Smooth factors of integers and elliptic curve based factoring with an oracle

Let $a_E(n)$ be the $n$-th coefficieint of the Hasse–Weil L-function $L(s,E)$ and $B=B(n)$ be relatively small. We deal with the factorization method with an oracle \lt em \gt PartOrd \lt /em \gt which for a point $Q\in E(\mathbb{Z}_n)$ returnes its maximal $B$-smooth factor. We analyse the conditions under which the related factoring can be successful even if the order of $Q$ is not a $B$-smooth number. Moreover we prove the conditional results on factorization of $n=pq$ if $a_E(n)$ is known, or $n=pqr$ if $a_E(n)$ and $a_E(p) qr + a_E(q)pr +a_E(r)pq$ are known.