Paper 2014/416

Adaptive Security of Constrained PRFs

Georg Fuchsbauer, Momchil Konstantinov, Krzysztof Pietrzak, and Vanishree Rao


Constrained pseudorandom functions have recently been introduced independently by Boneh and Waters (Asiacrypt'13), Kiayias et al. (CCS'13), and Boyle et al. (PKC'14). In a standard pseudorandom function (PRF) a key $K$ is used to evaluate the PRF on all inputs in the domain. Constrained PRFs additionally offer the functionality to delegate ``constrained'' keys $K_S$ which allow to evaluate the PRF only on a subset $S$ of the domain. The three above-mentioned papers all show that the classical GGM construction (J.ACM'86) of a PRF from a pseudorandom generator (PRG) directly yields a constrained PRF where one can compute constrained keys to evaluate the PRF on all inputs with a given prefix. This constrained PRF has already found many interesting applications. Unfortunately, the existing security proofs only show selective security (by a reduction to the security of the underlying PRG). To achieve full security, one has to use complexity leveraging, which loses an exponential factor $2^N$ in security, where $N$ is the input length. The first contribution of this paper is a new reduction that only loses a quasipolynomial factor $q^{\log N}$, where $q$ is the number of adversarial queries. For this we develop a new proof technique which constructs a distinguisher by interleaving simple guessing steps and hybrid arguments a small number of times. This approach might be of interest also in other contexts where currently the only technique to achieve full security is complexity leveraging. Our second contribution is concerned with another constrained PRF, due to Boneh and Waters, which allows for constrained keys for the more general class of bit-fixing functions. Their security proof also suffers from a $2^N$ loss, which we show is inherent. We construct a meta-reduction which shows that any ``simple'' reduction of full security from a non-interactive hardness assumption must incur an exponential security loss.

Note: Major revision improving presentation of the results.

Available format(s)
Secret-key cryptography
Publication info
A major revision of an IACR publication in ASIACRYPT 2014
Constrained PRFComplexity LeveragingFull SecurityMeta-Reduction
Contact author(s)
gfuchsbauer @ ist ac at
2015-01-28: revised
2014-06-05: received
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Creative Commons Attribution


      author = {Georg Fuchsbauer and Momchil Konstantinov and Krzysztof Pietrzak and Vanishree Rao},
      title = {Adaptive Security of Constrained {PRFs}},
      howpublished = {Cryptology ePrint Archive, Paper 2014/416},
      year = {2014},
      note = {\url{}},
      url = {}
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