An analysis on bitwise operations in the encrypted domain

When encrypted data need to be sent to an untrusted computer for processing, it is highly desirable to have a homomorphic cryptosystem that allows the untrusted computer to process the encrypted signals totally without decryption such that, when the encrypted domain result is decrypted, the decrypted value is the same as the equivalent plaintext domain operation. These operations range from basic arithmetic operations to complicated transformations. In this paper, we analyze the existence of equivalent operations in the encrypted domain for four of the common bitwise operations: OR, AND, NOR and NAND in plaintext domain. We will show that such equivalent operations should not exist. For otherwise, if such operations exist, the RSA cryptosystems can be broken by a low-complexity attack.