The Lempel--Ziv 78 (LZ78) factorization is a well-studied technique for data compression. It and its derivatives are used in compression formats such as "compress" or "gif". Although most research focuses on the factorization of plain data, not much research has been conducted on indexing the data for fast LZ78 factorization. Here, we study the LZ78 factorization and its derivatives in the substring compression model, where we are allowed to index the data and return the factorization of a substring specified at query time. In that model, we propose an algorithm that works in compressed space, computing the factorization with a logarithmic slowdown compared to the optimal time complexity.