Quantum Group Action

In quantum cryptography, there could be a new world, Microcrypt, where cryptography is possible but one-way functions (OWFs) do not exist. Many quantum analogues of OWFs and pseudorandom generators (PRGs) that are potentially weaker than OWFs have been introduced, and their useful applications have been demonstrated. However, Microcrypt still lacks enough foundations on which they are based compared with the classical cryptographic world. In classical cryptography, many hardness assumptions on concrete mathematical problems have been introduced, such as the discrete logarithm (DL) problems or the decisional Diffie-Hellman (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 concrete mathematical hardness assumptions with applications. Our assumptions are based on a quantum analogue of group actions. A group action is a tuple <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋆</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> of a group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> , a set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> , and an operation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⋆</mml:mo> <mml:mi>:</mml:mi> <mml:mi>G</mml:mi> <mml:mi>×</mml:mi> <mml:mi>S</mml:mi> <mml:mo>→</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> . We introduce a quantum analogue of group actions, which we call quantum group actions (QGAs), where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> is a subgroup of unitary operators, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> is a set of states, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⋆</mml:mo> </mml:mrow> </mml:math> 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 Diffie-Hellman (DDH) assumption and pseudorandom group actions. As an application, we construct classical-query pseudorandom function-like state generators (PRFSGs) based on these assumptions. PRFSGs are a quantum analogue of pseudorandom functions (PRFs), and have many applications such as IND-CPA SKE, EUF-CMA MAC, and private-key quantum money schemes. We also show that those assumptions are implied by non-adaptive pseudorandom unitaries (PRUs). Because classical group actions are instantiated with many concrete mathematical hardness assumptions, our QGAs could also have concrete (even OWFs-free) instantiations. We provide some candidate instantiations based on random circuits and IQP circuits.