Results 11 to 20 of about 12,236 (229)
Factoring with cyclotomic polynomials [PDF]
26th Annual Symposium on Foundations of Computer Science (sfcs 1985), 1985 This paper discusses some new integer factoring methods involving cyclotomic polynomials. There are several polynomials f ( X ) f(X) known to have the following property: given a multiple of f ( p ) f(p) , we can quickly split any composite number that has p ...
Bach, Eric, Shallit, Jeffrey
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Cyclotomic Completions of Polynomial Rings
Publications of the Research Institute for Mathematical Sciences, 2004 For a subset S ⊂ ℕ = \{1, 2, . . . \} and a commutative ring R with unit, let R[q]^S
Kazuo Habiro
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Coefficients of cyclotomic polynomials [PDF]
, 2009 Let $a(n, k)$ be the $k$-th coefficient of the $n$-th cyclotomic polynomial. Recently, Ji, Li and Moree \cite{JLM09} proved that for any integer $m\ge1$, $\{a(mn, k)| n, k\in\mathbb{N}\}=\mathbb{Z}$. In this paper, we improve this result and prove that for any integers $s>t\ge0$, $$\{a(ns+t, k)| n, k\in\mathbb{N}\}=\mathbb{Z}.$$
Pingzhi Yuan
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Cyclotomic Euler-Mahonian polynomials [PDF]
The cyclotomic Eulerian polynomials and the cyclotomic Mahonian polynomials have each been the subject of extensive studies in Combinatorics, with particular attention to their signed versions. In contrast, the joint study of cyclotomic Euler-Mahonian polynomials has received far less consideration.
Guo-Niu Han
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Resultants of Cyclotomic Polynomials [PDF]
Proceedings of the American Mathematical Society, 1995 The authors calculate the resultant \(\text{Res}_x (\Phi_a (x^b), \Phi_c (x^d))\) for positive integers \(a,b,c,d\), where \(\Phi_n (x)\) is the \(n\)-th cyclotomic polynomial of degree \(\varphi (n)\). After establishing the factorization \(\Phi_a (x^b) = \prod_{[m,b] = ab} \Phi_m(x)\), they apply a chain rule for resultants to reduce the problem to ...
Cheng, Charles Ching-an +2 more
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Honam Mathematical Journal, 2015 Summary: The \(n\)-th cyclotomic polynomial \({\Phi}_n(x)\) is irreducible over \(\mathbb{Q}\) and has integer coefficients. The degree of \({\Phi}_n(x)\) is \({\varphi}(n)\), where \({\varphi}(n)\) is the Euler Phi-function. In this paper, we define the semi-cyclotomic polynomial \(J_n(x)\).
Lee, Ki-Suk, Lee, Ji-Eun, Kim, Ji-Hye
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Mixing Additive and Multiplicative Masking for Probing Secure Polynomial Evaluation Methods
Transactions on Cryptographic Hardware and Embedded Systems, 2018 Masking is a sound countermeasure to protect implementations of block- cipher algorithms against Side Channel Analysis (SCA). Currently, the most efficient masking schemes use Lagrange’s Interpolation Theorem in order to represent any S- box by a ...
Axel Mathieu-Mahias, Michaël Quisquater
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Efficient Algorithm for Finding Roots of Error-Locator Polynomials
IEEE Access, 2021 A novel method for finding roots of polynomials over finite fields has been proposed. This method is based on the cyclotomic discrete Fourier transform algorithm. The improvement is achieved by using the normalized cyclic convolutions, which have a small
Sergei Valentinovich Fedorenko
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High-Precision Leveled Homomorphic Encryption for Rational Numbers
Mathematics, 2023 In most homomorphic encryption schemes based on RLWE, native plaintexts are represented as polynomials in a ring Zt[x]/xN+1, where t is a plaintext modulus and xN+1 is a cyclotomic polynomial with a degree power of two.
Long Nie, Shaowen Yao, Jing Liu
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On the k-error linear complexity of cyclotomic sequences [PDF]
, 2007 Exact values and bounds on the k-error linear complexity of p-periodic sequences which are constant on the cyclotomic classes are determined. This family of sequences includes sequences of discrete logarithms, Legendre sequences and Hall's sextic residue
Hassan, Aly +2 more
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