Card-Based Zero-Knowledge Proof Protocols for Pancake Sorting

Assume that, given a sequence of n integers from 1 to n arranged in random order, we want to sort them, provided that the only acceptable operation is a prefix reversal, which means to take any number of integers (sub-sequence) from the left of the sequence, reverse the order of the sub-sequence, and return them to the original sequence. This problem is called “pancake sorting,” and sorting an arbitrary sequence with the minimum number of operations restricted in this way is known to be NP-hard. In this paper, we consider applying the concept of zero-knowledge proofs to the pancake sorting problem. That is, we design card-based zero-knowledge proof protocols in which a user (the prover) who knows how to sort a given sequence with ℓ operations can convince another user (the verifier) that the prover knows this information without divulging it.