Abstract
Modern cryptography systems, especially public key encryption, protect everything from banking to private messages, but existing systems may be vulnerable to attacks mounted on future quantum computers. Post-quantum cryptography aims to protect against the threat of future quantum computers. My research explores two algebraic approaches to build up post‑quantum encryption: braid group cryptography and tensor‑based cryptography. By studying their mathematical structure and comparing their computational hardness, this project investigates whether tensor isomorphism could provide a more promising crypto-system for secure communication in a quantum era.
Blog Post
Every time we send a message, make an online payment, or log into a secure website, we rely on public‑key encryption. This mathematical tool allows people to securely send a message without first knowing a secret key. However, the arrival of quantum computing threatens many classical encryption systems, whose securities are largely based on the hardness of some mathematical problems. As researchers, we must ask: what mathematical problems will remain hard (inefficiently solved) even for quantum machines?
One direction I explored is braid‑based cryptography. Imagine several strands of string, initially parallel, are intertwined in complex patterns. These braids can be described using algebra, and certain problems about rearranging them are believed to be difficult to solve. Cryptographic systems can hide secret information inside these braid structures. But over time, many mathematicians have developed clever attacks that exploit some vulnerable structures within braid groups.
This motivated me to investigate a newer idea: tensor‑based cryptography. Tensors are generalizations of matrices. While matrices describe two‑dimensional arrays of numbers, tensors extend this idea to higher dimensions. When we apply transformations to tensors, we create complicated relationships between their components. Determining whether two tensors are related by such a transformation is called the tensor isomorphism problem. For higher dimensions, this problem appears significantly more complex than the matrix case.
Recent research suggests that certain weak choices of tensors may accidentally make the problem easier, so carefully initializing the keys is essential. I studied how researchers design stronger sampling methods to avoid these pitfalls. To gain intuition, I also ran a small computational experiment comparing naive attempts to solve tensor isomorphism with simple integer factoring. Even in small toy examples, the tensor problem grew in complexity very quickly.
Although these experiments are far from real‑world cryptographic scales, they highlight why higher‑dimensional algebra might offer fresh opportunities for post‑quantum security. My project does not claim to deliver a ready‑to‑use encryption scheme. Instead, it explores whether tensor structures could be a better candidate for future systems that remain secure in a world with quantum computers.
Mathematics often surprises us. Structures that look complex and theoretical can sometimes turn out to protect the global communication. By studying braids, tensors, and the opaque structures inside them, we move one more step closer to answering an important question: how can cryptography safeguard the information in the digital future?
Ziyan Chen
The University of Sydney
