Watermarking pseudorandom functions (PRF) allow an authority to embed an unforgeable and unremovable watermark into a PRF while preserving its functionality. In this work, we extend the work of Kim and Wu [Crypto'19] who gave a simple two-step construction of watermarking PRFs from a class of extractable PRFs satisfying several other properties – first construct a mark-embedding scheme, and then upgrade it to a message-embedding scheme.

While the message-embedding scheme of Kim and Wu is based on complex homomorphic evaluation techniques, we observe that much simpler constructions can be obtained and from a wider range of assumptions, if we forego the strong requirement of security against the watermarking authority. Concretely, we introduce a new notion called extractable PRGs (xPRGs), from which extractable PRFs (without security against authorities) suitable for the Kim-Wu transformations can be simply obtained via the Goldreich-Goldwasser-Micali (GGM) construction. We provide simple constructions of xPRGs from a wide range of assumptions such as hardness of computational Diffie-Hellman (CDH) in the random oracle model, as well as LWE and RSA in the standard model.