Recent stories in the press about potential mathematics breakthroughs are frightening computer security officers.
The gist of these stories is that "imminent" advances in mathematics would allegedly make it possible to crack encryption schemes based on the difficulty of factoring humongous numbers -- like RSA. This would destroy the primary methods of protecting passwords, virtual private networks and Internet-based secure transactions. It's the end of the world as we know it. Again.
However, there are good reasons for taking these stories with a grain of salt, due to the nature of mathematical research, the time involved and the difference between pure and applied mathematics.
P vs. NP and the Riemann zeta function problem are the two areas of mathematics mentioned most often in this regard. Real mathematicians will froth at the mouth as I describe these problems, because my descriptions lack mathematical rigor. So, real mathematicians, just do some deep breathing exercises for a moment.
The P versus NP problem involves the inherent difficulty of certain mathematical questions -- often those that have to do with optimization, such as the most efficient route for a delivery truck. No algorithm can solve such questions -- brute force guessing may eventually reveal a solution. Discovering the prime factors of huge numbers -- the basis of encryption schemes like RSA -- is a NP question. The news: NP problems really aren't as hard as we thought.
The Riemann zeta function is intimately related to the distribution of prime numbers, which are notorious for being whimsically distributed. Since the Riemann zeta function is related to the distribution of prime numbers, it can help narrow down the search for prime numbers, and thus speed up the search.
Still, keep in mind that mathematicians can consider a solution to be merely that an answer to a given question exists -- not what that answer might be or how to find it. A purely mathematical solution to these problems may be far from useful.
Also, consider that some of these problems have been known for centuries and, while progress has been made, they still haven't been solved. So, when mathematicians talk about a breakthrough, it does not mean a solution is looming. The solution might be found tomorrow -- or a century from now -- or never. There's no way to tell, so why worry about something that may be decades away?
Finally, the gulf between pure and applied mathematics is vast. Even if some genius announced tomorrow that an algorithm for finding factors of humongous numbers was possible, that wouldn't advance the search for that algorithm one nanometer. It's like an engineer announcing that it's possible to bridge a certain river: you wouldn't want to immediately drive up to the bank of the river and wait. Building that bridge will take time.
The bottom line is simple. Don't let announcements about possible mathematical breakthroughs throw you. They might take years to crack, their application to reality is iffy and any uses will take years to develop. Remember: Math is our friend.
About the author
Edmund X. DeJesus has degrees in mathematics and theoretical physics. He has worked on cryptographic solutions for Baltimore Technologies and has written extensively on encryption, mathematics and other technical topics.
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