  Formulating cryptographic definitions to protect against software piracy is
an important research direction that has not received much attention. Since
natural definitions using classical cryptography are impossible to achieve (as
classical programs can always be copied), this directs us towards using
techniques from quantum computing. The seminal work of Aaronson [CCC'09]
introduced the notion of quantum copy-protection precisely to address the
problem of software anti-piracy. However, despite being one of the most
important problems in quantum cryptography, there are no provably secure
solutions of quantum copy-protection known for any class of functions.
  We formulate an alternative definition for tackling software piracy, called
secure software leasing (SSL). While weaker than quantum copy-protection, SSL
is still meaningful and has interesting applications in software anti-piracy.
  We present a construction of SSL for a subclass of evasive circuits (that
includes natural implementations of point functions, conjunctions with wild
cards, and affine testers) based on concrete cryptographic assumptions. Our
construction is the first provably secure solution, based on concrete
cryptographic assumptions, for software anti-piracy. To complement our positive
result, we show, based on cryptographic assumptions, that there is a class of
quantum unlearnable functions for which SSL does not exist. In particular, our
impossibility result also rules out quantum copy-protection [Aaronson CCC'09]
for an arbitrary class of quantum unlearnable functions; resolving an important
open problem on the possibility of constructing copy-protection for arbitrary
quantum unlearnable circuits.
