The New Math of Quantum Cryptography

The original version of This story appeared in Quanta magazine.

Difficult problems are generally not a welcome show. But cryptographers love them. This is because some difficult mathematics problems underlie the safety of modern encryption. Any intelligent tip to solve them will condemn most forms of cryptography.

Several years ago, researchers found a radically new approach to encryption that does not have this weak potential point. The approach exploits the particular characteristics of quantum physics. But unlike the previous quantum encryption schemes, which only work for a few special tasks, the new approach can perform a much wider range of tasks. And that could work even if all the problems at the heart of ordinary “classic” cryptography are easily resolved.

But this striking discovery was based on unrealistic hypotheses. The result was “more proof of concept,” said Fermi MA, an cryptography researcher at the Simons Institute for Theory of Computing in Berkeley, California. “It is not a statement on the real world.”

Now, a new article by two cryptographers has opened a way to quantum cryptography without these bizarre hypotheses. “This article says that if some other conjectures are true, then quantum cryptography must exist,” said MA.

Castle in the sky

You can consider modern cryptography as a tower with three essential parts. The first part is the foundation deeply under the tower, which is made of hard mathematical problems. The tower itself is the second part – you can find specific cryptographic protocols that allow you to send private messages, sign digital documents, secrets of secret voting, etc.

Between the two, securing these daily applications in the mathematical base is a base made up of construction blocks called unidirectional functions. They are responsible for the asymmetry inherent in any encryption scheme. “It’s one -way because you can encrypt messages, but you cannot decipher them,” said Mark Zhandry, cryptographer at NTT Research.

In the 1980s, researchers proved that cryptography built at the top of unidirectional functions would ensure the safety of many different tasks. But decades later, they are still not sure that the foundation is strong enough to support it. The problem is that the rocky substratum is made up of particular difficult problems – in a technical manner known as NP problems – whose definition of functionality is that it is easy to check if a candidate solution is correct. (For example, dividing a number into its primary factors is an NP problem: difficult to do for a large number, but easy to check.)

Many of these problems seem intrinsically difficult, but computer scientists could not prove it. If someone discovers an ingenious algorithm to quickly solve the hardest NP problems, the foundation will collapse and the whole tower will collapse.

Unfortunately, you can’t just move your turn elsewhere. The Tower Foundation – works on a route – can only be seated on a basis of NP problems.

To build a tower on more difficult problems, cryptographers would need a new foundation which is not made of unidirectional functions. It seemed impossible until a few years ago, when researchers realized that quantum physics could help.