COMPX304-24A: Assignment 3

Individual Assignment

Submit on Moodle Deadline: 21:00, Sunday, 19th May 2024 Weight: 12.5%

Important

Please credit all sources you refer to. Students found plagiarising will be reported to the disciplinary committee. You are expected to follow the University’sguidelines here:

https://www.waikato.ac.nz/students/academic-integrity/student-information/plagiarism

Assignments will be checked against anti-plagiarism checkers.

Quantum Key Exchange

Symmetric-key cryptography is a popular method of securely communicating data between two  entities.  AES,  ChaCha20,  3DES  are  examples  of  symmetric   key  encryption  systems supported by TLS. While those are moderately complex algorithms a simple way to encrypt a message given a key is XOR encryption. This method repeatedly applies the XOR operation on the  message  using  the  key.  The  receiver  performs  the  same  operation  to  decipher  the message.

For  example,  if the  message  is  “0100110101100101”  and  the  key  is  “001”,  the  following processes take place:

Transmitter:

0100110101100101 (message)

0010010010010010 (key repeated)

0110100111110111 (XOR’ed cipher to be sent)

Transmitter ---------------------------------> Receiver

Receiver:

0110100111110111 (incoming message)

0010010010010010 (key repeated)

0100110101100101 (decipher message)

Every type of symmetric key encryption, however, requires a shared key to be known between the two parties. But how does one transmit a key over an insecure channel? Sending it as plain text means it can be intercepted by a malicious eavesdropper and we can’t encrypt it because it leads to a chicken-and-egg problem, since this operation would also require a key.  There are  many  solutions  proposed  to  solve  this  problem,  each  with  its  own  assumptions  and constraints. These solutions essentially build a secure channel within an insecure channel. A popular solution is the one used in TLS called the Diffie-Helman key-exchange algorithm.

Another innovative solution makes use of quantum computing. This solution is the subject of this assignment. Quantum computing manipulates quantum bits (qubits), instead of classical (regular) bits. Qubits are subject to quantum mechanical laws of physics. In this assignment, you’ll implement and test the Quantum Key Exchange (QKE) algorithm. This algorithm is also known as the BB84 algorithm.

QKE assumes the existence of a quantum communication channel over which qubits can be transferred. Qubits are sent as photons and can be encoded via a photon’s polarisation. For example, we can define counterclockwise polarization 。as 1, and clockwise polarisation ⟳ as 0. However, this is not the only option, a photon could also be polarised in a linear fashion; thus, we can define an upwards polarization ↑ as 1, and a downwards polarization ↓ as 0. Because  of  quantum  mechanics,  the  internal  state  of  a  qubit  can  only  be  measured  by interacting with it through a polarization filter of a specific type.

The key quantum mechanical observation is that these two types of polarization (circular vs. linear) are orthogonal to each other. This means that if a photon is polarized in a circular manner, it has an equal 50-50 chance to be measured as ↑ or ↓ when measured linearly (and vice-versa). Consequently, in software and for the purposes of this assignment, aqubit can be modelled as instances of the following class:

Based on these observations, the QKE algorithm works as follows (remember, the goal is to agree on a shared secret key):

1.  The transmitter sends a stream of qubits. For each qubit it records the value  and polarization type, which are both picked randomly with equal chances.

2. The receiver receives the stream of qubits. For each qubit it selects a random polarization type to measure it, and records the results.