No Constant-Cost Protocol for Point--Line Incidence

Abstract

Alice and Bob are given $n$-bit integer pairs $(x,y)$ and $(a,b)$, respectively, and they must decide if $y=ax+b$. We prove that the randomised communication complexity of this Point--Line Incidence problem is $Θ(\log n)$. This confirms a conjecture of Cheung, Hatami, Hosseini, and Shirley (CCC 2023) that the complexity is super-constant, and gives the first example of a communication problem with constant support-rank but super-constant randomised complexity.

Theorems & Definitions (14)

• Theorem 1
• Corollary 2
• Corollary 4
• Lemma 7: Line Lemma
• Lemma 8: Decomposition Lemma
• Lemma 8: Minor Arc Bound
• Lemma 9: Prime Bound
• Theorem 10: Siegel--Walfisz
• Claim 11
• proof : Proof of \ref{['clm:ap']}