The spectral norms of geometric circulant matrices with the generalized k-Horadam numbers

Shi, Beibei

doi:10.1186/s13660-017-1608-4

Cited by 14 publications

(4 citation statements)

References 8 publications

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“…Sometimes, f is called an input vector. It is easily seen that H * ( f ) is a more general form of the classical circulant matrix (see Davis (1994)) and r -circulant matrix (see Shi (2018), Yasin and Taskara (2013)). In fact, if…”

Section: Ideal Matricesmentioning

confidence: 99%

Cyclic Lattices, Ideal Lattices, and Bounds for the Smoothing Parameter

Zhiyong,

Fengxia,

Yunfan

et al. 2023

Financial Mathematics and Fintech

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Cyclic lattices and ideal lattices were introduced by Micciancio (2002), Lyubashevsky and Micciancio (2006), respectively, which play an efficient role in Ajtai’s construction of a collision resistant Hash function (see Ajtai (1996), Ajtai and Dwork (1997)) and in Gentry’s construction of fully homomorphic encryption (see Gentry (2009)). Let $$R=Z[x]/\langle \phi (x)\rangle $$ R = Z [ x ] / ⟨ ϕ ( x ) ⟩ be a quotient ring of the integer coefficients polynomials ring, Lyubashevsky and Micciancio regarded an ideal lattice as the correspondence of an ideal of R, but they neither explain how to extend this definition to whole Euclidean space $$\mathbb {R}^n$$ R n , nor exhibit the relationship of cyclic lattices and ideal lattices. In this chapter, we regard the cyclic lattices and ideal lattices as the correspondences of finitely generated R-modules, so that we may show that ideal lattices are actually a special subclass of cyclic lattices, namely, cyclic integer lattices. In fact, there is a one to one correspondence between cyclic lattices in $$\mathbb {R}^n$$ R n and finitely generated R-modules (see Theorem 4). On the other hand, since R is a Noether ring, each ideal of R is a finitely generated R-module, so it is natural and reasonable to regard ideal lattices as a special subclass of cyclic lattices (see Corollary 7). It is worth noting that we use a more general rotation matrix here, so our definition and results on cyclic lattices and ideal lattices are more general forms. As an application, we provide a cyclic lattice with an explicit and countable upper bound for the smoothing parameter (see Theorem 5). It is an open problem that is the shortest vector problem on cyclic lattice NP-hard (see Micciancio (2002)). Our results may be viewed as a substantial progress in this direction.

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Section: Ideal Matricesmentioning

confidence: 99%

Cyclic Lattices, Ideal Lattices, and Bounds for the Smoothing Parameter

Zhiyong,

Fengxia,

Yunfan

et al. 2023

Financial Mathematics and Fintech

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“…H f is a more general form of the classical circulant matrix (see [8]) and r-circulant matrix (see [9] and [10]). In fact, if ( )…”

Section: Ideal Matricesmentioning

confidence: 99%

Cyclic Lattices, Ideal Lattices and Bounds for the Smoothing Parameter

Zheng¹,

Liu²,

Yunfan³

et al. 2022

JIS

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In this article, we introduce the discrete subgroup in  n as preliminaries first. Then we provide some theories of cyclic lattices and ideal lattices. By regarding the cyclic lattices and ideal lattices as the correspondences of finitely generated R-modules, we prove our main theorem, i.e. the correspondence between cyclic lattices in  n and finitely generated R-modules is one-to-one. Finally, we give an explicit and countable upper bound for the smoothing parameter of cyclic lattices.

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“…Name of sequence Papers second order↓ second order↓ Fibonacci, Lucas [41,42,43,39,44,45,46,47,48,38,49 Also linear summing formulas of the generalized m-step Fibonacci sequences are required for the computation of various norms of circulant matrices circulant matrices with the generalized m-step Fibonacci sequences. We present some works on summing formulas of the numbers in the following Table 4.…”

Section: Table 3 Papers On the Normsmentioning

confidence: 99%