We introduce new forms of attack on expander-based cryptography, and in particular on Goldreich's pseudorandom generator and one-way function. Our attacks exploit low circuit complexity of the underlying expander's neighbor function and/or of the local predicate. Our two key conceptual contributions are:
1. We put forward the possibility that the choice of expander matters in expander-based cryptography. In particular, using expanders whose neighbour function has low circuit complexity might compromise the security of Goldreich's PRG and OWF in certain settings.
2. We show that the security of Goldreich's PRG and OWF over arbitrary expanders is closely related to two other long-standing problems: The existence of unbalanced lossless expanders with low-complexity neighbor function, and limitations on circuit lower bounds (i.e., natural proofs). In particular, our results further motivate the investigation of affine/local unbalanced lossless expanders and of average-case lower bounds against DNF-XOR circuits.
We prove two types of technical results. First, in the regime of quasipolynomial stretch (in which the output length of the PRG and the running time of the distinguisher are quasipolynomial in the seed length) we unconditionally break Goldreich's PRG, when instantiated with a specific expander whose existence we prove, and for a class of predicates that match the parameters of the currently-best ``hard'' candidates. Secondly, conditioned on the existence of expanders whose neighbor functions have extremely low circuit complexity, we present attacks on Goldreich's PRG in the regime of polynomial stretch. As one corollary, conditioned on the existence of the foregoing expanders, we show that either the parameters of natural properties for several constant-depth circuit classes cannot be improved, even mildly; or Goldreich's PRG is insecure in the regime of a large polynomial stretch for some expander graphs, regardless of the predicate used.
Incorporated helpful comments from journal reviewers: The presentation is improved, and Assumption 5.20 is refined.
We introduce new forms of attack on expander-based cryptography, and in particular on Goldreich's pseudorandom generator and one-way function. Our attacks exploit low circuit complexity of the underlying expander's neighbor function and/or of the local predicate. Our two key conceptual contributions are:
1. We put forward the possibility that the choice of expander matters in expander-based cryptography. In particular, using expanders whose neighbour function has low circuit complexity might compromise the security of Goldreich's PRG and OWF in certain settings.
2. We show that the security of Goldreich's PRG and OWF is closely related to two other long-standing problems: Specifically, to the existence of unbalanced lossless expanders with low-complexity neighbor function, and to limitations on circuit lower bounds (i.e., natural proofs). In particular, our results further motivate the investigation of affine/local unbalanced lossless expanders and of average-case lower bounds against DNF-XOR circuits.
We prove two types of technical results that support the above conceptual messages. First, we unconditionally break Goldreich's PRG when instantiated with a specific expander (whose existence we prove), for a class of predicates that match the parameters of the currently-best ``hard'' candidates, in the regime of quasi-polynomial stretch. Secondly, conditioned on the existence of expanders whose neighbor functions have extremely low circuit complexity, we present attacks on Goldreich's generator in the regime of polynomial stretch. As one corollary, conditioned on the existence of the foregoing expanders, we show that either the parameters of natural properties for several constant-depth circuit classes cannot be improved, even mildly; or Goldreich's generator is insecure in the regime of a large polynomial stretch, regardless of the predicate used.
Improved the exposition in the abstract and in Sec 1.1; elaborated on the connection to expander-based PRFs in Sec 2.4; various minor corrections and clarifications.
We introduce new forms of attack on expander-based cryptography, and in particular on Goldreich's pseudorandom generator and one-way function. Our attacks exploit low circuit complexity of the underlying expander's neighbor function and/or of the local predicate. Our two key conceptual contributions are:
* The security of Goldreich's PRG and OWF hinges, at least in some settings, on the circuit complexity of the underlying expander's neighbor function and of the local predicate. This sharply diverges from previous works, which focused on the expansion properties of the underlying expander and on the algebraic properties of the predicate.
* We uncover new connections between long-standing open problems: Specifically, we tie the security of Goldreich's PRG and OWF both to the existence of unbalanced lossless expanders with low-complexity neighbor function, and to limitations on circuit lower bounds (i.e., natural proofs).
We prove two types of technical results that support the above conceptual messages. First, we unconditionally break Goldreich's PRG when instantiated with a specific expander (whose existence we prove), for a class of predicates that match the parameters of the currently-best ``hard'' candidates, in the regime of quasi-polynomial stretch. Secondly, conditioned on the existence of expanders whose neighbor functions have extremely low circuit complexity, we present attacks on Goldreich's generator in the regime of polynomial stretch. As one corollary, conditioned on the existence of the foregoing expanders, we show that either the parameters of natural properties for several constant-depth circuit classes cannot be improved, even mildly; or Goldreich's generator is insecure in the regime of a large polynomial stretch, regardless of the predicate used.
In particular, our results further motivate the investigation of average-case lower bounds against DNF-XOR circuits of exponential size, and of the parameters that can be achieved by affine/local unbalanced expanders.
