Symplectic Lattices and GKP Codes -- Simple Randomized Constructions from Cryptographic Lattices

• URL: http://arxiv.org/abs/2509.10183v1
• Date: Fri, 12 Sep 2025 12:17:04 GMT
• Title: Symplectic Lattices and GKP Codes -- Simple Randomized Constructions from Cryptographic Lattices
• Authors: Johannes Blömer, Yinzi Xiao, Zahra Raissi, Stanislaw Soltan,
• Abstract summary: We construct good GKP codes from standard short integer solution lattices (SIS) as well as from ring SIS and module SIS lattices, R-SIS and M-SIS lattices.<n>Our construction yields GKP codes with distance $sqrtn/pi e$.<n>We present a simple decoding algorithm that, for many parameter choices, experimentally yields decoding results similar to the ones for NTRU-based codes.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We construct good GKP (Gottesman-Kitaev-Preskill) codes (in the sense of Conrad, Eisert and Seifert proposed) from standard short integer solution lattices (SIS) as well as from ring SIS and module SIS lattices, R-SIS and M-SIS lattices, respectively. These lattice are crucial for lattice-based cryptography. Our construction yields GKP codes with distance $\sqrt{n/\pi e}$. This compares favorably with the NTRU-based construction by Conrad et al. that achieves distance $\Omega(\sqrt{n/q}),$ with $n\le q^2/0.28$. Unlike their codes, our codes do not have secret keys that can be used to speed-up the decoding. However, we present a simple decoding algorithm that, for many parameter choices, experimentally yields decoding results similar to the ones for NTRU-based codes. Using the R-SIS and M-SIS construction, our simple decoding algorithm runs in nearly linear time. Following Conrad, Eisert and Seifert's work, our construction of GKP codes follows directly from an explicit, randomized construction of symplectic lattices with (up to constants $\approx 1$) minimal distance $(1/\sigma_{2n})^{1/2n}\approx \sqrt{\frac{n}{\pi e}}$, where $\sigma_{2n}$ is the volume of the 2n-dimensional unit ball. Before this result, Buser and Sarnak gave a non-constructive proof for the existence of such symplectic lattices.

Related papers

• On the Maximum Toroidal Distance Code for Lattice-Based Public-Key Cryptography [3.8559042217241566]
  We propose a maximum toroidal distance (MTD) code for lattice-based public-key encryption (PKE)<n>We show that the MTD code is essentially a variant of the Minal code recently introduced at IACR CHES 2025.
  arXiv  Detail & Related papers  (2026-01-13T11:27:15Z)
• Near-Asymptotically-Good Quantum Codes with Transversal CCZ Gates and Sublinear-Weight Parity-Checks [18.20811830109862]
  We construct the first known quantum codes with linear dimension and distance supporting non-Clifford gates.<n>We design an efficient decoding algorithm for these codes.<n>Our results can be viewed as a new generalization of Prony's method for reconstructing a function from partial access to its transform.
  arXiv  Detail & Related papers  (2025-10-08T09:27:41Z)
• Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes [0.9208007322096533]
  We construct an explicit infinite family of quantum LDPC codes supporting a $Cr-1Z$ gate with length $N$, dimension $Kgeq N1-epsilon$, distance $Dgeq N1/r/namepoly(log N)$, and stabilizer weight $wleqoperatorname(log N)$.
  arXiv  Detail & Related papers  (2024-10-18T17:52:59Z)
• Unifying error-correcting code/Narain CFT correspondences via lattices over integers of cyclotomic fields [0.0]
  We identify Narain conformal field theories that correspond to code lattices for quantum error-correcting codes (QECC) over integers of cyclotomic fields $Q(zeta_p)$ $(zeta_p=efrac2pi ip)$ for general prime $pgeq 3$.<n>This code-lattice construction is a generalization of more familiar ones such as Construction A$_C$ for ternary codes and (after the generalization stated below) Construction A for binary codes, containing them as special cases.
  arXiv  Detail & Related papers  (2024-10-16T12:08:04Z)
• Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates [23.22566380210149]
  We construct quantum codes that support $CCZ$ gates over qudits of arbitrary prime power dimension $q$. The only previously known construction with such linear dimension and distance required a growing alphabet size $q$.
  arXiv  Detail & Related papers  (2024-08-17T16:54:51Z)
• A shortcut to an optimal quantum linear system solver [55.2480439325792]
  We give a conceptually simple quantum linear system solvers (QLSS) that does not use complex or difficult-to-analyze techniques. If the solution norm $lVertboldsymbolxrVert$ is known exactly, our QLSS requires only a single application of kernel. Alternatively, by reintroducing a concept from the adiabatic path-following technique, we show that $O(kappa)$ complexity can be achieved for norm estimation.
  arXiv  Detail & Related papers  (2024-06-17T20:54:11Z)
• Perturb-and-Project: Differentially Private Similarities and Marginals [73.98880839337873]
  We revisit the input perturbations framework for differential privacy where noise is added to the input $Ain mathcalS$. We first design novel efficient algorithms to privately release pair-wise cosine similarities. We derive a novel algorithm to compute $k$-way marginal queries over $n$ features.
  arXiv  Detail & Related papers  (2024-06-07T12:07:16Z)
• Superposed Decoding: Multiple Generations from a Single Autoregressive Inference Pass [72.07642648108849]
  Superposed Decoding is a new decoding algorithm that generates $k$ drafts at the cost of one autoregressive inference pass. Superposed Decoding can be combined with other decoding strategies, resulting in universal coverage gains when scaling inference time compute.
  arXiv  Detail & Related papers  (2024-05-28T17:40:48Z)
• Near-optimal fitting of ellipsoids to random points [68.12685213894112]
  A basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = Omega(, d2/mathrmpolylog(d),)$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix.
  arXiv  Detail & Related papers  (2022-08-19T18:00:34Z)
• Quantum Resources Required to Block-Encode a Matrix of Classical Data [56.508135743727934]
  We provide circuit-level implementations and resource estimates for several methods of block-encoding a dense $Ntimes N$ matrix of classical data to precision $epsilon$. We examine resource tradeoffs between the different approaches and explore implementations of two separate models of quantum random access memory (QRAM) Our results go beyond simple query complexity and provide a clear picture into the resource costs when large amounts of classical data are assumed to be accessible to quantum algorithms.
  arXiv  Detail & Related papers  (2022-06-07T18:00:01Z)
• Divisible Codes for Quantum Computation [0.6445605125467572]
  Divisible codes are defined by the property that codeword weights share a common divisor greater than one. This paper explores how they can be used to protect quantum information as it is transformed by logical gates.
  arXiv  Detail & Related papers  (2022-04-27T20:18:51Z)
• On the Optimal Memorization Power of ReLU Neural Networks [53.15475693468925]
  We show that feedforward ReLU neural networks can memorization any $N$ points that satisfy a mild separability assumption. We prove that having such a large bit complexity is both necessary and sufficient for memorization with a sub-linear number of parameters.
  arXiv  Detail & Related papers  (2021-10-07T05:25:23Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.