Integer Factorization By Sieving The Delta

TL;DR

This paper proposes a novel factorization method based on sieving the difference between prime factors, which could potentially factorize semiprimes in constant time once certain conditions are met, and introduces a new trapdoor function.

Contribution

It introduces a new factorization approach based on sieving the difference of prime factors, avoiding reliance on congruence of squares or special properties of n.

Findings

Proposes a delta sieving factorization method.

Suggests constant-time factorization under specific conditions.

Introduces a new trapdoor encryption function.

Abstract

Let $n = \mathrm{p}\!\cdot\!\mathrm{q}$ (p < q) and $\Delta = \lvert p-q \rvert$, where p,q are odd integers, then, it is hypothesized that factorizing this composite n will take O(1) time once the steady state value is reached for any $\Delta$ in $zone_0$ of some observation deck (od) with specific dial settings. We also introduce a new factorization approach by looking for $\Delta$ in different $\Delta$ sieve zones. Once $\Delta$ is found and $n$ is already given, one can easily find the factors of this composite n from any one of the following quadratic equations: $p^2 + p\Delta -n = 0$ or $q^2 -q\Delta -n = 0$. The new factorization approach does not rely on congruence of squares or any special properties of n, p or q and is only based on sieving the $\Delta$. In addition, some other new factorization approaches are also discussed. Finally, a new trapdoor function is presented which is leveraged to encrypt and decrypt a message with different keys. The most fascinating part of the discovery is how addition is used in factorization of a semiprime number by making it yield the difference of its prime factors.