f o r K = 0 t o 232 - 1 // putative K O

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Given (plaintext,ciphertext) pairs (Pi,CZ), i

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= 0 , 1 , 2 , .. . , n -

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next i i f count[O] == n o r count[I] == n t h e n

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count[O] = count[1] = O for i =0 t o R - 1 j = bit computed in right-hand-side of (4.37) count [ j ] = count [ j ] 1

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Save K

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candidate for KO

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end i f next K

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The attack in Table 4.14 is feasible, but we can reduce the work factor considerably. Here, we only outline this improved attack-the details are left as an exercise. We first derive expressions analogous to those in (4.37), using (4.33), (4.34), and (4.35). Then by combining some of these, we obtain

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a = s5,13,21(LO C ROCB L4) @ Si5(Lo @ L4 Q R4) B

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@ S15F(LO @

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ROCE K O ) ,

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(4.38)

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where a is a fixed, but unknown, constant. Now let

= ((K0)O 7 @ (K0)8 15, (K0)16 ...23 @ (KO)24 ...31). ... ...

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From the first line in (4.26), we see that S15F(Lo@Ro@Ko) (4.38) depends of only on the bits (J ~)~...15,17...~3. In addition. it follows from (4.30) that bits 9 and 17 of I o are XORed in the right-hand side of (4.38), so these bits can be taken to the left-hand side of (4.38) and treated as constant (but unknown) values. Then we are left with an expression that depends only on the twelve unknown key bits ( ~ 0 ) 1 ~ . . . 1 5 , 1 8 . . . This allows for an exhaustive search for ~3. twelve bits of PO. Similar expressions can be derived that allow for an extremely efficient attack to recover almost all of the bits of K O ,and the few remaining bits are easily found by a final exhaustive search. The overall work factor for this attack is far less than the 232 required for the attack given in Table 4.14. The linear crytanalytic attack on FEAL-4 described here is explored further in the problems at the end of the chapter. Specifically, Problems 33 through 35 deal with this attack.

4.7.4 Confusion and Diffusion

In his classic paper [133],Shannon discusses confusion and diffusion the in context of symmetric ciphers. These two fundamental concepts are still guiding principles of symmetric cipher design. Roughly speaking, confusion obscures the relationship between the plaintext and the ciphertext, while diffusion spreads the plaintext statistics through the ciphertext. The simple substitution and the one-time pad can be viewed as confusion-only ciphers, while transposition ciphers are of the diffusion-only variety. Within each block, any reasonable block cipher employs both confusion and diffusion. To see, for example, where confusion and diffusion occur in FEAL-4, first note that FEAL-4 is a Feistel Cipher (see Problem 26), where the Feistel round function is simply F ( X i @ K i ) , with F illustrated in Figure 4.17, and defined in (4.26). The FEAL-4 function F does employ both confusion and diffusion, but only to a very limited degree. The diffusion is a result of the shifting within each byte, and also the shifting of thc bytes themselves (represented by the horizontal arrows in Figure 4.17). The confusion is primarily due to the XOR with the key, and, to a lesser extent, the modulo 256 addition that occurs within each Gi function. However, in FEAL-4, both the confusion and diffusion are extremely weak as evidenced by the relatively simple linear and diffcrential attacks presented above. Later members of the FEAL family of ciphers improved on FEAL-4, with the stronger versions having better confusion and diffusion properties, thereby making linear and differential attacks more difficult. However, attacks exist for a,ll versions of FEAL, indicating that the cipher design itself is fundament,ally flawed.

4.8 SUMMARY

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Summary

Block cipher design is relatively well understood. Consequently, it is not too difficult to design a plausible block cipher-although, by Kerckhoffs Principle, such a cipher would not be trusted until it had received extensive peer review. For example, if we create a Feistel Cipher with a round function that has reasonable confusion and diffusion properties, and we iterate the round function a large number of times, it is likely that any attack will be nontrivial. However, things are much more challenging if we try to design a block cipher that is as efficient as possible. Two of the three ciphers discussed in this chapter are weak primarily because they were designed for extreme efficiency-Akelarre is a notable exception, since it is weak regardless of the number of rounds.