NTRU-GKP code[1]
Description
Multi-mode GKP code whose underlying lattice is utilized in variations of the NTRU cryptosystem [2]. Randomized constructions yield constant-rate GKP code families whose largest decodable displacement length scales as \(O(\sqrt{n})\) with high probability.
The integer-valued \(q\)-symplectic Gram matrix for an \(n\)-mode \(k\)-qubit good NTRU-GKP code is \begin{align} A = \sqrt{\frac{2}{q}}\begin{pmatrix}I & Q\\ 0 & qI \end{pmatrix}~, \tag*{(1)}\end{align} where \(Q\) is a circulant matrix constructed from coefficients of a cyclic polynomial used in the NTRU cryptosystem, and \(I\) is the \(n\)-dimensional identity matrix [1; Prop. 2].
Rate
Randomized constructions yield constant-rate GKP code families whose largest decodable displacement length scales as \(O(\sqrt{n})\) with high probability.Decoding
Babai’s nearest plane algorithm [3] can be used for bounded-distance decoding.An NTRU-based decoder against stochastic displacement noise is efficient because the decoding problem is equivalent to decrypting the NTRU cryptosystem with knowledge of the encoder.Code Capacity Threshold
A lower bound on the threshold for displacement noise can be formulated in terms of code parameters [1; Appx. B].Realizations
Public-key NTRU-based quantum communication protocol [1].Cousin
- Random quantum code— Several NTRU lattices come from randomized constructions, yielding constant-rate GKP code families whose largest decodable displacement length scales as \(O(\sqrt{n})\) with high probability.
Primary Hierarchy
References
- [1]
- J. Conrad, J. Eisert, and J.-P. Seifert, “Good Gottesman-Kitaev-Preskill codes from the NTRU cryptosystem”, Quantum 8, 1398 (2024) arXiv:2303.02432 DOI
- [2]
- J. Hoffstein, J. Pipher, and J. H. Silverman, “NTRU: A ring-based public key cryptosystem”, Lecture Notes in Computer Science 267 (1998) DOI
- [3]
- L. Babai, “On Lovász’ lattice reduction and the nearest lattice point problem”, Combinatorica 6, 1 (1986) DOI
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2023-03-27)
Cite as:
“NTRU-GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/ntru_gkp, arXiv:2606.11484