Maths, encryption, and quantum computing

Are you worried about someone listening into your calls, reading your emails, or watching your video chats?

You’re not alone.

The internet has changed the face of communication and how communication can be stolen, spied upon or manipulated, and it always gets harder to protect ourselves as technology evolves. This becomes even more difficult as we face a future of quantum computers, which will be so powerful they’ll make the security we use now look like mere child’s play.

But thankfully, there is a huge amount of research into cybersecurity, and complex mathematics acts as our invisible protector. This is the field of cryptography – the study of secure communications.

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Internet communication is significantly harder to keep secure because it’s difficult to make sure you’re communicating with the right person or system.

“[At an] event, you see new people,” explains Josef Pieprzyk, a cryptographer and senior principal research scientist at CSIRO Data61. “If you would like to get to know someone, you can get a friend to introduce the new person to you.

Josef pieprzyk pic
Professor Josef Pieprzyk.

 “All this social interaction is more or less face-to-face – all boxed – so such things like identification of the person is probably relatively easy. Once you know the person you can identify the person wherever you meet them again.

“The problem with the internet is that this is not the reality. You are talking to somebody you think you know, but you may not actually be talking to the right person.”

In this case, extra steps need to be taken to ensure that identification is correct – except a lot of this happens behind the scenes, so we barely know it is happening. This is where encryption comes in.


Like secret code words used to get into a clubhouse as a kid, if two people, and two people alone, share a secret password to identify each other, false messages are less likely to make it through the door. Better yet, if that message is coded on its way over, and only the receiver knows the cypher, the message is further protected.

Encryption works in a similar way. The sender and the receiver both have special keys to identify whether the message came from the correct source and reached the correct destination. The message is written and sent off, but the contents are scrambled and become unreadable – that is, a cryptogram. The message can only be translated when the receiver has the second key.

The keys are a collection of algorithms, and they can be a single key or two different keys, depending on need. Asymmetric encryption has two different keys; one is a public key that many people can use, which renders the data down into the secret code. The second key is not shared but is held by only one person or system, and it is responsible for reading the coded data and translating it into something we can understand.

Asymmetric encryption
Encryption. Credit: Cosmos Magazine

The well-know asymmetric RSA encryption, named after their inventors: Rivest, Shamir and Adleman, first converts messages into integers and next rises a message/integer to the power, which is the public key. The calculations are done modulo a long integer N, whose factorization is known to the receiver only. The security of RSA relies on the difficulty of factoring long integers.

When the number is long, for example, 240 digits, there are so many integer combinations that make the code, it would take up to 800 years of computing power to break.

All of this happens behind the scenes, and it can be used to store or send confidential information so that people can’t steal it or eavesdrop.

“This provides some sort of confidentiality,” says Pieprzyk. “Sometimes you would like to remain private. In a sense, anonymous.”

But what if we had a faster, quantum computer?

The faster the computer, the easier it is to break the encryption.

“Cryptography actually always tries to keep sync with the developments, and you can actually [see] that’s happening in the quantum world,” says Piepryzk.

“Factorisation, which is used for the current classical public key cryptography, is easy [to break] on quantum computers. factorisation is simple.

“You can factor long integers and break RSA on Quantum. It’s quite easy.

“So now we are trying to design the cryptography, which will be resistant against quantum computing.”

Instead of using integer factorisation, other mathematical approaches need to be used to circumvent the sheer ‘brain’ power quantum computers will possess. One of mathematical tools that are being used to construct quantum-resistant encryption is Geometry of Numbers or Lattice Theory.

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In this case, a mathematical lattice is built, where the encryption key moves points of lattices (or messages) in such a way that their decryption is easy if you know the matching secret key. If you do not, then decryption is difficult and equivalent to finding the shortest vector in lattices.

“Finding the shortest vectors in lattices is relatively easy for two-dimensional spaces, but if the size of the space is, let’s say, hundreds or thousands, suddenly this problem becomes really difficult and quantum computers will not [crack it without the key],” says Piepryzk.

“And that is how we keep up with the evolution of a technology, because the quantum computers will be probably the next big, big thing if they happen.”

But there isn’t a robust way of testing security yet – we have to rely on mathematical theory, because frameworks need to be established before we even have quantum computers.

Thankfully, the codes are based in mathematics, where the logic doesn’t change, and that logical framework can still provide a solid security protocol that will be implemented into future quantum computers because the logic itself can be thoroughly tested.

Regardless, the invisible mathematics that protects our data will be just as important in a quantum future as it is now.

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