Quantum Cheques

Publicly-verifiable quantum money has been a central and challenging goal in quantum cryptography. To this day, no constructions exist based on standard assumptions. In this study, we propose an alternative notion called quantum cheques (QCs) that is more attainable and technologically feasible. A quantum cheque can be verified using a public-key but only by a single user. Specifically, the payer signs the quantum cheque for a particular recipient using their ID, and the recipient can validate it without the assistance of the bank, ensuring that the payer cannot assign the same cheque to another user with a different ID. Unlike quantum money, QCs only necessitate quantum communication when a cheque is issued by the bank, meaning all payments and deposits are entirely classical! We demonstrate how to construct QCs based on the well-studied learning-with-errors (LWE) assumption. In the process, we build two novel primitives which are of independent interest. Firstly, we construct signatures with publicly-verifiable deletion under LWE. This primitive enables the signing of a message $m$ such that the recipient can produce a classical string that publicly proves the inability to reproduce a signature of $m$. We then demonstrate how this primitive can be used to construct 2-message signature tokens. This primitive enables the production of a token that can be used to sign a single bit and then self-destructs. Finally, we show that 2-message signature tokens can be used to construct QCs.