What the security of public key asymetric cryptography is based upon?

Answer 1

The security of asymmetric cryptography depends on the difficulty - of determining the private key given the public key. For example: an asymmetric encryption algorithm might rely on the difficulty of factoring a very large number into two prime factors. If the cost of computational power and time to perform the factoring exceeds the value of the encrypted information, then the encryption would be considered sufficiently secure for practical purposes. An asymmetric encryption algorithm may also rely on the difficulty of computing discrete logarithms. Both the "factoring products of primes" and "calculating discrete logarithms" approaches to recovering the "private" key of an asymmetric encryption are assumed to rely on the difficulty of brute force attacks. Sometimes algorithms are improperly implemented, the keys are too short, or an inherent weakness is found that allows faster recovery of the key by restricting the key space that must be searched. In these cases, the security of the asymmetric encryption may be significantly less than would be otherwise assumed - i.e. it may be possible to break the encryption fast enough to make it worth the effort.