Sparse Learning Parity with Noise ⇒ LPN

Sparse Learning Parity with Noise implies LPN.

Statement

Migrated verbatim from learning-parity-with-noise § Sparse Learning Parity with Noise:

is negligible. Note that Sparse LPN with $d=k$ reduces to standard LPN, so sparse hardness is a stronger assumption for smaller $d$.

Notes

source: folklore: the claim carried no citation on the page it was migrated from, and none was invented.

class: unstated: no citing page says which notion of reduction is meant. Recording a class the wiki does not state would add a claim.

Recorded during migration and not fixed — these are claims about the source text, not changes to it:

• SUSPECTED MATHEMATICAL ERROR: with d = k every row of Hamming weight exactly d is the all-ones vector, which is not the uniformly random matrix of standard LPN; the intended statement is presumably about d near k/2 or an asymptotic approximation.
• reduces to is directionally ambiguous as written, and the follow-on (so sparse hardness is a stronger assumption for smaller d) is an unstated second relation.
• No citation.