DES (Data Encryption Standard) although an elegantly designed cipher has become old. Its n = 56 bit.

DES (Data Encryption Standard) although an elegantly designed cipher has become old. Its n = 56 bit key is being challenged by the present day computation power. As an alternative, it was thought of applying DES twice, i.e in creating a product cipher DES0 = DES × DES. If the key space of DES was K = {0, 1} n, the key size of the product cipher is expected to be K1 × K2 = (K1, K2), where K1, K2 ? K. The plaintext of the cipher is denoted by P = {0, 1} m and the cipher is endomorphic (the plaintext and the ciphertext are the same set). In regard to this composed cipher answer the following questions: i. What is the property in the DES construction which helps to increase the key length by performing such composition? (Another way of asking the question is: why is DES not idempotent?) ii. Using the DES cipher an attacker obtains l pairs of plaintexts and ciphertexts: (p1, c1), . . . ,(pl , cl). The key is say (K1, K2) but unknown to the attacker (obviously, else why will he/she be an attacker).