Fast Multiplication and the PLWE-RLWE Equivalence for an Infinite Family of Cyclotomic Subextensions

• URL: http://arxiv.org/abs/2410.00792v1
• Date: Tue, 1 Oct 2024 15:32:02 GMT
• Title: Fast Multiplication and the PLWE-RLWE Equivalence for an Infinite Family of Cyclotomic Subextensions
• Authors: Joonas Ahola, Iván Blanco-Chacón, Wilmar Bolaños, Antti Haavikko, Camilla Hollanti, Rodrigo Martín Sánchez-Ledesma,
• Abstract summary: We prove the equivalence between the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems. We also describe a fast integers for computing the product of two elements in the ring of the maximal real subfield.
• Score: 6.487242614495099
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We prove the equivalence between the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems for the maximal totally real subfield of the $2^r 3^s$-th cyclotomic field for $r \geq 3$ and $s \geq 1$. Moreover, we describe a fast algorithm for computing the product of two elements in the ring of integers of these subfields. This multiplication algorithm has quasilinear complexity in the dimension of the field, as it makes use of the fast Discrete Cosine Transform (DCT). Our approach assumes that the two input polynomials are given in a basis of Chebyshev-like polynomials, in contrast to the customary power basis. To validate this assumption, we prove that the change of basis from the power basis to the Chebyshev-like basis can be computed with $\mathcal{O}(n \log n)$ arithmetic operations, where $n$ is the problem dimension. Finally, we provide a heuristic and theoretical comparison of the vulnerability to some attacks for the $p$-th cyclotomic field versus the maximal totally real subextension of the $4p$-th cyclotomic field for a reasonable set of parameters of cryptographic size.

Related papers

• Parallel Quantum Signal Processing Via Polynomial Factorization [3.1981483719988235]
  We develop a Quantum Parallel Signal Processing algorithm. Our algorithm parallelizes the computation of $texttr (P(rho)$ over $k$ systems and reduces the query depth to $d/k$, thus enabling a family of time-space tradeoffs for QSP. This furnishes a property estimation suitable for quantum computers, and is realized at the expense of increasing the number of measurements by factor $O(textpoly(d) 2(k) )$.
  arXiv  Detail & Related papers  (2024-09-27T17:54:30Z)
• A Novel Finite Fractional Fourier Transform and its Quantum Circuit Implementation on Qudits [0.0]
  We present a new number theoretic definition of discrete fractional Fourier transform (DFrFT) The DFrFT is defined as the $N times N$ dimensional unitary representation of the generator of the arithmetic rotational group $SO_2[mathbbZ_pn]$.
  arXiv  Detail & Related papers  (2024-09-09T16:15:53Z)
• On the Constant Depth Implementation of Pauli Exponentials [49.48516314472825]
  We decompose arbitrary exponentials into circuits of constant depth using $mathcalO(n)$ ancillae and two-body XX and ZZ interactions. We prove the correctness of our approach, after introducing novel rewrite rules for circuits which benefit from qubit recycling.
  arXiv  Detail & Related papers  (2024-08-15T17:09:08Z)
• On Maximal Families of Binary Polynomials with Pairwise Linear Common Factors [1.249418440326334]
  We characterize maximal families over the binary field $mathbbF$. Our findings prompt several more open questions, which we plan to address in an extended version of this work.
  arXiv  Detail & Related papers  (2024-05-14T16:30:28Z)
• Polynomial-Time Solutions for ReLU Network Training: A Complexity Classification via Max-Cut and Zonotopes [70.52097560486683]
  We prove that the hardness of approximation of ReLU networks not only mirrors the complexity of the Max-Cut problem but also, in certain special cases, exactly corresponds to it. In particular, when $epsilonleqsqrt84/83-1approx 0.006$, we show that it is NP-hard to find an approximate global dataset of the ReLU network objective with relative error $epsilon$ with respect to the objective value.
  arXiv  Detail & Related papers  (2023-11-18T04:41:07Z)
• Using 1-Factorization from Graph Theory for Quantum Speedups on Clique Problems [0.0]
  We provide new Quantum oracle designs based on the 1-factorization of complete graphs. We also discuss the usage of one of these oracles in bringing the Triangle Finding time complexity down to $O(n2.25 poly(log n))$.
  arXiv  Detail & Related papers  (2023-08-31T15:59:35Z)
• Quantum and classical low-degree learning via a dimension-free Remez inequality [52.12931955662553]
  We show a new way to relate functions on the hypergrid to their harmonic extensions over the polytorus. We show the supremum of a function $f$ over products of the cyclic group $exp(2pi i k/K)_k=1K$. We extend to new spaces a recent line of work citeEI22, CHP, VZ22 that gave similarly efficient methods for learning low-degrees on hypercubes and observables on qubits.
  arXiv  Detail & Related papers  (2023-01-04T04:15:40Z)
• Average-Case Complexity of Tensor Decomposition for Low-Degree Polynomials [93.59919600451487]
  "Statistical-computational gaps" occur in many statistical inference tasks. We consider a model for random order-3 decomposition where one component is slightly larger in norm than the rest. We show that tensor entries can accurately estimate the largest component when $ll n3/2$ but fail to do so when $rgg n3/2$.
  arXiv  Detail & Related papers  (2022-11-10T00:40:37Z)
• Robustly Learning any Clusterable Mixture of Gaussians [55.41573600814391]
  We study the efficient learnability of high-dimensional Gaussian mixtures in the adversarial-robust setting. We provide an algorithm that learns the components of an $epsilon$-corrupted $k$-mixture within information theoretically near-optimal error proofs of $tildeO(epsilon)$. Our main technical contribution is a new robust identifiability proof clusters from a Gaussian mixture, which can be captured by the constant-degree Sum of Squares proof system.
  arXiv  Detail & Related papers  (2020-05-13T16:44:12Z)
• SURF: A Simple, Universal, Robust, Fast Distribution Learning Algorithm [64.13217062232874]
  SURF is an algorithm for approximating distributions by piecewises. It outperforms state-of-the-art algorithms in experiments.
  arXiv  Detail & Related papers  (2020-02-22T01:03:33Z)

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.