Linear regression
1. [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.
2. [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.
Sample Solution
Linear regression is a basic and commonly used type of predictive analysis. The overall idea of regression is to examine two things: (1) does a set of predictor variables do a good job in predicting an outcome (dependent) variable? (2) Which variables in particular are significant predictors of the outcome variable, and in what way do they, indicated by the magnitude and sign of the beta estimated, impact the outcome variable? These regression estimates are used to explain the relationship between one dependent variable and one or more independent variables. The simplest form of regression equation with one dependent and one independent variable is defined by the formula y = c + b*x, where y = estimated dependent variable score, c = constant, b = regression coefficient, and x = score on the independent variable.
