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Pn must set some parameters. They choose a large prime p so that p = 2q + 1 and q is also prime; they also pick one element g ∈ Z∗p of order q. Using the key generation protocol described in , players jointly generate a public key y = ni=1 yi . Each player Pi keeps her corresponding share αi of the private key and publishes yi = g αi . 2 Deck Generation Each card is represented by a value jointly computed by all players in Protocol 1. We ﬁrst explain what Protocol 1 does and then describe the protocol in detail.