Primes, on a base level, appear to be quite simple, even useless, so why go to all the effort discussed in the previous post to find them?
Prime numbers are actually very relevant and quite important in mathematics, but also in the real world, and today we'll be looking at their use in cryptography and cyber-security.
The search for monster primes(for more info on this watch Adam Spencer's Ted Talk) is not just some strange hobby practised by mathematical fanatics. Mega prime numbers actually play a major role in the security of many nations and organisations. Because primes are so rare, the chances of multiple organisations finding the same one is very low, and makes the number a close to impenetrable barrier when used as a key to a system or firewall.
However, for extreme security, the standard method far more advanced then even that. An organisation will find 2 monster primes, and then multiply them together to produce the end key. The chances of this being cracked is astronomically low, as whoever is trying to break through must identify not one but two prime numbers, and they must be the exact two used. Say an organisation was just using 4 digit prime numbers to encrypt their software. Using method #1, in which they merely found a random 4-digit prime and used it as a password, the chances of finding it first try would be 1/1061, but if method 2 was used, the chance of finding it first try would be 1/562330, and that requires that all 4 digit primes were already known by whoever was trying to break in, meaning the realistic chance would probably be lower still. But then consider the fact that the prime number used in cyber security are millions of digits long, and even finding one, without even knowing if its the right one, can take a supercomputer an extremely long time. Furthermore, the precise length of the required primes isn't certain. Say that somehow there was exactly one prime at 1m digits long, one at 1m and 1 digits, and so on, all the way up to 100m digits, and somehow someone knew them all (consider it would realistically be even harder than this). The chances of them breaking through a Method 2 system in one attempt would be 1/4900499950500000. Of course, all these values and scenarios are entirely theoretical, but they pretty well convey just how useful primes are in the world of cyber security.
However, this could well change very soon, as the creation of a quantum computer would undermine the entire system. This is because quantum computers are so incredibly fast, even compared to the fastest of supercomputers, that they could pretty much scan through all numbers and find all primes not instantly, but not too far off. Sure, maybe that's a bit of an exaggeration, but our post on quantum mechanics will explain why they are so fast, and just how fast they are.
The search for monster primes(for more info on this watch Adam Spencer's Ted Talk) is not just some strange hobby practised by mathematical fanatics. Mega prime numbers actually play a major role in the security of many nations and organisations. Because primes are so rare, the chances of multiple organisations finding the same one is very low, and makes the number a close to impenetrable barrier when used as a key to a system or firewall.
However, for extreme security, the standard method far more advanced then even that. An organisation will find 2 monster primes, and then multiply them together to produce the end key. The chances of this being cracked is astronomically low, as whoever is trying to break through must identify not one but two prime numbers, and they must be the exact two used. Say an organisation was just using 4 digit prime numbers to encrypt their software. Using method #1, in which they merely found a random 4-digit prime and used it as a password, the chances of finding it first try would be 1/1061, but if method 2 was used, the chance of finding it first try would be 1/562330, and that requires that all 4 digit primes were already known by whoever was trying to break in, meaning the realistic chance would probably be lower still. But then consider the fact that the prime number used in cyber security are millions of digits long, and even finding one, without even knowing if its the right one, can take a supercomputer an extremely long time. Furthermore, the precise length of the required primes isn't certain. Say that somehow there was exactly one prime at 1m digits long, one at 1m and 1 digits, and so on, all the way up to 100m digits, and somehow someone knew them all (consider it would realistically be even harder than this). The chances of them breaking through a Method 2 system in one attempt would be 1/4900499950500000. Of course, all these values and scenarios are entirely theoretical, but they pretty well convey just how useful primes are in the world of cyber security.However, this could well change very soon, as the creation of a quantum computer would undermine the entire system. This is because quantum computers are so incredibly fast, even compared to the fastest of supercomputers, that they could pretty much scan through all numbers and find all primes not instantly, but not too far off. Sure, maybe that's a bit of an exaggeration, but our post on quantum mechanics will explain why they are so fast, and just how fast they are.
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