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176 From the definition of F in (4.26), we see that if A
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We can use this fact to determine a set of A = ( 2 ,ao, a l , 2 ) that are candidate values for M ( K 3 ) . This allows us to, in effective, determines 16 bits of K3. IJsing four pairs of chosen plaintexts, we expect to reduce the number of such candidates to a small number. For the secondary phase, each survivors of the primary phase is further tested and in the process we determine the entire subkey K3. Given a siirvivor A = ( z , a o , a l , z ) from the primary phase, we can easily determine the full K3 as follows. First, generate a 16-bit value B = ( b o , b l ) , and test a putativ: subkey I 3 = (bo, a0 @ b ~a1 @ b ~b , ~ as discussed previously. That is, , I ) use K3 as the putative subkey K:3 and compute 2 and 21, test whether 0 and the condit,ion 2 = 2 @ 2 1 holds. If so, then save E ( 3 as a possible K3, 0 otherwise we know that K3 # K 3 and we select another B . In this way, we recover K3, or a small number of candidate subkeys. Assuming that a single chosen plaintext pair PO and PI is, used, the primary phase of this differential attack appears in Table 4.12 and the secondary phase is given in Table 4.13. However, using only a single pair of chosen plaintexts, the number of putative K3 will be large. As mentioned above, we need to use four chosen plaintext pairs to reduce the number of putative K3 to one (or a very small number). When more than one chosen plaintext pair is used, the primary and secondary attacks in Tables 4.12 and 4.13, respectively, both require slight modifications. In the primary phase, we want to save only those (ao, u l ) that satisfy the necessary conditions for all plaintext and ciphertext pairs. Then in the secondary phase we will have a small number of survivors (ideally, only one) and these, again, must satisfy the necessary conditions for all of the plaintext and ciphertext pairs. The precise details of this attack are left as an exercise; see Problem 30. After successful completion of this differential attack, we will have recovered K3, or a small number of putative K3 values. For simplicity, we assume a single K3 is obtained. Now we must recover the remaining subkey values. This can be accomplished by unzipping the cipher to successively obtain K2, K1, K O , and, finally, K4 and Kg. With K3 available, we can determine the input and the output to the third F function in Figure 4.16 and we can then determine K2 in a similar manner as was used to find K3. Once K3 and K2 are known, we can then effectively rcniove the last two rounds of the cipher and attack K1, and so on. There are several subtle points to this attack that we leave as exercises; see Problem 31 for more details. Before we move on to consider linear cryptanalysis, there is one issue regarding differential cryptanalysis that is worth pondering. In this differential attack on FEAL-4, the characteristic we used to determine K:i occurs with
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Table 4.12: Primary Phase of Differential Attack for K3
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// Characteristic is 0x8080000080800000 PO= random 64-bit value = P @ 0~8080000080800000 o // Given corresponding ciphertexts // c = (Lo,Ro) and c = (L1,Rl) o 1 Y = Lo CB Ro o Yi = Li CB Ri L = Lo @ L1 2 = L @ 0x02000000 for (ao, a l ) = (OXOO,OXOO) to (Oxff,Oxff) Qo = F ( M ( Y 0 ) (0x00, (~1,oxoO)) CB ao, &I = F(M(Yi) @ (0x00, a i r0x00)) ao, if (Qo CB Q l ) 8 ...23 == (2 )8 ...23 then Save (ao,al) end if
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probability one. Since the invention of differential cryptanalyis, block ciphers have been designed with differential attacks in mind. Consequently, differcntials that occur with a high probability are unlikely to be found in practice. Nevertheless, given a differential that occurs with some positive probability p , it is still possible to determine information about the subkey. However, the smaller the value of p , the larger the number of chosen plaintexts that will be required to determine subkey bits (that is, the larger the amount of data that is required) and the higher the work factor. Ideally, the designer of a block cipher would like to make the work for any differential attack at least as high as that of an exhaustive key search.
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