Searchable Encryption schemes allow searching within encrypted data without prior decryption. Various index-based schemes have been proposed in the past, which are only adequate for certain use cases. There is a lack of schemes with exact pattern matching capabilities. We introduce Symmetric Searchable Encryption for Exact Pattern Matching, a new class of searchable encryption schemes. To this end, we define the XPM-SSE primitive and two privacy notions for the new primitive. Our own construction, SEDAWG, is a XPM-SSE scheme which uses Directed Acyclic Word Graphs. We discuss and prove its properties.