2014 | OriginalPaper | Chapter
On the Impossibility of Basing Public-Coin One-Way Permutations on Trapdoor Permutations
Author : Takahiro Matsuda
Published in: Theory of Cryptography
Publisher: Springer Berlin Heidelberg
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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.