This paper introduces a new algorithm for factoring positive integers using elliptic curves. The method is derived from Pollard's $p-1$-method by replacing the multiplicative group with the group of points on a random elliptic curve. The algorithm aims to find a non-trivial divisor of a composite number $n$ in expected time at most $K(p! \log n)^2$, where $p$ is the least prime dividing $n$ and $K$ is a function with $\log K(x) = \sqrt{(2 + o(1)) \log \log \log x}$ for $x \to \infty$. The algorithm is particularly efficient for small prime factors $p$. The paper also discusses the analysis of the algorithm, including the probability of success and the expected running time. The method is compared with other factoring algorithms, such as the class group method and the quadratic sieve, highlighting its advantages in terms of running time and the size of prime factors.This paper introduces a new algorithm for factoring positive integers using elliptic curves. The method is derived from Pollard's $p-1$-method by replacing the multiplicative group with the group of points on a random elliptic curve. The algorithm aims to find a non-trivial divisor of a composite number $n$ in expected time at most $K(p! \log n)^2$, where $p$ is the least prime dividing $n$ and $K$ is a function with $\log K(x) = \sqrt{(2 + o(1)) \log \log \log x}$ for $x \to \infty$. The algorithm is particularly efficient for small prime factors $p$. The paper also discusses the analysis of the algorithm, including the probability of success and the expected running time. The method is compared with other factoring algorithms, such as the class group method and the quadratic sieve, highlighting its advantages in terms of running time and the size of prime factors.