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Beginning with the work of Groth and Sahai, there has been much interest in transforming pairing-based schemes in composite-order groups to equivalent ones in prime-order groups. A method for achieving such transformations has recently been proposed by Freeman, who identied two properties of pairings using composite-order groups | \cancelling" and \projecting" | on which many schemes rely, and showed how either of these properties can be obtained using prime-order groups. In this paper, we give evidence for the existence of limits to such transformations. Specically, we show that a pairing generated in a natural way from the Decision Linear assumption in primeorder groups can be simultaneously cancelling and projecting only with negligible probability. As evidence that these properties can be helpful together as well as individually, we present a cryptosystem whose proof of security makes use of a pairing that is both cancelling and projecting. Our example cryptosystem is a simple round-optimal blind signature scheme that is secure in the common reference string model, without random oracles, and based on mild assumptions; it is of independent interest.

Limitations on Transformations from Composite-Order to Prime-Order Groups: The Case of Round-Optimal Blind Signatures