Imagine a world where digital communication and data transfer was a minefield of danger and uncertainty. Every time you sent a message or transmitted data, you risked interception by malicious actors seeking to steal your information. But then came the heroes of modern cryptography - prime numbers.
These seemingly ordinary numbers turned out to be the key to securing digital communications and data transfer in the modern age. In particular, the RSA algorithm became one of the most widely used cryptographic algorithms, relying heavily on prime numbers to ensure its security.
But how do these unassuming numbers provide such incredible security?
Why are prime numbers so important?
Most of us know what prime numbers are - those enigmatic numbers that can only be divided by 1 and themselves. For example, the number 7 is a prime number because it can only be divided by 1 and 7. But did you know that these seemingly ordinary numbers are the unsung heroes of modern cryptography?
Prime numbers have unique mathematical properties that make them ideal for use in cryptographic algorithms. One such property is their mathematical irreducibility, which means that they cannot be expressed as the product of two smaller numbers. This makes it difficult for attackers to factor large numbers into their prime factors - a process known as prime factorization.
For example, the number 15 can be expressed as the product of 3 and 5, which are both prime numbers. But the number 17 cannot be expressed as the product of two smaller numbers, making it a prime number.
Another important property of prime numbers is their unpredictability. Because there is no discernible pattern to the distribution of prime numbers, they can be used as the basis for generating random numbers, which are essential to many cryptographic protocols.
Figure 1: Distribution of Prime Numbers in Polar Plot (Source: www.3blue1brown.com/)
Without both of these features (irreducibility and unpredictability) many of the cryptographic algorithms that secure our digital communications and data transfer would be far less effective.
How do prime numbers work in cryptography?
At first glance, it’s probably hard to imagine why prime numbers would be anything but perfect for use in cryptography. However, there is one major issue that arises when working with prime numbers - they don’t play well with computers.
For starters, prime numbers are notoriously difficult to generate using computer algorithms. While it is relatively easy to identify prime numbers that are small, once we start looking at larger prime numbers, things get exponentially more difficult.
This is because there is no known mathematical formula that can generate prime numbers. Instead, we rely on algorithms that check for the divisibility of numbers by smaller primes. However, these algorithms become less efficient as the size of the prime numbers increases.
Another issue with prime numbers is that they require more memory to store than non-prime numbers. This is because prime numbers cannot be easily compressed or represented using fewer bits, which can make them a challenge to work with in certain cryptographic applications.
So while computers may not always like prime numbers, they are an essential part of the digital security infrastructure that underpins our modern world.
How do we generate prime numbers?
Generating prime numbers can be a challenging task, but it is essential for many cryptographic applications. There are several algorithms and methods for generating prime numbers, ranging from brute-force methods to more sophisticated probabilistic algorithms:
- Sieve of Eratosthenes - a simple algorithm, relatively efficient for generating small prime number,
- Miller-Rabin primality test - a probabilistic algorithm for generating large prime numbers,
- Lucas-Lehmer test
Let’s take a closer look at how prime numbers underpin one of the most widely used cryptographic algorithms: the RSA algorithm.
RSA Algorithm
Imagine a locked box containing a secret message that you want to send to someone else. The box is secured with a lock that can only be opened with a special key. You want to send the key to your recipient, but you don’t want anyone else to be able to intercept the key and open the box.
This is the problem that the RSA algorithm was designed to solve. The RSA algorithm is an asymmetric cryptographic algorithm that uses a pair of keys to encrypt and decrypt messages. One key is used to encrypt the message, while the other key is used to decrypt the message.
The public key can be freely shared with anyone, while the private key is kept secret. When you want to send a message to someone else, you encrypt it using their public key. Only the person with the corresponding private key can decrypt the message and read its contents.
How do prime numbers come into play here?
The RSA algorithm relies heavily on the fact that it is extremely difficult to factorize the product of two large prime numbers. This fact is used to create the keys that are used to encrypt and decrypt messages.
To create an RSA key pair, we start by selecting two large prime numbers, p and q (they should be kept secret). We then multiply these two primes together to get a product, n = p * q. This product is used as the modulus for the RSA keys (both the public and private keys).
Next, we select a number e that is relatively prime to (p-1) * (q-1). This number is used as the public key exponent. Finally, we compute the private key exponent, d, such that d = e^-1 (mod (p-1) * (q-1)).
Now we have a public key (n, e) and a private key (n, d) that can be used to securely transmit information over an insecure channel.
Why it works?
The RSA algorithm’s security is based on the formidable challenge of factoring the product of two substantial prime numbers. If an attacker could factorize the modulus n and compute the private key exponent d, they would be able to decrypt any messages encrypted with the public key.
However, the RSA algorithm is designed to make this process extremely difficult. In fact, it is so difficult that it is currently believed to be infeasible to factorize the product of two 1024-bit prime numbers.
So the next time you send a secure message using the RSA algorithm, remember that it is the power of prime numbers that is keeping your message safe and secure.