The article concerns the selected approaches to integer factorization problem of positive integers n with the aid of elliptic curves E over \(\mathbb {Z}_n\) . It is focusing on the investigation of partially smooth numbers and their application to elliptic-curve factoring (ECF) approach by Lenstra and the ECF with oracle (ECFO), both in the probabilistic and deterministic approaches. We define the separating and non-separating decomposition witnesses for n, which play the significant role in both ECF and ECFO approaches and apply them for semiprime numbers \(n=pq\) and admissible elliptic curves E over \(\mathbb {Z}_n\) .