This chapter considers various techniques for public-key encryption, also referred to as asymmetric encryption. As introduced previously (§1.8.1), in public-key encryption systems each entity A has a public key e and a corresponding private key d. In secure systems, the task of computing d given e is computationally infeasible. The public key defines an encryption transformation E _e , while the private key defines the associated decryption transformation D _d . Any entity B wishing to send a message m to A obtains an authentic copy of A’s public key e, uses the encryption transformation to obtain the ciphertext c = E _e (m), and transmits c to A. To decrypt c, A applies the decryption transformation to obtain the original message m = D _d (c).