In this work we first provide a framework for defining a large subset of pairing-based digital signature schemes which we call Partially Structure-Preserving Signature (PSPS) schemes. PSPS schemes are similar in nature to structure-preserving signatures with the exception that in PSPS schemes messages are scalars from Zp instead of being group elements. This class encompasses various existing schemes which have a number of desirable features which makes them an ideal building block for many privacy-preserving cryptographic protocols. Such schemes include the widely-used schemes of Camenisch-Lysyanskaya (CRYPTO 2004) and Pointcheval-Sanders (CT-RSA 2016). We then provide various impossibility and lower bound results for variants of this class. Our results include bounds for the signature and verification key sizes as well as lower bounds for achieving strong unforgeability. We also give a generic framework for transforming variants of PSPS schemes into structure-preserving ones. As part of our contribution, we also give a number of optimal PSPS schemes which may be of independent interest. Our results aid in understanding the efficiency of pairing-based signature schemes and show a connection between this class of schemes and structure-preserving ones.
Ghadafi, E. (in press). Partially structure-preserving signatures: Lower bounds, constructions and more