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Cryptology

[50 Points] Suppose that the S-Box of Example 4.1 is replaced by the S-Box defined by the
following substitution πS′:
z 0 1 2 3 4 5 6 7 8 9 A B C D E F
πS′(z) 8 4 2 1 C 6 3 D A 5 E 7 F B 9 0
(a) [21 Points] Compute the linear approximation table NL (as defined in Definition 4.1)
for this S-Box.
(b) [21 Points] Find a linear approximation using the active S-Boxes S
1
4
, S
2
1
and S
3
1
, and
use the piling-up lemma to estimate the bias of the random variable
X16 ⊕ U
4
1 ⊕ U
4
9
(c) [8 Points] Describe a linear attack, analogous to Algorithm 4.2, that will find eight
subkey bits in the last round.
[50 Points] Suppose that the S-Box of Example 4.1 is replaced by the S-Box defined by
following substitution πS′′:
z 0 1 2 3 4 5 6 7 8 9 A B C D E F
πS′′(z) E 2 1 3 D 9 0 6 F 4 5 A 8 C 7 B
(a) [21 Points] Compute the table of values ND (as defined in Definition 4.3) for this
S-Box.
(b) [21 Points] Find a differential trail using four active S-Boxes, namely, S
1
1
, S
1
4
, S
2
4
, and
S
3
4
, that has propogation ratio 27
2048 .
(c) [8 Points] Describe a differential attack, analogous to Algorithm 4.3, that will find
eight subkey bits in the last round.

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