Paper 2024/1578
Quantum Group Actions
Abstract
In quantum cryptography, there could be a new world, Microcrypt, where cryptography is possible but oneway functions (OWFs) do not exist. Although many fundamental primitives and useful applications have been found in Microcrypt, they lack ``OWFsfree'' concrete hardness assumptions on which they are based. In classical cryptography, many hardness assumptions on concrete mathematical problems have been introduced, such as the discrete logarithm (DL) problems or the decisional DiffieHellman (DDH) problems on concrete group structures related to finite fields or elliptic curves. They are then abstracted to generic hardness assumptions such as the DL and DDH assumptions over group actions. Finally, based on these generic assumptions, primitives and applications are constructed. The goal of the present paper is to introduce several abstracted generic hardness assumptions in Microcrypt, which could connect the concrete mathematical hardness assumptions with applications. Our assumptions are based on a quantum analogue of group actions. A group action is a tuple $(G,S,\star)$ of a group $G$, a set $S$, and an operation $\star:G\times S\to S$. We introduce a quantum analogue of group actions, which we call quantum group actions (QGAs), where $G$ is a set of unitary operators, $S$ is a set of states, and $\star$ is the application of a unitary on a state. By endowing QGAs with some reasonable hardness assumptions, we introduce a natural quantum analogue of the decisional DiffieHellman (DDH) assumption and pseudorandom group actions. Based on these assumptions, we construct classicalquery pseudorandom functionlike state generators (PRFSGs). PRFSGs are a quantum analogue of pseudorandom functions (PRFs), and have many applications such as INDCPA SKE, EUFCMA MAC, and privatekey quantum money schemes. Because classical group actions are instantiated with many concrete mathematical hardness assumptions, our QGAs could also have some concrete (even OWFsfree) instantiations.
Metadata
 Available format(s)
 Category
 Foundations
 Publication info
 Preprint.
 Keywords
 quantum cryptographygroup actionsmicrocrypt
 Contact author(s)

tomoyuki morimae @ yukawa kyotou ac jp
keita xagawa @ tii ae  History
 20241008: approved
 20241007: received
 See all versions
 Short URL
 https://ia.cr/2024/1578
 License

CC BY
BibTeX
@misc{cryptoeprint:2024/1578, author = {Tomoyuki Morimae and Keita Xagawa}, title = {Quantum Group Actions}, howpublished = {Cryptology {ePrint} Archive, Paper 2024/1578}, year = {2024}, url = {https://eprint.iacr.org/2024/1578} }