> the RSA cryptosystem (and indeed, many asymmetric-key cryptosystems) is based on the notion that multiplying two gigantic prime numbers together to get anothe

> the RSA cryptosystem (and indeed, many asymmetric-key cryptosystems) is based on the notion that multiplying two gigantic prime numbers together to get another gigantic non-prime number is easy; but taking a gigantic non-prime number and figuring out which two prime numbers were multiplied together is incredibly hard. While it is true that if we can factor a number that is a composite of two primes is sufficient to break RSA, the security of RSA is based on a weaker property (cf. https://en.wikipedia.org/w/index.php?title=RSA_(cryptosystem)&oldid=805975016#Integer_factorization_and_RSA_problem https://en.wikipedia.org/w/index.php?title=RSA_(cryptosystem...): Given two prime number p,q >= 3, let n := p * q. Let lambda(n) = lcm(p-1, q-1) (Carmichael's totient function). A public key then is (n,e), where 1 < e < lambda(n). If one knows p,q one can compute the corresponding private key (n,d), where d * e = 1 mod lambda(n) (which is considered to be a hard problem if one does not know p,q). The RSA problem asks that given a c \in {0, ..., n-1}, find an m \in {0, ..., n-1} such that c = m^e mod n. It is easy to check how this problem becomes simple if we can factor n (then we simply can find the private key). But it is an open problem whether solving the RSA problem is as hard as factoring or not.