on algorithm (DEA) and by ISO (International Standards Organization) under the name of DEA-1. Classical and quantum cryptography In 3DES, three independent secret keys are used; one can simply iterate DES with three keys. A different approach, referred to by ANSI as TDEA (triple data encryp- tion algorithm) is to use a combination of encryption and decryption with the three keys: if K1, K2, K3 are the three keys and EK and DK the encryption and decryp- tion functions with a given key K, the cipher C and message M are defined by = EK3 {DK2 [EK1 (M)]} and M = DK1 {Ek2 [DK3 (C)]}, respectively. With 3DES, the complexity is really a whopping 212. In spite of the improvements introduced by double and triple encryption, DES further evolved into the advanced encryption standard (AES), with the prospect of lasting use in future decades. It is beyond the scope of this chapter to enter into the algorithmic features of DES and AES. For a rapid but detailed introduction of the DEA and AES, see, for instance, my earlier publication. 13 However, it is worthwhile highlighting here some of the innovating features of AES. Basically, AES is a symmetric encryption algorithm using blocks of 128 bit (twice the size of DES), and key sizes of 128, 192 or 256 bits (twice or four times the 64-bit DES key size). The ground-breaking innovation in AES is the introduction of byte-to-byte multiplication, which comes as a supplemental operation resource to the Boolean XOR addition of previous DES. For the bytes to keep a constant size under multiplication, the operation requires the bytes to be expressed in a polynomial representation, and the multiplication to be performed modulo some irreducible polynomial. Although we will not have any use of polynomial multiplication in this chapter, its principle is described in Appendix Y, for the sake of education and also for the curious or demanding. The AES algorithm is completed by various stages of mingling together the plaintext and key "sub-bytes," shifting the result by rows and columns, and eff