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# AppnBytes: RSA Encryption & Prime Numbers

We’re back with another AppByte! This time, we’re exploring the fascinations of Emile Nieuwoudt, an entry-level dev at Appenate.

If the coalescence of encryption and math lore interests you, stick around.

## Context

As an Entry-Level Full-Stack-Developer with a background in Mathematics that’s recently joined the IT-Industry, it warmed my heart to find an old friend from years past when I was studying number theory and discrete methods.

This occurred while checking the RSA key of an SSL certificate test (server side) for one of my code changes, which showed me one of the practical linkages between mathematics and programming, RSA encryption

## What is RSA Encryption?

Well, RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem widely used for secure data transmissions all over the world.

An RSA user creates and publishes a public key based on two large prime numbers and an auxiliary value. The prime numbers are kept secret. Anyone can encrypt messages via the public key but it can only be decoded by someone who knows the prime numbers.

## Why is it so secure?

The reason is that some numbers cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. These numbers are called ‘prime’, and they play an important role, both in pure mathematics and its applications, in this case, encryption. So far, we have been unable to find any regular pattern that follows the distribution of such prime numbers among all-natural numbers.

The security of RSA relies on the practical difficulty of factoring the product of two large prime numbers. Thus if you do not have at least one prime number, it is almost impossible to decrypt (even with infinite resources and time,) with existing computing technology. One estimate holds; it would take 300 trillion years to “brute force” an RSA 2048-bit key.

## The Riemann Hypothesis

But there is a Hypothesis that, if solved, will cause some issues and revolutionise how we encrypt and decrypt information. The Riemann Hypothesis: This makes use of the prime number theorem that determines the average distribution of the primes, and then looks at the deviation from this average.

The German mathematician G.F.B. Riemann (1826 – 1866) observed that the frequency of prime numbers is very closely related to the behaviour of an elaborate function.

ζ(s) = 1 + 1/2s + 1/3s + 1/4s + …

Called the Riemann Zeta function.

Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers.

The Riemann hypothesis and some of its generalizations, along with Goldbach’s conjecture and the twin prime conjecture, make up Hilbert’s eighth problem in David Hilbert’s list of twenty-three unsolved problems; it is also one of the Clay Mathematics Institute’s Millennium Prize Problems, which offers a million dollars to anyone who solves any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.

Proof that it is true for every intersecting solution would shed light on many of the mysteries surrounding the distribution of prime numbers- and a large leap forward for encryption and the world as a whole.

## Just A Byte

There you have it, a quick breakdown from Emile on RSA encryption and the Riemann hypothesis. We hope you enjoyed this snapshot, and if you ever attempt to solve one of Hilbert’s problems, we wish you all the best. 😉