[Paper Review] The Existence of the Tau One-Way Functions Class as a Proof that P != NP

Javier A. Arroyo-Figueroa|arXiv (Cornell University)|Apr 12, 2016

Coding theory and cryptography4 references3 citations

TL;DR

This paper claims to prove P ≠ NP by introducing a new class of functions called 'Tau,' which are shown to be one-way functions—computable in polynomial time but infeasible to invert using any polynomial-time probabilistic algorithm. The key contribution is the existence of such functions, which implies that NP problems cannot be solved in polynomial time, thus establishing P ≠ NP.

ABSTRACT

We prove that P != NP by proving the existence of a class of functions we call Tau, each of whose members satisfies the conditions of one-way functions. Each member of Tau is a function computable in polynomial time, with negligible probability of finding its inverse by any polynomial probabilistic algorithm. We also prove that no polynomial-time algorithm exists to compute the inverse of members of Tau, and that the problem of computing the inverse of Tau cannot be reduced to FSAT in polynomial time.

Motivation & Objective

• To establish a rigorous proof that P ≠ NP.
• To define and demonstrate the existence of a new class of one-way functions called 'Tau'.
• To show that inverting Tau functions cannot be achieved by any polynomial-time probabilistic algorithm.
• To prove that the inverse computation of Tau functions cannot be reduced to FSAT in polynomial time.
• To provide a foundation for separating P and NP using cryptographic primitives.

Proposed method

• Introduces a novel class of functions named 'Tau' that are computable in polynomial time.
• Demonstrates that the inverse of any Tau function has negligible probability of being found by any polynomial-time probabilistic algorithm.
• Proves that no polynomial-time algorithm can compute the inverse of Tau functions.
• Establishes that the problem of inverting Tau functions is not polynomial-time reducible to FSAT.
• Uses complexity-theoretic arguments to show that Tau functions satisfy the formal definition of one-way functions.
• Relies on computational complexity and cryptographic hardness assumptions to derive the separation of P and NP.

Experimental results

Research questions

• RQ1 Does there exist a class of functions that are efficiently computable but infeasible to invert in polynomial time?
• RQ2 Can the existence of such functions be used to prove that P ≠ NP?
• RQ3 Is the problem of inverting these functions reducible to FSAT in polynomial time?
• RQ4 Can any polynomial-time probabilistic algorithm invert a Tau function with non-negligible probability?
• RQ5 Do these functions satisfy the formal criteria for one-way functions in computational complexity theory?

Key findings

• The Tau function class exists and consists of functions computable in polynomial time.
• The inverse of any Tau function cannot be computed by any polynomial-time probabilistic algorithm with non-negligible probability.
• No polynomial-time algorithm exists to invert members of the Tau class.
• The problem of inverting Tau functions cannot be reduced to FSAT in polynomial time.
• The existence of Tau functions implies that P ≠ NP.
• The paper concludes that the separation of P and NP is established through the cryptographic hardness of the Tau class.