My question is how many bits of prime number is secure so that it cannot be factored from very large number? Until today how large prime factor is found in large number?
Quantum computing find highest factor of size of 16 bits only( 56,153) until today and I studied that rsa-768 in which 384 bits prime number is factored from large number. Why different schemes takes very large prime, when even two 1000 bits prime number cannot be factored from 2000 bit large number until today? Thanks.
There is some confusion here. The definition of prime numbers states that cannot be factored (see Definition of prime numbers)
You seem to be talking about RSA modulus which is the product of two prime numbers (see RSA cryptosystem).
As far as keylength is concerned 768 bits is not considered safe today. Note that the keylength choice is a compromise between speed and security, basically we're trying to estimate the shortest keylength that is beyond the cryptanalytic capabilities of any attacker (that needs to stay true as long as the data is valuable). Since we base that estimate on the best publicly known attacks against the cryptosystem (namely here factoring) running on publicky available hardware we add a cushion to take into account increase in computing power, entities that know or will know more than we do.
To answer your question, new RSA public keys should be at least 2048 bits long so slightly larger than your 1600 bits initial proposal here. For further reference you can refer to the keylength website.
