The Existence of One-Way Functions

Under the assumption that there exist one-way functions, then we obtain a contradiction following a solid argumentation and therefore, one-way functions do not exist by contraposition. Hence, function problems such as the integer factorization of two large primes can be solved efficiently. In this way, we prove that is not safe many of the encryption and authentication methods such as the public-key cryptography. It could be the case that $P = NP$ or $P \neq NP$, even though there are no one-way functions. However, this result proves that $P = UP$.