A cryptographic processing system and method to decrease the sizes of public parameters and a master secret key, and shorten the time taken for the generation process of the secret key to be supplied to the user and the encryption process. A generation device generates a basis B and a basis B* that form public parameters and a master secret key using a sparse matrix in which each row and each column have at least one value other than 0. An encryption device generates a vector in the basis B, the vector being embedded with predetermined information, as a cipher vector. A decryption device, by treating a predetermined vector in the basis B* as a key vector, conducts a pairing operation for the cipher vector and the key vector, to decrypt the cipher vector.