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Yes, $E$ will be always be secure. This follows from a standard type of proof called a hybrid argument. Giving the full details would be tedious, so here is a sketch in case you are familiar with hybrid arguments: We define games $H_0,H_1,H_2$. We let $H_0$ be the IND-CPA game, but with the game's secret bit hardcoded to $0$. So the game always outputs ...

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