Editing Copy Protection

== Functionality Description ==

Copy Protection is a functionality first defined by Aaronson [[Copy Protection #References|[1] ]] that enables a Vendor to send a program (a circuit) to a Client so that the Client cannot duplicate it.

Classically, this functionality has been proven impossible. However, it is possible to copy protect some families of programs using quantum computation.

'''Tags:''' Quantum Functionality, Two Party Protocols, Universal Task, Computational Security

== Use-cases ==

* Any kind of software licence protection

== Protocols ==

Currently, protocols for Copy Protection are only known for a few families of circuits.

* Quantum Copy Protection for Compute and Compare functions [[Copy Protection #References|[3] ]] [[Copy Protection #References|[4] ]]
* Quantum Copy Protection for Pseudo Random Number Generators [[Copy Protection #References|[5] ]]

== Properties ==

A Copy Protection protocol for a family of circuits is made of two algorithms:

* '''Protect''', which takes as input a classical description of a circuit <math display="inline">C</math> and outputs a quantum encoding <math display="inline">\rho_C</math> of this circuit.
* '''Eval''', which takes as input a quantum state and an classical input, and returns a classical output.

A Copy Protection scheme for a family of circuits has <math display="inline">\varepsilon</math>-<math display="inline">correctness</math> if for any circuit <math display="inline">C</math> of this family and for any input <math display="inline">x</math> for this circuit, <math display="block">Pr[\mathbf{Eval}(\rho_C, x) = f(x);\ \rho_C \gets \mathbf{Protect}(C)] \geq 1 - \varepsilon</math>

A Copy Protection scheme for a family of circuits has <math display="inline">\delta</math>-<math display="inline">security</math> if no polynomially bounded quantum adversary can efficiently copy a protected program, more formally if for any such adversary, her probability of winning the following game is lower than <math display="inline">1 - \delta</math>:

* A Challenger samples a circuit C in the family and sends Protect(C) to the Adversary
* The Adversary runs any polynomial computation she wants on Protect(C) and sends two quantum states, respectively <math display="inline">\psi_A</math> and <math display="inline">\psi_B</math> to two of her agents, respectively Alice and Bob
* The Challenger samples two inputs <math display="inline">x_A, x_B</math> for the circuit and sends <math display="inline">x_A</math> to Alice and <math display="inline">x_B</math> to Bob.
* Alice sends <math display="inline">y_A</math> to the Challenger and Bob sends <math display="inline">y_B</math> to the Challenger.
* The Adversary wins iff <math display="inline">C(x_A) = y_A</math> and <math display="inline">C(x_B) = y_B</math>

We assume that Alice and Bob cannot communicate with each other.

== Further Information ==

Even with quantum computation, Copy Protection is not possible for all families of circuits. Currently, it has been proven impossible for all ''learnable functions'' and ''de-quantumizable functions'' [[Copy Protection #References|[2] ]].

==Knowledge Graph==
{{graph}}

== References ==

#[https://doi.org/10.1109/ccc.2009.42 Aaronson (2009)] proposed a fist definition of Quantum Copy Protection.
#[http://arxiv.org/abs/2005.05289 Ananth, Prabhanjan, and La Placa (2020)] constructed a family of unlearnable circuits that cannot be copy protected.
#[http://arxiv.org/abs/2009.13865 Coladangelo et al. (2020)] proposed a copy protection construction for Compute and Compare functions in the QROM.
#[http://arxiv.org/abs/2101.12739 Broadbent et al. (2021)] proposed a copy protection construction for Compute and Compare functions without assumptions but with a weaker adversary model.
#[http://arxiv.org/abs/2107.05692 Coladangelo et al. (2021)] proposed a copy protection construction for PRNGs based on coset states.