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A Cramer-Shoup public key has the form: $B = g_1^{b_1} g_2^{b_2}; \quad C = g_1^{c_1} g_2^{c_2}; \quad D = g_1^{d_1} g_2^{d_2}$ and an encryption of $m$ has the form: $g_1^r, ~ g_2^r, ~ m B^r, ~ (CD^\mu)^r$ where $\mu$ is a hash of the first 3 components. CCA security proof breaks down: In the CCA security proof, you observe that if you know ...

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