Much of the information transferred over the internet is encrypted using methods which rely on the fact that currently, very large numbers are extremely difficult to factor. Imagine that in the near future, developments in mathematics result in algorithms can factor numbers in a fraction of a second (which is actually possible). How much would "life as we know it", with online banking, communication, etc. change? Would these industries be able to quickly develop other methods of encryption, or would the world get kicked back to pre-internet days?
closed as too broad by Aify, JDługosz, Hohmannfan, bilbo_pingouin, a CVn♦ Jun 19 '16 at 14:02
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There is no one answer to this, because it is too complicated of a topic. The key to the answer would be in global dynamics. How does China respond? How does Russia respond? How does ISIS respond? How does Anonymous respond?
Rest assured, very little encryption relies on the difficulty of factoring large composite numbers. Most symmetric algorithms rely on other proofs for their security. The facet of encryption that would be pounded would be public-key encryption, where RSA is the current reigning champion.
There are other public-key encryptions out there. Some even rely on lattice based techniques that are immune to Shor's algorithm, making them particularly resilient against quantum computers. In the greater scheme of things, the internet can survive someone determining how to rapidly factor a large composite number.
However, in that short period during the changeover, there would be a lot of turmoil. The individual players on this global scene would have a lot to say about how things play out.
I really like Cort's answer and I think it is the correct one. This one is just me bringing more info to the table.
There is a matter of scale involved. We usually hear about encryption keys with some amount of bits attached. That's the size of the key, and the longer it is, the more computing it takes to break.
Adding four extra bits to an encryption alghoritm will, on back-of-napkin calculations, make it one order of magnitude harder to break. Now look how we went from 512 bits keys to 1024 bits ones.
No matter how much mathematics advance, we are still limited to processing power. Even if you figure an easier way to break a key, it still involves computing. So if you suddenly develop an alghoritm that makes it possoble to break a 2048 bits key in one minute, I'll just start using 4096 bits. I'll take whatever performance overhead that costs me, but your alghoritm will take eons to break the new key.
Again, back-of-napkin calculations have a brute force attack taking approximately 10^2045 (the numer one, followed by two thousand and forty-five zeroes) longer to break a 4096 bits key than a 2048 one. I may be a little off there, but the amount of zeroes will be close enough to give you an idea.
Okay, I expect your attack to be a non-brute force one, which scales more favorably than that, but even then, scaling the key up may make any attacks unfeasible for a long while - until better processors are developed, and anoter math genius comes up with another, clever method of encryption breaking.