Reductions are the common technique to prove security of cryptographic constructions based on a primitive. They take an allegedly successful adversary against the construction and turn it into a successful adversary against the underlying primitive. To a large extent, these reductions are black-box in the sense that they consider the primitive and/or the adversary against the construction only via the input-output behavior, but do not depend on internals like the code of the primitive or of the adversary. Reingold, Trevisan, and Vadhan (TCC, 2004) provided a widely adopted framework, called the RTV framework from hereon, to classify and relate different notions of black-box reductions. Having precise notions for such reductions is very important when it comes to black-box separations, where one shows that black-box reductions cannot exist. An impossibility result, which clearly specifies the type of reduction it rules out, enables us to identify the potential leverages to bypass the separation. We acknowledge this by extending the RTV framework in several respects using a more fine-grained approach. First, we capture a type of reduction-frequently ruled out by so-called meta-reductions-which escapes the RTV framework so far. Second, we consider notions that are "almost black-box", i.e., where the reduction receives additional information about the adversary, such as its success probability. Third, we distinguish explicitly between efficient and inefficient primitives and adversaries, allowing us to determine how relativiz-ing reductions in the sense of Impagliazzo and Rudich (STOC, 1989) fit into the picture.
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