The Existence of One-Way Functions

Frank Vega (1)

1 Datys

We assume there are one-way functions and obtain a contradiction following a solid argumentation, and therefore, one-way functions do not exist applying the reductio ad absurdum method. Indeed, for every language $L$ that is in $EXP$ and not in $P$, we show that any configuration, which belongs to the accepting computation of $x \in L$ and is at most polynomially longer or shorter than $x$, has always a non-polynomial time algorithm that find it from the initial or the acceptance configuration on a deterministic Turing Machine which decides $L$ and has always a string in the acceptance computation that is at most polynomially longer or shorter than the input $x \in L$. Next, we prove the existence of one-way functions contradicts this fact, and thus, they should not exist. Hence, function problems such as the integer factorization of two large primes can be solved efficiently. In this way, this work proves that is not safe many of the encryption and authentication methods such as the public-key cryptography. It could be the case of $P = NP$ or $P \neq NP$, even though there are no one-way functions. However, we prove that $P = UP$.

Mots clés: Complexity classes; one-way function; UP; NP; EXP

Domaines: Complexité [cs.CC]

Frank Vega. The Existence of One-Way Functions. 2014. ⟨hal-01020088v2⟩