Revisiting Security of Symmetric-Key Encryption with Imperfect Randomness and Correlation Between Source and Keys

In the ideal environment, a secret key for a cryptosystem is generated with perfect randomness and is independent of other keys or sources. It is well-known that such an ideal condition gives the perfect security to many encryption schemes. However, in practice, it is not uncommon that due to limited resources, we can not always guarantee such ideal conditions. Thus, it is very important to know the amount of security we can guarantee under a non-ideal condition in practice. In this paper, we are interested in investigating the security of symmetric encryption schemes under non-ideal conditions. Especially, we focus on a wide category of encryption schemes with the following property: for any plaintext, there is no collision between ciphertexts generated from the encryption of the plaintext by different secret keys. As the main contribution, we derive the sufficient and the necessary condition for guaranteeing the security of any such encryption scheme even in the non-ideal cases where: (1) the source (plaintext) and the secret key used in the encryption process are correlated, and (2) the secret key is not perfectly random.