NP-Completeness and Physical Zero-Knowledge Proof of Hotaru Beam

TL;DR

It is shown that Hotaru Beam is NP-complete and present a physical zero-knowledge proof (i.e. implementable using physical items) for proving that one knows a solution to the puzzle.

Abstract

Hotaru Beam is a logic puzzle which objective is to connect circles placed on a grid by drawing only lines with specified starting points and numbers of bends. A zero-knowledge proof is a communication protocol that allows one player to persuade the other that they are in possession of a certain piece of information without actually revealing it. We show that Hotaru Beam is NP-complete and present a physical zero-knowledge proof (i.e. implementable using physical items) for proving that one knows a solution to the puzzle.

Figures (10)

• Figure 1: An instance of Hotaru Beam (left) and a solution to it (right)
• Figure 2: An instance of the planar monotone 3-SAT problem (the vertical segments are drawn using dashed lines in order to improve clarity)
• Figure 3: Gadget simulating a variable. Beams from fireflies with bending number 0 (drawn using solid lines) are fixed in any solution. Note that there are only two ways how to draw the beams from the fireflies with bending numbers 1 and 2. The possibility drawn using the dashed lines corresponds to setting the variable to true (the other case is symmetric). The shaded fireflies function as potential entry points for beams from the clauses in which this variable appears. In this case, only positive clauses (placed above) are going to be able to connect to the variable.
• Figure 4: Gadget simulating a positive clause. The shaded fireflies represent the literals; a possible set of beams shot from them is drawn using dashed lines. In this example, only the second literal is going to be connected to the gadget simulating its corresponding variable.
• Figure 5: Card representation of the instance in Figure \ref{['fig.instance']}

Theorems & Definitions (1)

• Theorem 1