In recent years, content identification based on digital fingerprinting attracts a lot of attention in different emerging applications. At the same time, the theoretical analysis of digital fingerprinting systems for finite length case remains an open issue. Additionally, privacy leaks caused by fingerprint storage, distribution and sharing in a public domain via third party outsourced services cause certain concerns in the cryptographic community. In this paper, we perform an information-theoretic analysis of finite length digital fingerprinting systems in a private content identification setup and reveal certain connections between fingerprint based content identification and Forney's erasure/list decoding. Along this analysis, we also consider complexity issues of fast content identification in large databases on remote untrusted servers.