The problem of factoring a composite integer into the product of two distinct primes (the factoring problem) is one of the famous hard problems on which the security of many cryptographic primitives relies. In this paper, we introduce a new cryptographic hard problem (RSA-polynomial problem) and prove that solving the RSA-polynomial problem is at least as hard as solving the factoring problem. As applications of the RSA-polynomial problem, we propose a commitment scheme. The proposed scheme is free of any group-exponentiation and outperforms the previous commitment schemes. Also, using the lattice basis reduction techniques and the RSA-polynomial problem, we propose a method to factor composite integers that are the product of two distinct primes.

将复合整数因式分解为两个不同素数的乘积的问题（因式分解问题）是许多密码原语的安全性所依赖的著名难题之一。在本文中，我们引入了一个新的密码学难题（RSA 多项式问题），并证明解决 RSA 多项式问题至少与解决因式分解问题一样困难。作为 RSA 多项式问题的应用，我们提出了一个承诺方案。所提出的方案没有任何群求幂，并且优于之前的承诺方案。此外，使用格基约简技术和 RSA 多项式问题，我们提出了一种分解作为两个不同素数乘积的复合整数的方法。