Prime numbers and cybersecurity

Would you like to see an ingenious example of how the world of mathematics can have unexpected ramifications on the world?

You may be aware of the role of the special numbers e = 2.718 …, pi = 3.14 …, and the golden ratio Phi = 1.618 …, in our world. It turns out that prime numbers, numbers that cannot be divided or reduced into small numbers, also have a special property: they are ideal for helping to create a secure banking system.

You see, security systems that allow you to safely use ATM or online banking, and allow you to send information securely over public networks, use a form of encryption or encryption that is based on prime numbers.

Surprisingly, 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, and was able to explain why this simple property is true. However, at the time, his discovery had no obvious application: it was simply 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 securely. They used Fermat’s little theorem as part of a “recipe” for coding numbers, the RSA algorithm.

Without going into too much detail, what happens when a system uses the RSA algorithm or a similar algorithm, for example when you access the ATM: the ATM stores your debit card information and PIN number as a real number, a string of 0 and 1? . It then scrambles this number using a “key” known only to the ATM and the bank.

The ATM then sends the debit card information to the bank using this “key,” and if the message is seen by a spy, criminal, or snooper, it is encrypted. To decode the message, they would have to know the “key”, and to determine the key, they would have to factor a number several hundred digits long. This is very difficult, almost impossible, even for the fastest and most advanced 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 eventually 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 links to the physical universe, many unexpected applications that are sometimes not apparent for even centuries.