  Secure software leasing (SSL) is a quantum cryptographic primitive that
enables users to execute software only during the software is leased. It
prevents users from executing leased software after they return the leased
software to its owner. SSL can make software distribution more flexible and
controllable. Although SSL is an attractive cryptographic primitive, the
existing SSL scheme is based on public key quantum money, which is not
instantiated with standard cryptographic assumptions so far. Moreover, the
existing SSL scheme only supports a subclass of evasive functions. In this
work, we present SSL schemes based on the learning with errors assumption
(LWE). Specifically, our contributions consist of the following.
  - We construct an SSL scheme for pseudorandom functions from the LWE
assumption against quantum adversaries.
  - We construct an SSL scheme for a subclass of evasive functions from the LWE
assumption against sub-exponential quantum adversaries.
  - We construct SSL schemes for the functionalities above with classical
communication from the LWE assumption against (sub-exponential) quantum
adversaries. SSL with classical communication means that entities exchange only
classical information though they run quantum computation locally.
  Our crucial tool is two-tier quantum lightning, which is introduced in this
work and a relaxed version of quantum lighting. In two-tier quantum lightning
schemes, we have a public verification algorithm called semi-verification and a
private verification algorithm called full-verification. An adversary cannot
generate possibly entangled two quantum states whose serial numbers are the
same such that one passes the semi-verification, and the other also passes the
full-verification. We show that we can construct a two-tier quantum lightning
scheme from the LWE assumption.
