Prime numbers and cyber security
Would you like to see a great example of how the world of mathematics can have unexpected ramifications on the world?
You may be aware of the role that the special numbers e=2.718… , pi=3.14… , and the golden section Phi=1.618… , have in our world. It turns out that prime numbers – numbers that cannot be divided or reduced to smaller numbers – also have a special property: they are ideal for helping to create a secure banking system.
You see – the security systems that allow you to use the ATM or online banking securely and allow you to send information securely over public networks – use a form of cryptography or coding that is based on prime numbers.
Amazingly, most of the algorithms—in other words, the methods—for encoding your information are based on a 300-year-old discovery about prime numbers, Fermat’s Little Theorem.
The French mathematician Fermat discovered a relatively simple property about the way prime numbers behave when multiplied together, and was able to explain why this simple property is true. At the time, however, his discovery had no obvious application—it was just an interesting fact about prime numbers.
Then, in the mid-20th century, a team of cryptographers—people whose job it is to help encode information—found a way to use Fermat’s Little Theorem, this discovery about prime numbers—to send information safely and securely. They used Fermat’s Little Theorem as part of the “recipe” for encoding numbers, the RSA algorithm.
Without going into too much detail, what happens when a system uses the RSA algorithm or a similar algorithm – say, when you walk into an ATM: the ATM stores your debit card and PIN information as an actual number – a string of 0’s and 1’s. It then encodes that number using a “key” that only the ATM and the bank know.
The ATM then sends the debit card information to the bank using this “key” – and if a spy or criminal or eavesdropper sees the message – it is encrypted. To decode the message, they would need to know the “key”, and to determine the key, they would need to decompose a number several hundred digits long. This is very difficult, almost impossible, even for the fastest and most modern computers, so your information is safe.
The remarkable thing about this is that it is all based on the 300-year-old discovery of the mathematician Fermat. At the time, Fermat had no idea that what he discovered would ultimately be the key to keeping information safe in the 21st century.
This is one of the many remarkable properties of the world of mathematics – it has many unexpected connections with the physical universe, many unexpected applications that are sometimes not apparent even for centuries.
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