The Equivalence of Two Quadratic Based IBEs

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In this short note, we show that two identity-based encryption schemes, introduced by Joye and Zhao et al., which appear different, are essentially the same scheme. The only difference between them is that one prioritizes speed (Zhao et al.), while the other prioritizes bandwidth (Joye). We also show how to speed-up Joye's scheme at the cost of adding at most one integer to the public key, thereby achieving a better encryption complexity, while having the same bandwidth requirements.