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Rabi and Sherman [RS97,RS93] proved that the hardness of factoring is a sufficient condition for there to exist one-way functions (i.e., p-time computable, honest, p-time noninvertible functions) that are total, commutative, and associative but not strongly noninvertible. In this paper we improve the sufficient condition to P ≠ NP. More generally, in this paper we completely characterize which types of one-way functions stand or fall together with (plain) one-way functions—equivalently, stand or fall together with P ≠ NP. We look at the four attributes used in Rabi and Sherman’s seminal work on algebraic properties of one-way functions (see [RS97,RS93]) and subsequent papers—strongness (of noninvertibility), totality, commutativity, and associativity—and for each attribute, we allow it to be required to hold, required to fail, or “don’t care.” In this categorization there are 34 = 81 potential types of one-way functions. We prove that each of these 81 feature-laden types stand or fall together with the existence of (plain) one-way functions.

Enforcing and Defying Associativity, Commutativity, Totality, and Strong Noninvertibility for One-Way Functions in Complexity Theory