Suppose that the S - Box of Example 4.1 is replaced by the S - Box defined by the following substitution πS...

Question

Suppose that the S - Box of Example 4.1 is replaced by the S - Box defined by the following substitution πS:

(a) [21 Points] Compute the linear approximation table N1 (as defined in Definition 4.1) for this S - Box.

(b) [21 Points] Find a linear approximation using four active S - Boxes that will find eight subkey bits in the last round and use the piling-up lemma to estimate the bias of the random variable

$x_0 = x_0_0 + U_1^4_0 + U_2^4_0 + U_3^4_0$

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) | 7 | 3 | A | 5 | 4 | 8 | 9 | 2 | E | 6 | C | 1 | D | B | 0 | F  |

(a) [21 Points] Compute the linear approximation table N1 (as defined in Definition 4.1) for this S-Box.

(b) [21 Points] Find a linear approximation using four active S-Boxes that will find eight subkey bits in the last round and use the piling-up lemma to estimate the bias of the random variable

$X_7 \oplus X_8 \oplus U_1^4 \oplus U_4 \oplus U_1^{4_3} \oplus U_1^{4_6}$

CramX · Accepted Answer

Step 1: I'll solve part (a) of the problem here.. I'll solve part (b) in a separate response to ensure each part is well-explained and easy to follow. Here's the table: |-------|-------|-------|----------|---------------------|-----------------------------------| | 0000  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 0001  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 0010  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 0011  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 0100  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 0101  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 0110  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 0111  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 1000  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 1001  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 1010  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 1011  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 1100  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 1101  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 1110  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 1111  | 0000  | 0000  | 0000     | 0000                | 0000                              | | 0000  | 0001  | 0001  | 0000     | 0000                | 0000                              | | 0001  | 0001  | 0001  | 0001     | 0001                | 0000                              | | 0010  | 0001  | 0010  | 0000     | 0000                | 0000                              | | 0011  | 0001  | 0011  | 0001     | 0001                | 0000                              | | 0100  | 0001  | 0100  | 0000     | 0000                | 0000                              | | 0101  | 0001  | 0101  | 0001     | 0001                | 0000                              | | 0110  | 0001  | 0111  | 0000     | 0001                | 0001                              | | 0111  | 0001  | 0110  | 0001     | 0000                | 0001                              | | 1000  | 0001  | 1001  | 0000     | 0001                | 0001                              | | 1001  | 0001  | 1000  | 0001     | 0000                | 0001                              | | 1010  | 0001  | 1011  | 0000     | 0001                | 0001                              | | 1011  | 0001  | 1010  | 0001     | 0000                | 0001                              | | 1100  | 0001  | 1101  | 0000     | 0001                | 0001                              | | 1101  | 0001  | 1100  | 0001     | 0000                | 0001                              | | 1110  | 0001  | 1111  | 0000     | 0001                | 0001                              | | 1111  | 0001  | 1110  | 0001     | 0000                | 0001                              | | 0000  | 0010  | 0010  | 0000     | 0000                | 0000                              | | 0001  | 0010  | 0011  | 0001     | 0001                | 0000                              | | 0010  | 0010  | 0000  | 0000     | 0000                | 0000                              | | 0011  | 0010  | 0001  | 0001     | 0001                | 0000                              | | 0100  | 0010  | 0100  | 0000     | 0000                | 0000                              | | 0101  | 0010  | 0101  | 0001     | 0001                | 0000                              | | 0110  | 0010  | 0110  | 0001     | 0001                | 0000                              | | 0111  | 0010  | 0111  | 0000     | 0000                | 0000                              | | 1000  | 0010  | 1000  | 0000     | 0001                | 0001                              | | 1001  | 0010  | 1001  | 0001     | 0000                | 0001                              | | 1010  | 0010  | 1010  | 0001     | 0001                | 0000                              | | 1011  | 0010  | 1011  | 0000     | 0000                | 0000                              | | 1100  | 0010  | 1100  | 0000     | 0001                | 0001                              | | 1101  | 0010  | 1101  | 0001     | 0000                | 0001                              | | 1110  | 0010  | 1110  | 0001     | 0001                | 0000                              | | 1111  | 0010  | 1111  | 0000     | 0000                | 0000                              | | 0000  | 0011  | 0011  | 0001     | 0001                | 0000                              | | 0001  | 0011  | 0010  | 0001     | 0001                | 0000                              | | 0010  | 0011  | 0001  | 0001     | 0001                | 0000                              | | 0011  | 0011  | 0000  | 0001     | 0001                | 0000                              | | 0100  | 0011  | 0101  | 0001     | 0001                | 0000                              | | 0101  | 0011  | 0100  | 0001     | 0001                | 0000                              | | 0110  | 0011  | 0111  | 0000     | 0001                | 0001                              | | 0111  | 0011  | 0110  | 0000     | 0001                | 0001                              | | 1000  | 0011  | 1001  | 0001     | 0000                | 0001                              | | 1001  | 0011  | 1000  | 0000     | 0000                | 0000                              | | 1010  | 0011  | 1011  | 0001     | 0001                | 0000                              | | 1011  | 0011  | 1010  | 0001     | 0001                | 0000                              | | 1100  | 0011  | 1101  | 0000     | 0000                | 0000                              | | 1101  | 0011  | 1100  | 0000     | 0000                | 0000                              | | 1110  | 0011  | 1111  | 0001     | 0001                | 0000                              | | 1111  | 0011  | 1110  | 0001     | 0001                | 0000                              | | 0000  | 0100  | 0100  | 0000     | 0000                | 0000                              | | 0001  | 0100  | 0101  | 0001     | 0001                | 0000                              | | 0010  | 0100  | 0110  | 0000     | 0001                | 0001                              | | 0011  | 0100  | 0111  | 0001     | 0001                | 0000                              | | 0100  | 0100  | 0000  | 0000     | 0000                | 0000                              | | 0101  | 0100  | 0001  | 0001     | 0001                | 0000                              | | 0110  | 0100  | 0010  | 0001     | 0001                | 0000                              | | 0111  | 0100  | 0011  | 0000     | 0000                | 0000                              | | 1000  | 0100  | 1000  | 0000     | 0001                | 0001                              | | 1001  | 0100  | 1001  | 0001     | 0000                | 0001                              | | 1010  | 0100  | 1010  | 0001     | 0001                | 0000                              | | 1011  | 0100  | 1011  | 0000     | 0001                | 0001                              | | 1100  | 0100  | 1100  | 0000     | 0001                | 0001                              | | 1101  | 0100  | 1101  | 0001     | 0000                | 0001                              | | 1110  | 0100  | 1110  | 0001     | 0001                | 0000                              | | 1111  | 0100  | 1111  | 0000     | 0000                | 0000                              | | 0000  | 0101  | 0101  | 0001     | 0001                | 0000                              | | 0001  | 0101  | 0100  | 0001     | 0001                | 0000                              | | 0010  | 0101  | 0111  | 0001     | 0001                | 0000 Final Answer: | a     | b     | c     | a * S(b) | c * S(b $\oplus$ 1) | a * S(b) $\oplus$ c * S(b $\oplus$ 1) |

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QuestionAnatomy and Physiology

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Step 1:
I'll solve part (a) of the problem here.

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