Good Gottesman-Kitaev-Preskill codes from the NTRU cryptosystem
1Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
2Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany
3Fraunhofer Heinrich Hertz Institute, Einsteinufer 37, 10587 Berlin, Germany
4Electrical Engineering and Computer Science Department, Technische Universität Berlin, Straße des 17. Juni 135, 10587 Berlin, Germany
5Fraunhofer Institute for Secure Information Technology, Rheinstraße 75, 64295 Darmstadt, Germany
| Published: | 2024-07-04, volume 8, page 1398 |
| Eprint: | arXiv:2303.02432v4 |
| Doi: | https://doi.org/10.22331/q-2024-07-04-1398 |
| Citation: | Quantum 8, 1398 (2024). |
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Abstract
We introduce a new class of random Gottesman-Kitaev-Preskill (GKP) codes derived from the cryptanalysis of the so-called NTRU cryptosystem. The derived codes are $good$ in that they exhibit constant rate and average distance scaling $\Delta \propto \sqrt{n}$ with high probability, where $n$ is the number of bosonic modes, which is a distance scaling equivalent to that of a GKP code obtained by concatenating single mode GKP codes into a qubit-quantum error correcting code with linear distance. The derived class of NTRU-GKP codes has the additional property that $decoding$ for a stochastic displacement noise model is equivalent to $decrypting$ the NTRU cryptosystem, such that every random instance of the code naturally comes with an efficient decoder. This construction highlights how the GKP code bridges aspects of classical error correction, quantum error correction as well as post-quantum cryptography. We underscore this connection by discussing the computational hardness of decoding GKP codes and propose, as a new application, a simple public key quantum communication protocol with security inherited from the NTRU cryptosystem.
Featured image: Outline of the private quantum channel established using the NTRU-GKP code.
Popular summary
Lattices — periodic arrangements of points in space — are one of the simplest mathematical structures, ubiquitous in the sciences and everyday life. If one were to stack a pile of oranges — or cannonballs — the most efficient way to arrange them are often given by the clear periodic pattern of a lattice. The same holds for the atoms that make up a solid or a crystal, where the specific shape of the lattice controls how light refracts or electrons hop through the solid when a voltage is applied. In either of these cases, a core feature of a lattice is the shortest vector length: the smallest distance that separates two lattice points. It quantifies how tightly those oranges can be packed together in a periodic way. The larger the shortest vector length, the less space is wasted in the arrangement.
While we are accustomed to stacking oranges (hopefully not cannonballs) in three dimensional space, it turns out that in higher, n-dimensional space these packings get more and more efficient as lattices can be found with a shortest vector length that grows as a square root of the dimension of the space.
As the dimensions get higher, however, lattice problems, like the closest vector problem that asks to find a lattice point closest to any given point in space, become increasingly hard. They are so hard, in fact, that they are assumed to be intractable even for quantum computers, so that many proposals for classical cryptographic protocols that should remain safe even at a time where quantum computers become widely accessible are constructed based on such problems.
When attempting to design a quantum computer that should faithfully store and process information in a noisy world, we are confronted with a similar problem as when we tried to stack oranges. We are given a physical Hilbert space, representing the degrees of freedom of a quantum system, and wish to pack it with logical states. Just as oranges and cannonballs cannot overlap due to their intrinsically hard shell, now we need to arrange the logical states in Hilbert space such that overlaps between them are as small as possible even in the presence of noise that kicks them around by a bounded amount.
The way we decide to pack these states in the Hilbert space is described by a quantum error correcting code. Gottesman-Kitaev-Preskill (GKP) codes realize this packing within a special phase-space representation of the Hilbert space in the precise form of a lattice, such that the length of the shortest lattice vector (the efficiency of the packing) also quantifies the robustness to noise of this quantum error correcting code.
In this work we knit all of these aspects and applications of lattices together. We use a post-quantum cryptosystem, called NTRU, to derive lattices whose shortest vector length grows with the spatial dimension and tweak them into a form compatible with the GKP code. In this way, we establish a new connection between classical cryptography and quantum information theory. The result is a quantum error correcting code, an NTRU-GKP code, whose robustness (also called distance) scaling grows optimally with the size of the physical system; we obtain a good quantum error correcting code.
When a state of the system is perturbed by some small amount of noise, the task of decoding is to find a correction that takes us back to the state we had prior to the application of noise. In GKP codes, this task translates into finding the closest lattice point relative to a point measured post-noise which for NTRU-GKP codes is exactly the same as attempting to decrypt the NTRU cryptosystem!
Public key cryptosystems, such as NTRU, are constructed such that each instance comes with a public and a secret key. The public key allows anybody to encrypt information, but one needs the secret key to decrypt (unless they want to spend the rest of their lives waiting for a computer to solve the decryption problem).
As such, the NTRU-GKP code is not just a quantum error correcting code with good parameter scalings, but it also lays out an interesting cryptographic scheme, where only parties trusted with the secret key can recover logical information from corrupted states. In this work, this idea allows us to propose a so-called private quantum channel, which is a cryptographic scheme that protects the transmission of quantum information through a public channel. The proposed idea is more generally to build quantum cryptographic setups where untrusted parties are given the power to encode information in a quantum error correcting code, but the power to decode is kept to only a trusted few.
While this idea is only exemplified using the NTRU GKP code for a private quantum channel, the basic idea can be translated to other quantum error correcting codes and it is likely that more interesting applications can be found.
► BibTeX data
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