## Cryptology ePrint Archive: Report 2014/416

**Adaptive Security of Constrained PRFs**

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

**Abstract: **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
gives 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
get 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 novel 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.
We construct a meta-reduction which shows
that any ``simple'' reduction that proves full security of this construction from a non-interactive hardness assumption must incur an exponential security loss.

**Category / Keywords: **secret-key cryptography / Constrained PRF, Complexity Leveraging, Full Security, Meta-Reduction

**Date: **received 2 Jun 2014

**Contact author: **krzpie at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20140605:203716 (All versions of this report)

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