Studying the Bounds on Required Samples Numbers for Solving the General Approximate Common Divisors Problem

Yu, Xiaowei; Wang, Yuntao; Xu, Chungen; Takagi, Tsuyoshi

doi:10.1109/icisce.2018.00117

Cited by 3 publications

(7 citation statements)

References 9 publications

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“…Both ( 15) and ( 16) are all shaped ∨, so they are called OL-∨ algorithm. This kind of algorithm can be referred to [ [30], [31], [41], [42]].…”

Section: Orthogonal Lattice (Ol) Based Approachmentioning

confidence: 99%

“…To make the equation u t = 0 true, Ding et al [30] and Yang et al [41] gave the bounds of the short vectors of the lattice L 2 (α), they are shown in ( 28) and ( 29) respectively,…”

Section: (I) Whenmentioning

confidence: 99%

“…The original papers [5,16] presented a few possible lattice attacks on this problem, including orthogonal lattices (OL), simultaneous diophantine approximation (SDA) and multivariate polynomial equations (MP). Further cryptanalytic work was done by [23][24][25]30,31,36,37,[41][42][43][44]. This paper surveys and compares the known lattice algorithms for the ACD problem.…”

Section: Introductionmentioning

confidence: 99%

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Alogrithms for the ACD Problem and Cryptanalysis of ACD-based FHE schemes, Revisited

Yin-xia¹,

Pan²,

Wang³

2023

Preprint

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The security of several full homomorphic encryption (FHE) schemes depends on the hardness of the approximate common divisor (ACD) problem. The analysis of attack and defense against the system is one of the frontiers of cryptography research. In this paper, the performance of existing algorithms, including orthogonal lattice, simultaneous diophantine approximation, multivariate polynomial and sample pre-processing are reviewed and analyzed for solving the ACD problem. Orthogonal lattice (OL) algorithms are divided into two categories (OL-$\land$ and OL-$\vee$) for the first time. And an improved algorithm of OL-$\vee$ is presented to solve the GACD problem. This new algorithm works well in polynomial time if the parameter satisﬁes certain conditions. Compared with Ding and Tao's OL algorithm, the lattice reduction algorithm is used only once, and when the error vector $\mathbf{r}$ is recovered in Ding et al.'s OL algorithm, the possible difference between the restored and the true value of $p$ is given. It is helpful to expand the scope of OL attacks.

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“…Both ( 15) and ( 16) are all shaped ∨, so they are called OL-∨ algorithm. This kind of algorithm can be referred to [ [30], [31], [41], [42]].…”

Section: Orthogonal Lattice (Ol) Based Approachmentioning

confidence: 99%

“…To make the equation u t = 0 true, Ding et al [30] and Yang et al [41] gave the bounds of the short vectors of the lattice L 2 (α), they are shown in ( 28) and ( 29) respectively,…”

Section: (I) Whenmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Alogrithms for the ACD Problem and Cryptanalysis of ACD-based FHE schemes, Revisited

Yin-xia¹,

Pan²,

Wang³

2023

Preprint

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show abstract

“…The original papers [1,2] presented a few possible lattice attacks on the GACD problem, including the orthogonal lattices (OL) method, simultaneous diophantine approximation (SDA) method, and multivariate polynomial equations (MP) method. During the past decade, several related improvements and cryptanalytic works were proposed [6][7][8][9][10][11][12][13][14]. Detailed comparisons of these methods are summarized in Table 1.…”

Section: Introductionmentioning

confidence: 99%

“…In this sequel, we mainly focus on this improved OL method. According to the shape of the basis of the working lattice L(α), this kind of OL method can be further divided into two sub-categories: OL-∧, with a lower triangular matrix as the working lattice basis [8,9], and OL-∨, with an upper triangular matrix as the working lattice basis [7,9,10]. • MP methods.…”

mentioning

confidence: 99%

AIOL: An Improved Orthogonal Lattice Algorithm for the General Approximate Common Divisor Problem

Ran,

Pan,

Wang

et al. 2023

Mathematics

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The security of several fully homomorphic encryption (FHE) schemes depends on the intractability assumption of the approximate common divisor (ACD) problem over integers. Subsequent efforts to solve the ACD problem as well as its variants were also developed during the past decade. In this paper, an improved orthogonal lattice (OL)-based algorithm, AIOL, is proposed to solve the general approximate common divisor (GACD) problem. The conditions for ensuring the feasibility of AIOL are also presented. Compared to the Ding–Tao OL algorithm, the well-known LLL reduction method is used only once in AIOL, and when the error vector r is recovered in AIOL, the possible difference between the restored and the true value of p is given. Experimental comparisons between the Ding-Tao algorithm and ours are also provided to validate our improvements.

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Revisiting the Orthogonal Lattice Algorithm in Solving General Approximate Common Divisor Problem

Wang

et al. 2022

IEICE Trans. Fundamentals

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Studying the Bounds on Required Samples Numbers for Solving the General Approximate Common Divisors Problem

Cited by 3 publications

References 9 publications

Alogrithms for the ACD Problem and Cryptanalysis of ACD-based FHE schemes, Revisited

Alogrithms for the ACD Problem and Cryptanalysis of ACD-based FHE schemes, Revisited

AIOL: An Improved Orthogonal Lattice Algorithm for the General Approximate Common Divisor Problem

Revisiting the Orthogonal Lattice Algorithm in Solving General Approximate Common Divisor Problem

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