Watermarking PRFs against Quantum Adversaries

TL;DR

This paper introduces the concept of watermarking pseudorandom functions (PRFs) resilient against quantum attacks, proposing constructions based on quantum hardness assumptions and developing quantum extraction techniques.

Contribution

It defines secure quantum watermarking PRFs and provides two constructions, one privately extractable and one publicly extractable, using quantum hardness assumptions and indistinguishability obfuscation.

Findings

Constructed privately extractable watermarking PRF from quantum LWE

Developed publicly extractable watermarking PRF using IO and quantum LWE

Introduced quantum extraction technique for information retrieval from quantum states

Abstract

We initiate the study of software watermarking against quantum adversaries. A quantum adversary generates a quantum state as a pirate software that potentially removes an embedded message from a classical marked software. Extracting an embedded message from quantum pirate software is difficult since measurement could irreversibly alter the quantum state. In this work, we define secure watermarking PRFs for quantum adversaries (unremovability against quantum adversaries). We also present two watermarking PRFs as follows. - We construct a privately extractable watermarking PRF against quantum adversaries from the quantum hardness of the learning with errors (LWE) problem. The marking and extraction algorithms use a public parameter and a private extraction key, respectively. The watermarking PRF is unremovable even if adversaries have (the public parameter and) access to the extraction oracle, which returns a result of extraction for a queried quantum circuit. - We construct a publicly extractable watermarking PRF against quantum adversaries from indistinguishability obfuscation (IO) and the quantum hardness of the LWE problem. The marking and extraction algorithms use a public parameter and a public extraction key, respectively. The watermarking PRF is unremovable even if adversaries have the extraction key (and the public parameter). We develop a quantum extraction technique to extract information (a classical string) from a quantum state without destroying the state too much. We also introduce the notion of extraction-less watermarking PRFs as a crucial building block to achieve the results above by combining the tool with our quantum extraction technique.