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Generalized Selective Decryption (GSD), introduced by Panjwani [TCC’07], is a game for a symmetric encryption scheme \(\mathsf{Enc}\) that captures the difficulty of proving adaptive security of certain protocols, most notably the Logical Key Hierarchy (LKH) multicast encryption protocol. In the GSD game there are n keys \(k_1,\ldots ,k_n\), which the adversary may adaptively corrupt (learn); moreover, it can ask for encryptions \(\mathsf{Enc}_{k_i}(k_j)\) of keys under other keys. The adversary’s task is to distinguish keys (which it cannot trivially compute) from random. Proving the hardness of GSD assuming only IND-CPA security of \(\mathsf{Enc}\) is surprisingly hard. Using “complexity leveraging” loses a factor exponential in n, which makes the proof practically meaningless.

A Quasipolynomial Reduction for Generalized Selective Decryption on Trees