On: Graduate Student’s Side Project Proves Prime Numbe...

[vishnugupta]

Oops; you are right. If my memory serves right, showing prime factorisation in P will imply P == NP; which would have been a sensational news at a different level altogether.

[q-big → vishnugupta]

> showing prime factorisation in P will imply P == NP

Whether "Factorization is in P" actually implies "P=NP" is another open problem (most researchers in this area don't believe that this is the case).

What does hold is that if we found a "fast" algorithm for factorization, this would break some cryptosystems. The most well-known example is RSA, but other, more academic cryptosystems would be broken, too.

In its "Public key cryptography" template (https://en.wikipedia.org/wiki/Template:Cryptography_public-k... click "[show]"), Wikipedia lists the following cryptosystems to be dependent on the hardness of integer factorization:

  * Benaloh
  * Blum–Goldwasser
  * Cayley–Purser
  * Damgård–Jurik
  * GMR
  * Goldwasser–Micali
  * Naccache–Stern
  * Paillier
  * Rabin
  * RSA
  * Okamoto–Uchiyama
  * Schmidt–Samoa

[Ar-Curunir → vishnugupta]

No, factoring being in P would not imply P == NP. Factoring is not NP-hard.

[Laakeri → Ar-Curunir]

Well, it is also not proved to not be NP-hard, so it could also be NP-hard to our current knowledge. But you are right that researchers believe that it's not NP-hard.

[Ar-Curunir → Laakeri]

Good catch, you’re right, nobody has yet proven that factoring is NP-hard, but a ton of evidence indicates that it is highly unlikely to be NP-hard