# Related-Key Security for Pseudorandom Functions Beyond the Linear Barrier

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## Abstract

Related-key attacks (RKAs) concern the security of cryptographic primitives in the situation where the key can be manipulated by the adversary. In the RKA setting, the adversary’s power is expressed through the class of related-key deriving (\(\mathrm {RKD}\)) functions which the adversary is restricted to using when modifying keys. Bellare and Kohno (EUROCRYPT 2003, volume 2656 of LNCS, Springer, Heidelberg, pp 491–506, 2003) first formalized RKAs and pinpointed the foundational problem of constructing RKA-secure pseudorandom functions (RKA-PRFs). To date there are few constructions for RKA-PRFs under standard assumptions, and it is a major open problem to construct RKA-PRFs for larger classes of \(\mathrm {RKD}\) functions. We make significant progress on this problem. We first show how to repair the framework for constructing RKA-PRF by Bellare and Cash (CRYPTO 2010, volume 6223 of LNCS, Springer, Heidelberg, pp 666–684, 2010) and extend it to handle the more challenging case of classes of \(\mathrm {RKD}\) functions that contain claws. We apply this extension to show that a variant of the Naor–Reingold function already considered by Bellare and Cash is an RKA-PRF for a class of affine \(\mathrm {RKD}\) functions under the Decisional Diffie–Hellman (DDH) assumption, albeit with a blowup that is exponential in the PRF input size. We then develop a second extension of the Bellare–Cash framework and use it to show that the same Naor–Reingold variant is actually an RKA-PRF for a class of degree *d* polynomial \(\mathrm {RKD}\) functions under the stronger decisional *d*-Diffie–Hellman inversion assumption. As a significant technical contribution, our proof of this result avoids the exponential-time security reduction that was inherent in the work of Bellare and Cash and in our first result. In particular, by setting \(d = 1\) (affine functions), we obtain a construction of RKA-secure PRF for affine relation based on the polynomial hardness of DDH.

## Keywords

Related-key security Pseudorandom functions Polynomial RKD functions## Notes

### Acknowledgements

We thank Susan Thomson for bringing the issues in the original Bellare–Cash framework to our attention, and for useful comments on the paper. Michel Abdalla, Fabrice Benhamouda, and Alain Passelègue were supported in part by the French ANR-10-SEGI-015 PRINCE Project, the *Direction Générale de l’Armement* (DGA), the CFM Foundation, the European Commission through the FP7-ICT-2011-EU-Brazil Program under Contract 288349 SecFuNet, and the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement 339563 – CryptoCloud). Alain Passelègue was supported in part from a DARPA/ARL SAFEWARE award, NSF Frontier Award 1413955, NSF Grants 1619348, 1228984, 1136174, and 1065276, BSF Grant 2012378, a Xerox Faculty Research Award, a Google Faculty Research Award, an equipment grant from Intel, and an Okawa Foundation Research Grant. This material is based upon work supported by the Defense Advanced Research Projects Agency through the ARL under Contract W911NF-15-C-0205. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense, the National Science Foundation, or the US Government. Kenneth G. Paterson was supported by an EPSRC Leadership Fellowship, EP/H005455/1.

## Supplementary material

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