In this paper, we study the generic hardness of the inversion problem on a ring, which is a problem to compute the inverse of a given prime c by just using additions, subtractions and multiplications on the ring. If the characteristic of an underlying ring is public and coprime to c, then it is easy to compute the inverse of c by using the extended Euclidean algorithm. On the other hand, if the characteristic is hidden, it seems difficult to compute it. For discussing the generic hardness of the inversion problem, we first extend existing generic ring models to capture a ring of an unknown characteristic. Then we prove that there is no generic algorithm to solve the inversion problem in our model when the underlying ring is isomorphic to ${\mathbb{Z}}_p$ for a randomly chosen prime p assuming the hardness of factorization of an unbalanced modulus. We also study a relation between the inversion problem on a ring and a self-bilinear map. Namely, we give a construction of a self-bilinear map based on a ring on which the inversion problem is hard, and prove that natural complexity assumptions including the multilinear computational Diffie-Hellman (MCDH) assumption hold w.r.t. the resulting sef-bilinear map.

在本文中，我们研究了环上反演问题的一般硬度，这是仅通过在环上使用加法，减法和乘法来计算给定素数c的逆的问题。如果基础环的特征是公共的并且是c的素数，那么很容易计算c的逆通过使用扩展的欧几里得算法。另一方面，如果特征被隐藏，则似乎很难对其进行计算。为了讨论反演问题的通用硬度，我们首先扩展现有的通用环模型以捕获未知特性的环。然后我们证明当基础环同构为同构时，模型中没有通用算法可以解决反演问题${\mathbb{ž}}_p$对于随机选择的素数p，假设不平衡模量的因式分解的硬度。我们还研究了环上的反演问题和自双线性映射之间的关系。即，我们给出了一个基于环的自双线性图的构造，在该环上反演问题比较困难，并证明了自然复杂度假设（包括多线性计算Diffie-Hellman（MCDH）假设）在所得的sef-双线性图中都成立。