# Structure-Preserving Signatures on Equivalence Classes and Constant-Size Anonymous Credentials

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## Abstract

Structure-preserving signatures (SPS) are a powerful building block for cryptographic protocols. We introduce SPS on equivalence classes (SPS-EQ), which allow joint randomization of messages and signatures. Messages are projective equivalence classes defined on group-element vectors, so multiplying a vector by a scalar yields a different representative of the same class. Our scheme lets one adapt a signature for one representative to a signature for another representative without knowledge of any secret. Moreover, given a signature, an adapted signature for a different representative is indistinguishable from a fresh signature on a random message. We propose a definitional framework for SPS-EQ and an efficient construction in Type-3 bilinear groups, which we prove secure against generic forgers. We also introduce set-commitment schemes that let one open subsets of the committed set. From this and SPS-EQ, we then build an efficient multi-show attribute-based anonymous credential system for an arbitrary number of attributes. Our ABC system avoids costly zero-knowledge proofs and only requires a short interactive proof to thwart replay attacks. It is the first credential system whose bandwidth required for credential showing is independent of the number of its attributes, i.e., constant-size. We propose strengthened game-based security definitions for ABC and prove our scheme anonymous against malicious organizations in the standard model; finally, we discuss a concurrently secure variant in the CRS model.

## Keywords

Public-key cryptography Pairing-based cryptography Structure-preserving signatures Attribute-based anonymous credentials Set commitments## Notes

### Acknowledgements

Work started while the first author was at IST Austria and supported by the European Research Council, ERC Starting Grant (259668-PSPC); now supported by the French ANR EfTrEC project (ANR-16-CE39-0002). Work has been done while the second and third authors were at IAIK, Graz University of Technology. The second author has been supported by the European Commission through projects FP7-MATTHEW (GA No. 610436) and FP7-FutureID (GA No. 318424). The work of the last author has been supported by the European Commission through project FP7-FutureID (GA No. 318424) and by EU Horizon 2020 through project Prismacloud (GA No. 644962).

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