A Note on Public-key Cryptosystems and Their Underlying Mathematical Problems

While the classic symmetric encryption systems require a single key for both encryption and decryption, public-key systems are based on the existence of two distinct keys, one private and one public, and on the concept that, while the private key is never transmitted over any channel, and is therefore known only by its owner, the public key is made publicly known. Public-key systems are thus extremely useful in open network scenarios, where not all users are known in advance, or where it is simply impractical to establish a secure channel with any of them over which to exchange symmetric keys for the ensuing communications protection. Asymmetric systems are very interesting from a mathematical point of view, since they are based on one-way trapdoor functions, which are invertible functions that are “easy” to compute in one direction and “difficult” to compute in the opposite direction, with the additional condition of being “easy” to compute in that direction if additional information (the trap) is available.