Cryptography Week 2 - Problem Set

$2^{128}\approx3.4e+38$ $$\begin{align*} m = 0^{64}:& \\ &L_1 = 0^{32} R_1 \\ &R_1 = 0^{32} \oplus F(k_1, 0^{32}) \\ &L_2 = 0^{32} \oplus F(k_1, 0^{32}) \\ &R_2 = 0^{32} \oplus F(k_2, 0^{32}\oplus F(k_1,0^{32})) \\ m = 1^{32}0^{32}:& \\ &L_1 = 0^{32} R_1 \\ &R_1 = 1^{32} \oplus F(k_1, 0^{32}) \\ &L_2 = 1^{32} \oplus F(k_1, 0^{32}) \\ &R_2 = 0^{32} \oplus F(k_2, 1^{32}\oplus F(k_1,0^{32})) \\ \end{align*}$$

Therefore:

$$\underset{m = 0^{64}}{L_2} \oplus \underset{m = 1^{32}0^{32}}{L_2} \equiv 1^{32}$$

$$\begin{align*} c' &= F(k, IV \oplus m_1) \\ &= F(k, F(k, c_0) \oplus c_0 \oplus c_0) \\ &= F(k, F(k, c_0) \oplus c_0 \oplus F(k, c_0)) \\ &= F(k, c_0) \end{align*}$$

Therefore:

$$c_1 = c'_0$$