Algorithms pp 31-38

# Structural analyses on the complexity of inverting functions

• Osamu Watanabe
• Seinosuke Toda
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 450)

## Abstract

In this paper we investigate the complexity of inverting polynomial-time computable functions by methods developed in structural complexity theory. We first analyze upper bounds of the complexity of inverse functions by using complexity classes of functions. We prove the following: (1) NP/bit (the class of functions whose each bit is NP computable) is an upper bound for inverting honest and one-to-one functions, and (2) relative to almost all oracle, the class PF tt NP (the class of functions that are polynomial time computable by asking non-adaptive queries to an NP oracle) is an upper bound for inverting honest functions. Next we investigate relative complexity of inverse functions by using polynomial-time reducibility of functions. We prove that an honest function is NP/bit invertible if the class of its inverse functions possesses the least element under polynomial-time non-adaptive one-query reducibility.

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## Authors and Affiliations

• Osamu Watanabe
• 1
• Seinosuke Toda
• 2
1. 1.Department of Computer ScienceTokyo Institute of TechnologyTokyoJapan
2. 2.Department of Computer ScienceUniversity of Electro-CommunicationsTokyoJapan