Talk: Post-Quantum Cryptography and Crypto Agility

Mario Duhanic, 2024-10-24

Introduction

On October 24, 2024, I was invited to give a talk at SysEleven (for the OWASP community) on the topic of crypto-agility in times of quantum computing. That was a wonderful event with a great discussion afterwards.

The idea was to motivate the experts in the audience to keep post-quantum cryptography and its dangers on their radar. The talk did not go deep into the mathematics behind it, but was intended to be more of a top-level overview, albeit for a technical audience.

To keep the topic from being too dry, I embedded a few nerd T-shirts.

Presentation

Read on if you want to know about some nerdy basics (you don’t need them for the presentation) or click through the presentation below.

Basics (For the Mathematician inside You)

Here are a few basics to help you understand some of the basic ideas behind qubits. If you are interested in math, this should be digestible, if not, no worries: the presentation is not based on it.

Bra-ket

When reading about quantum mechanics you will often encounter angle brackets. What do they mean and how do you pronounce them?

This Bra-ket notation, also known as Dirac notation, is a notation widely used in quantum mechanics in linear algebra (on complex vector spaces). The notation was introduced to simplify the notation of quantum mechanical expressions. The name alludes to the English “bracket”.

In quantum mechanics and quantum computing, this notation is used to denote so called quantum states. The notation uses angle brackets ⟨ and ⟩.

Ket

A ket is “just” a complex vector and is written \(|\varphi\rangle\) representing a state of a quantum system. Read “ket phi”.

Bra

A bra \(\langle \psi|\) (read “bra psi”) can be understood as a vector of the dual (complex) vector space. Each \(|\varphi\rangle\) (“Ket phi”) then corresponds to a \(\langle \psi|\) (“Bra psi”). Yes, that sounds complicated and there are good books on the subject. For us here, that’s enough for now.

Bits and Qubits

In contrast to a classic bit, a qubit can assume more than two states, theoretically even (uncountably) an infinite number.

But let’s first clear up a common misconception. When we talk about a qubit, it’s not that it’s a 1 and a 0 at the same time. Instead, think of it as being partly a 1 and partly a 0. It’s like a dimmer switch for probability, not an on/off switch. Before we measure it, a qubit exists in a superposition of these states, meaning it has a certain probability of being a 0 and a certain probability of being a 1.

Now, how do we visualize this “partly” state? That’s where the Bloch sphere comes in handy. Imagine a regular sphere. The north pole represents the state where the qubit is definitely a 0, and the south pole represents the state where it’s definitely a 1. Any other point on the surface of this sphere represents a possible superposition state of the qubit. The closer a point is to the north pole, the more “0-like” the qubit is; the closer to the south pole, the more “1-like.”

So, how do we translate a point on this Bloch sphere into a mathematical description? That’s where the bra-ket notation steps in. It’s a super elegant and compact way to represent quantum states.

Think of the definite state of \(0\) as being written like this: \(∣0⟩\). This is called a “ket.” Similarly, the definite state of \(1\) is written as: \(∣1⟩\).

Now, if our qubit is in a superposition, meaning it’s somewhere on the Bloch sphere that’s not directly at the poles, we can describe its state as a combination of these \(∣0⟩\) and \(∣1⟩\) kets. For example, a general qubit state ∣ψ⟩ can be written as:

Here, α and β are complex numbers that tell us the “amount” of \(∣0⟩\) and \(∣1⟩\) in our qubit’s state. The absolute square of these numbers, \(∣α∣^2\) and \(∣β∣^2\), gives us the probabilities of measuring the qubit in the state \(∣0⟩\) or \(∣1⟩\) respectively. And importantly, these probabilities must add up to 1 (because when we measure, we have to get either a 0 or a 1).

So, in a nutshell, the Bloch sphere gives us a nice visual of all the possible superposition states of a qubit, and the bra-ket notation provides a precise mathematical language to describe these states, showing how much of the definite \(∣0⟩\) and \(∣1⟩\) states are “mixed” together. The position on the Bloch sphere directly corresponds to the values of α and β in the bra-ket notation!

Well, welcome to the world of qubits!