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I am currently writing a math paper regarding the importance of prime numbers in RSA encryption. I understand that generating q x p = N (where p and q are prime numbers) is simple for a computer however factoring N into its two primes is improbable within a reasonable amount of time.

As mentioned before I am addressing the importance of prime numbers. What I think the reason for their importance is that if RSA used composite numbers they can be broken up into smaller numbers as they are composite making it easier to factor however I am unsure if this is correct reasoning.

I would greatly appreciate it if someone could help me understand the true importance of prime numbers beyond the simple reasoning that it is because factoring N is highly improbable in a reasonable amount of time.

Jackwannsee
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In order to create a private, public key pair we need to know the factorization of N. If we pick N as a random composite number we are no better off then an attacker. If we can find the factors so can the attacker.

If we pick p and q as random numbers we will need to factorize them in order to find the factors of N, this may be easy or hard. But in the end we want N to be hard to factor. One thing we want is for N to have no small factors which will aid in factorizing it.

In order to ensure: a. The generator of the secret key knows the factors of N b. An attacker can not easily find any factor of N

We choose random large primes p,q and multiply them to produce N.

There are also multi-prime variants of RSA with more than 2 primes used to create N but we still start with large primes.

Meir Maor
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