Abstract
A desirable property of one-way functions is that they be total, one-to-one, and onto—in other words, that they be permutations. We prove that one-way permutations exist exactly if P ≠ UP ∩ coUP. This provides the first characterization of the existence of one-way permutations based on a complexity-class separation and shows that their existence is equivalent to a number of previously studied complexity-theoretic hypotheses.
We also study permutations in the context of witness functions of nondeterministic Turing machines. A language is in PermUP if, relative to some unambiguous, nondeterministic, polynomial-time Turing machine accepting the language, the function mapping each string to its unique witness is a permutation of the members of the language. We show that, under standard complexity-theoretic assumptions, PermUP is a nontrivial subset of UP.
We study SelfNP, the set of all languages such that, relative to some nondeterministic, polynomial-time Turing machine that accepts the language, the set of all witnesses of strings in the language is identical to the language itself. We show that SAT ∊ SelfNP, and, under standard complexity-theoretic assumptions, SelfNP ≠ NP.
Supported in part by Dept. of Education (GAANN program) grant EIA-0080124, and by grant NSF-INT-9815095/DAAD-315-PPP-gü-ab.
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Homan, C.M., Thakur, M. (2002). One-Way Permutations and Self-Witnessing Languages. In: Baeza-Yates, R., Montanari, U., Santoro, N. (eds) Foundations of Information Technology in the Era of Network and Mobile Computing. IFIP — The International Federation for Information Processing, vol 96. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-35608-2_21
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