeny path-finding problem of 3-power isogeny degrees on supersingular elliptic curves. This is a brief summary of [16] with implementation codes

写真アルバムを「開く」 -1930年代製作のアルバムを事例として-

departmental bulletin pape

Weak rate of convergence of the Euler-Maruyama scheme for stochastic differential equations with non-regular drift

International audienceWe consider an Euler-Maruyama type approximation method for a stochastic differential equation (SDE) with a non-regular drift and regular diffusion coefficient. The method regu-larizes the drift coefficient within a certain class of functions and then the Euler-Maruyama scheme for the regularized scheme is used as an approximation. This methodology gives two errors. The first one is the error of regularization of the drift coefficient within a given class of parametrized functions. The second one is the error of the regularized Euler-Maruyama scheme. After an optimization procedure with respect to the parameters we obtain various rates, which improve other known results

On Weak Approximation of Stochastic Differential Equations with Discontinuous Drift Coefficient

This is an abridged version submitted in a conference proceedings.International audienceIn this paper, weak approximations of multi-dimensional stochastic differential equations with discontinuous drift coefficients are considered. Here as the approximated process, the Euler-Maruyama approximation of SDEs with approximated drift coefficients is used, and we provide a rate of weak convergence of them. Finally we present a rate of weak convergence of the Euler-Maruyama approximation of the original SDEs with constant diffusion coefficients