Hello there, I need some help on this cryptanalysis related problem.

A simple 3-round substitution permutation network cipher is shown in **Figure 1.**

The cipher operates on 8-bit blocks. The 8-bit plaintext block P is XOR-ed bitwise with the 8-bit key K1 before the resulting 8-bit block enters the two first round S- boxes. The remaining key mixing operations are handled similarly.

A substitution box (S-box) is shown in **Figure 2**. This S-box is used throughout the cipher shown in Figure 1, all 3 S-boxes are identical.

The permutation part of the first two rounds is as shown in **Figure 1.** The final (third) round does not implement any permutation; the outputs from the final round S-boxes are simply XOR-ed bitwise with the key K4 to produce ciphertext C .

256 plaintext-ciphertext (PC) pairs have been generated using the 3-round cipher and four secret keys (K1, K2, K3, K4). The 256 PC pairs are given in the file **PtoC.txt** attached below. Plaintexts and ciphertexts are given as integers with the natural binary interpretation. For example the integer 5 represents the 8-bit block 00000101.

The Question: Use Differential Cryptanalysis to recover the final round key K4.

i. develop one or more suitable 2-round differential approximations involving bits of the plaintexts P and bits of intermediate ciphertexts U3 (as shown in Figure 1).

ii. indicate any active S-boxes in your approximations and their biases. Indicate any tables (or other sources) you have used to calculate the biases

iii. give the absolute value of the bias of any 2-round approximation derived above and show how all such biases were calculated.

iv. Explain your specific choice of approximation(s).

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