The Hardness of LPN over Any Integer Ring and Field for PCG Applications

Learning parity with noise (LPN) has been widely studied and used in cryptography. It was recently brought to new prosperity since Boyle et al. (CCS’18), putting LPN to a central role in designing secure multi-party computation, zero-knowledge proofs, private set intersection, and many other protocols. In this paper, we thoroughly studied the security of LPN problems in this particular context. We found that some important aspects have long been ignored and many conclusions from classical LPN cryptanalysis do not apply to this new setting, due to the low noise rates, extremely high dimensions, various types (in addition to \(\mathbb F _2\) ) and noise distributions. Finally, we provide an all-in-one estimator tool for the bit security of LPN parameters in the context of PCG, incorporating the recent advanced attacks.