Reducibility between different cryptographic primitives is a fundamental problem in modern cryptography. As one of the primitives, traitor tracing systems help content distributors recover the identities of users that collaborated in the pirate construction by tracing pirate decryption boxes. We present the first negative result on designing efficient traitor tracing systems via black-box constructions from symmetric cryptographic primitives, e.g. one-way functions. More specifically, we show that there is no secure traitor tracing scheme in the random oracle model, such that $\ell_k\cdot\ell_c^2\ge\widetilde{\Omega}(n)$, where $\ell_k$ is the length of user key, $\ell_c$ is the length of ciphertext and $n$ is the number of users, under the assumption that the scheme does not access the oracle to generate user keys. To our best knowledge, almost all the practical (non-artificial) cryptographic schemes (not limited to traitor tracing systems) via black-box constructions satisfy this assumption. Thus, our negative results indicate that most of the standard black-box reductions in cryptography cannot help construct a more efficient traitor tracing system.

We prove our results by extending the connection between traitor tracing systems and differentially private database sanitizers to the setting with random oracle access. After that, we prove the lower bound for traitor tracing schemes by constructing a differentially private sanitizer that only queries the random oracle polynomially many times. In order to reduce the query complexity of the sanitizer, we prove a large deviation bound for decision forests, which might be of independent interest.