Topic
Divisor
About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.
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24 Aug 2020TL;DR: The peculiarity of the construction of n is used to build an improved lattice-based attack in cases where n is composite with an odd divisor, and the estimated complexity is reduced from 2 to 2.
Abstract: In 2019, Gu Chunsheng introduced Integer-RLWE, a variant of RLWE devoid of some of its efficiency flaws. Most notably, he proposes a setting where n can be an arbitrary positive integer, contrarily to the typical construction \(n = 2^k\). In this paper, we analyze the new problem and implement the classical meet-in-the-middle and lattice-based attacks. We then use the peculiarity of the construction of n to build an improved lattice-based attack in cases where n is composite with an odd divisor. For example, for parameters \(n = 2000\) and \(q = 2^{33}\), we reduce the estimated complexity of the attack from \(2^{288}\) to \(2^{164}\). We also present reproducible experiments confirming our theoretical results.
1 citations
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TL;DR: In this paper, the relative duality isomorphism of a mixed-characterized complete discrete valuation ring with a projective morphism between its special fibers was studied and proved to be commutative with Frobenius.
Abstract: Let $\V$ be a mixed characteristic complete discrete valuation ring, let $\X$ and $\Y$ be two smooth formal $\V$-schemes, let $f_0$ : $X \to Y$ be a projective morphism between their special fibers, let $T$ be a divisor of $Y$ such that $T_X := f_0 ^{-1} (T) $ is a divisor of $X$ and let $\M \in D ^\mathrm{b}_\mathrm{coh} (\D ^\dag_{\X} (\hdag T_X)_{\Q})$. We construct the relative duality isomorphism $ f_{0T +} \circ \DD_{\X, T_X} (\M) \riso \DD_{\Y, T} \circ f_{0T +} (\M)$. This generalizes the known case when there exists a lifting $f : \X \to \Y$ of $f_{0}$. Moreover, when $f_0$ is a closed immersion, we prove that this isomorphism commutes with Frobenius.
1 citations
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TL;DR: In this article, the Smarandache-Coman divisors of order k of a composite integer n with m prime factors are defined, a notion that seems to have promising applications, at first glance at least in the study of absolute and relative Fermat pseudoprimes, Carmichael numbers and Poulet numbers.
Abstract: We will define in this paper the Smarandache-Coman divisors of order k of a composite integer n with m prime factors, a notion that seems to have promising applications, at a first glance at least in the study of absolute and relative Fermat pseudoprimes, Carmichael numbers and Poulet numbers.
1 citations
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01 Feb 2013
TL;DR: In this paper, a disclosed multiple-modulus divider has a divisor loader, a multiplemodulus dividing circuit, and a modulus controller, which can operate in either a close-loop state or an open-loop states.
Abstract: Disclosed are multiple-modulus dividers and related control methods. A disclosed multiple-modulus divider has a divisor loader, a multiple-modulus dividing circuit, and a modulus controller. When a load signal indicates the start of a divisor period, the divisor loader downloads a divisor. The multiple-modulus dividing circuit has several divider cells connected in cascade, providing an output frequency according to the divisor and an input frequency. The divider cells output modulus output signals, respectively. Each divider cell operates in either a close-loop state or an open-loop state. Based on the divisor, the modulus controller selects and controls one of the divider cells, assuring it to stay in the open-loop state at the end of a divisor period. The load signal corresponds to one of the modulus output signals.
1 citations
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TL;DR: In this paper, it was shown that there is an isogeny whose kernel is supported on the Eisenstein maximal ideals of the Hecke algebra acting on $J_0(65)$, and moreover the odd part of the kernel is generated by a cuspidal divisor of order $7.
Abstract: Let $J^{65}$ be the Jacobian of the Shimura curve attached to the indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $65$. We study the isogenies $J_0(65)\rightarrow J^{65}$ defined over $\mathbb{Q}$, whose existence was proved by Ribet. We prove that there is an isogeny whose kernel is supported on the Eisenstein maximal ideals of the Hecke algebra acting on $J_0(65)$, and moreover the odd part of the kernel is generated by a cuspidal divisor of order $7$, as is predicted by a conjecture of Ogg.
1 citations