On the Impossibility of Basing Public-Coin One-Way Permutations on Trapdoor Permutations

One of the fundamental research themes in cryptography is to clarify what the minimal assumptions to realize various kinds of cryptographic primitives are, and up to now, a number of relationships among primitives have been investigated and established. Among others, it has been suggested (and sometimes explicitly claimed) that a family of one-way trapdoor permutations (TDP) is sufficient for constructing almost all the basic primitives/protocols in both “public-key” and “private-key” cryptography. In this paper, however, we show strong evidence that this is not the case for the constructions of a one-way permutation (OWP), one of the most fundamental primitives in private cryptography. Specifically, we show that there is no black-box construction of a OWP from a TDP, even if the TDP is ideally secure, where, roughly speaking, ideal security of a TDP corresponds to security satisfied by random permutations and thus captures major security notions of TDPs such as one-wayness, claw-freeness, security under correlated inputs, etc. Our negative result might at first sound unexpected because both OWP and (ideally secure) TDP are primitives that implement a “permutation” that is “one-way”. However, our result exploits the fact that a TDP is a “secret-coin” family of permutations whose permutations become available only after some sort of key generation is performed, while a OWP is a publicly computable function which does not have such key generation process.