# Research into Carmichael Numbers

### Reasoning for this Divergence

Pursuing a career into the academia, I wish to be able to combine Mathematics and Cybersecurity. One aspect I saw this would succeed is in Cryptology. Though Cryptology is the parent to Cryptography and Cryptanalysis, it is important to look at even the minor aspects of Cryptology. One such aspect I have taken inspriration from is Daniel Larsen's interview by Quanta Magazine.

To go into detail on what he has said, in the world of cryptography, the formation of any encrypted message is preluded with the creation or rediscovery of a large number. This large number can be summed up in hex. This creates messages that are encrypted in blockchains or two key encryption. Even outside of messages, this holds true for password holding, or 2FA, digital signitures. All of these use prime numbers.

Primes hold our internet security together. It is easy to multiply two primes together, however it is extremely difficult to do the reverse. Finding larger and larger primes is the key to ensuring security, and websites like FactorDB is a testament as to why we must ensure this. FactorDB counted its way up number by number and multiplying them together to get new larger numbers. This hurts security due to becoming easier to find large prime number factorizations. A number that is 50 digits long, that is almost impossible to communicate how large it is, is already achievable to be factored by this website. However, it does not excuse us from the largest known prime which is almost a million digits.

Public keys hold the result of two primes multiplied together, and this ensures encryption. On the otherhand, private keys hold the two factors that make up the public key, allowing decryption. Such a simple concept contains our security, and it is at risk of being harmed due to better computers. Here is where Carmichael Numbers come in.

If an algorithm mistakes a carmichael composite number for a prime, then security is ensured to fail sooner than later.