Enforcing and defying associativity, commutativity, totality, and strong noninvertibility for worst-case one-way functions

Abstract

Rabi and Sherman [M. Rabi, A. Sherman, An observation on associative one-way functions in complexity theory, Information Processing Letters 64 (5) (1997) 239–244; M. Rabi, A. Sherman, Associative one-way functions: A new paradigm for secret-key agreement and digital signatures, Tech. Rep. CS-TR-3183/UMIACS-TR-93-124, Department of Computer Science, University of Maryland, College Park, MD, 1993] 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; this paper is in the worst-case model, not the average-case model) 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 [M. Rabi, A. Sherman, An observation on associative one-way functions in complexity theory, Information Processing Letters 64 (5) (1997) 239–244; M. Rabi, A. Sherman, Associative one-way functions: A new paradigm for secret-key agreement and digital signatures, Tech. Rep. CS-TR-3183/UMIACS-TR-93-124, Department of Computer Science, University of Maryland, College Park, MD, 1993]) 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 3 4 = 81 potential types of one-way functions. We prove that each of these 81 feature-laden types stands or falls together with the existence of (plain) one-way functions.