Cryptography - Sage

The Advanced Encryption Standard allows Rijndael with key lengths 128, 192, or 256 bits.The eight-bit byte blocks which form the matrix entries are interpreted as elements ofthe finite field of 2 8 = 256 elements. The finite field is represented by the quotient ringF 2 8 = F 2 [X]/(X 8 + X 4 + X 3 + X + 1),whose elements are polynomials c 7 X 7 + c 6 X 6 + c 5 X 5 + c 4 X 4 + c 3 X 3 + c 2 X 2 + c 1 X + c 0 .We denote by BS, SR, MC, and ARK these four basic steps. There exist correspondinginverse operations IBS, ISR, IMC, IARK. The flow of the algorithms for enciphering anddeciphering are as follows:1. ARK2. BS, SR, MC, ARK.3. BS, SR, MC, ARK4. BS, SR, ARK1. ARK2. IBS, ISR, IMC, IARK.3. IBS, ISR, IMC, IARK4. IBS, ISR, ARKByteSub. The ByteSub operation is given by the S-box look-up table. Alternatively theS-box has a description in terms of the structure of the finite fields and linear algebra. Letx ′ be the inverse of x in F 2 8 if x ≠ 0 and set x ′ = x = 0 otherwise. Then the ByteSub stepis given by x ↦→ X 6 + X 5 + X + 1 + x ′ A where A is the matrix:⎛⎞1 1 1 1 1 0 0 00 1 1 1 1 1 0 00 0 1 1 1 1 1 0A =0 0 0 1 1 1 1 11 0 0 0 1 1 1 1⎜ 1 1 0 0 0 1 1 1⎟⎝ 1 1 1 0 0 0 1 1 ⎠1 1 1 1 0 0 0 15.4 Modes of OperationBlock ciphers can be applied to longer ciphertexts using one of various modes of operation.We assume that the input is plaintext M = M 1 M 2 . . ., the block enciphering map for givenkey K is E K , and the output is C = C 1 C 2 . . .. The possible block cipher modes of operationwhich we treat are identified by the acronyms ECB, CBC, CFB, and OFB. In each case weassume that we have a cipher of block length n, with enciphering maps E K and decipheringmaps D K for each key K.5.4. Modes of Operation 41