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Soft Question: What would be the ramifications if there was an easy method found to factor any number into the product of its primes? When I say easy I mean a method where the time it takes to apply the method to factor the number is independent of the size of the number.

What would be some of the farther reaching effects into different areas of mathematics or other areas like cryptography?

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    This [book](https://books.google.fr/books?id=cbl_BAAAQBAJ&pg=PA323&lpg=PA323&dq=mathematical+alternative+to+rsa&source=bl&ots=qEhCe3B0Ha&sig=wKESfvKDs8srn3q0jzMymMIjZ4I&hl=fr&sa=X&ved=0ahUKEwicv-2FoPfRAhWEsxQKHUT3B00Q6AEIKzAC#v=onepage&q=mathematical%20alternative%20to%20rsa&f=false) looks good. I'd say it is not so easy to analyze the mathematical properties of public key encryption algorithms and if RSA is difficult or hard to break has an effect on some [alternatives to RSA](https://en.wikipedia.org/wiki/Elliptic_curve_cryptography) – 2017-02-04
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    You could crack the most common codes in the same amount of time. I'm pretty sure the people whose data you're decrypting would catch on eventually, and they'd come up with strong encryption that doesn't depend on it being difficult to factor large numbers. It would be very exciting if such an algorithm existed, but the complexity class of the problem is simply not known. Finding out that it's actually easy wouldn't really have any implications in computational complexity theory at least. – 2017-02-04

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