Whole factorization and cryptography

Question

I know that public key cryptography uses primes, they also know that two large (for example, 100 digits) primes (P, Q) are used as the private key, the product is the public key N = P * Q, and the use of primes the fact that factoring N to get P, Q is a difficult task and takes a lot of time, I'm fine, but I’m puzzled why not just use the usual big complicated numbers for P, Q and therefore factoring N will still be difficult because that is because now there are not only two but a factor, but even more.

thanks....

Answer 1

I am not a specialist in cryptography.

why not just use regular non-empty numbers for P, Q

Because there will be more factors. Integer factorization is an attack on public-key encryption. This attack uses this same attitude.

It would be easier to find attitudes and possible meanings with more common factors. It comes down to algebra.

N = P * Q

if P and Q are both Prime, then N has 4 factors {NPQ 1}

But! if P and Q both have a coefficient of 2

N / 4 = P / 2 * Q / 2

If N could be 0..2 ^ 4096, then now 0..2 ^ 4094, and since 2 was a factor, a large number was also a factor.

This means that I could find a scalar multiple, P ', Q' from P, Q ST P ', Q' P, Q

I don’t quite understand the concept myself, but I think it shows where I am going with this.

You need to look for less space until you press a key.

Answer 2

It is perfectly possible to use RSA with a module N consisting of more than two prime factors P and Q, but two things must be noted:

• You must know the exact meaning of all these factors, otherwise you cannot get the private key from the public key when generating the keys. The equation for two-phase RSA is 1 = D * E (mod LCM (P-1, Q-1)). If you do know these simple factors, you can do the calculation. If you don’t know the main factors, you cannot perform this calculation, which, by the way, is safe to make the public key E, N public - you cannot get the private key D if you have only information that is easily obtained from the public key, if only you cannot determine N.

• RSA security is effectively limited by the second largest prime factor of the RSA N module. Finding small prime factors that are less than 2 ^ 32 can be done for a split second on a modern computer, simply trying to divide module N into each such prime number and check if the remainder is equal zero (which means that N is divisible by this number) or not (this means that the number is not a factor of N). If N consists only of such small prime factors multiplied by one large prime factor Q, it would be trivial to find that Q, simply by dividing N by all small coefficients, to get N 'and check N' for simplicity. If N 'is a prime, this is the last prime factor of Q.

Answer 3

I am not very good at cryptology (therefore, if I am mistaken, just tell me in the comment and I will delete this answer immediately), but I think because if you just use random large numbers, you can get easily factorizable (i.e. e. you do not need to get up on really big primes to get their primary factors). Thus, really large, guaranteed primes are used.