We study a static analysis problem on k-secrecy, which is a metric for the security against inference attacks on XML databases. Intuitively, k-secrecy means that the number of candidates of sensitive data of a given database instance or the result of unauthorized query cannot be narrowed down to k-1 by using available information such as authorized queries and their results. In this paper, we investigate the decidability of the schema k-secrecy problem defined as follows: for a given XML database schema, an authorized query and an unauthorized query, decide whether every database instance conforming to the given schema is k-secret. We first show that the schema k-secrecy problem is undecidable for any finite k>1 even when queries are represented by a simple subclass of linear deterministic top-down tree transducers (LDTT). We next show that the schema ∞-secrecy problem is decidable for queries represented by LDTT. We give an algorithm for deciding the schema ∞-secrecy problem and analyze its time complexity. We show the schema ∞-secrecy problem is EXPTIME-complete for LDTT. Moreover, we show similar results LDTT with regular look-ahead.