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

Lane A. Hemaspaandra, Joerg Rothe+1 more|ArXiv.org|Mar 21, 2005

Computability, Logic, AI Algorithms25 references4 citations

TL;DR

This paper establishes that the existence of one-way functions with any combination of four algebraic properties—strong noninvertibility, totality, commutativity, and associativity—depends precisely on the complexity class separation P ≠ NP. It proves that for all 81 possible combinations of these properties (each either required, forbidden, or ignored), such one-way functions exist if and only if P ≠ NP.

ABSTRACT

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; 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 does not equal 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 not equaling 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 3^4 = 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.

Motivation & Objective

• To determine the complexity-theoretic conditions under which one-way functions with specific algebraic properties exist.
• To resolve the open question of whether P ≠ NP is sufficient to guarantee the existence of one-way functions with mixed properties, including non-associative or non-commutative ones.
• To extend prior results by Rabi and Sherman, which relied on the hardness of factoring, by showing that P ≠ NP suffices as a condition.
• To fully characterize the 81 possible combinations of the four properties (strong noninvertibility, totality, commutativity, associativity) in terms of their existence relative to P ≠ NP.

Proposed method

• The authors analyze all 81 combinations of the four properties (Y = required, N = forbidden, * = indifferent) for one-way functions.
• They use reductions and constructions to show that for each combination, the existence of such a one-way function is equivalent to P ≠ NP.
• Key constructions include transforming a total one-way function into a noncommutative, associative, non-strong one-way function using a carefully designed binary operation.
• The proof leverages known results from complexity theory, including the equivalence of one-way function existence and P ≠ NP, and applies closure properties under reductions.
• They analyze and refute flawed constructions from prior work, such as Rabi and Sherman’s proposed method to add totality, showing it would imply UP = NP.
• The argument uses logical closure under implications and reductions to show that if P ≠ NP, then all 81 types of one-way functions exist.

Experimental results

Research questions

• RQ1 Is P ≠ NP a sufficient condition for the existence of one-way functions that are total, commutative, associative, but not strongly noninvertible?
• RQ2 Can one-way functions exist with the combination of non-totality, non-commutativity, and non-associativity, while still being non-strongly noninvertible?
• RQ3 Does the existence of one-way functions with any specific combination of the four algebraic properties depend solely on P ≠ NP?
• RQ4 Can the hardness of factoring be replaced as a sufficient condition for the existence of certain types of one-way functions, with P ≠ NP being a stronger and more general condition?
• RQ5 What is the impact of requiring or forbidding strong noninvertibility on the existence of one-way functions under P ≠ NP?

Key findings

• For every one of the 81 possible combinations of the four properties (strong noninvertibility, totality, commutativity, associativity), the existence of such a one-way function is equivalent to P ≠ NP.
• The paper improves upon Rabi and Sherman’s result by showing that P ≠ NP, not just the hardness of factoring, is sufficient for the existence of non-strong, total, commutative, associative one-way functions.
• A construction is provided that transforms a total one-way function into a noncommutative, associative, non-strong one-way function, proving the existence of (N,Y,Y,Y)-OWFs under P ≠ NP.
• The paper demonstrates that Rabi and Sherman’s proposed method to add totality to a partial one-way function is invalid unless UP = NP, which is considered unlikely.
• The existence of (N,Y,N,Y)-OWFs (non-strong, total, non-commutative, associative) is also shown to be equivalent to P ≠ NP.
• The result fully characterizes the landscape of one-way functions with algebraic properties, showing that no combination is more or less likely to exist than any other under the same complexity assumption.