Correlated Pseudorandom Functions from Variable-Density LPN

Correlated secret randomness is a useful resource for many cryptographic applications. We initiate the study of pseudorandom correlation functions (PCFs) that offer the ability to securely generate virtually unbounded sources of correlated randomness using only local computation. Concretely, a PCF is a keyed function
$F_{k}$
such that for a suitable joint key distribution (
$k_{0}, k_{1}$
), the outputs
$(f_{k_{0}}(x), f_{k_{1}}(x))$
are indistinguishable from instances of a given target correlation. An essential security requirement is that indistinguishability hold not only for outsiders, who observe the pairs of outputs, but also for insiders who know one of the two keys. We present efficient constructions of PCFs for a broad class of useful correlations, including oblivious transfer and multiplication triple correlations, from a variable-density variant of the Learning Parity with Noise assumption (VDLPN). We also present several cryptographic applications that motivate our efficient PCF constructions. The VDLPN assumption is independently motivated by two additional applications. First, different flavors of this assumption give rise to weak pseudorandom function candidates in depth-2
$\text{AC}^{0}[\oplus]$
that can be conjectured to have subexponential security, matching the best known learning algorithms for this class. This is contrasted with the quasipolynomial security of previous (higher-depth)
$\text{AC}^{0}[\oplus]$
candidates. We support our conjectures by proving resilience to several classes of attacks. Second, VDLPN implies simple constructions of pseudorandom generators and weak pseudorandom functions with security against XOR related-key attacks.