MATH 195: CRYPTOGRAPHY HOMEWORK #5

Problem 3.15. In SDES, a key K = (k_{1}, …, k_{10}) ∈ 𝔽_{2}^{10} = (ℤ/2ℤ)^{10} gives rise to two subkeys K_{1}, K_{2} ∈ 𝔽_{2}^{8}. Express K_{1} and K_{2} directly in terms of K. [Hint: Use the text, pp. 52–53.]

Problem 3.1. Refer to Figure 3.2, which depicts key generation for SDES. (a) How important is the initial P10 permutation function?

(b) How important are the two LS-1 shift functions?

Problem 3.3. Using SDES, decrypt the string (10100010) using the key

(0111111101)

by hand. Show intermediate results after each function (IP, F_{K}, SW, F_{K}, IP^{−1}). Then decode the first 4 bits of the plaintext string to a letter and the second 4 bits to another letter where we encode A through P in base 2 (i.e., A = 0000, B = 0001, …, P = 1111). [Hint: As a midway check, after the application of SW, the string should be (00010011).]

Problem 3.16. In DES, one has K_{i} = τλ^{n_{i}}σ(K) for 1 ≤ i ≤ 16 with τ, λ, σ and n_{1}, …, n_{16} as explained in class. Prove: K_{17−i} = τρ^{n_{i}−1}σ(K), where ρ = λ^{−1}. From which property of Table 3.4(c) does this follow?

Date: February 26, 2002.

3.15, 3.1, 3.3, 3.16.