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The multiple polynomial quadratic sieve [PDF]
Mathematics of Computation, 1987 A modification, due to Peter Montgomery, of Pomerance’s Quadratic Sieve for factoring large integers is discussed along with its implementation. Using it, allows factorization with over an order of magnitude less sieving than the basic algorithm. It enables one to factor numbers in the 60-digit range in about a day, using a large minicomputer.
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Planar orthogonal polynomials as Type II multiple orthogonal polynomials [PDF]
Journal of Physics A: Mathematical and Theoretical, 2019 We show that the planar orthogonal polynomials with $l$ logarithmic singularities in the potential are the multiple orthogonal polynomials (Hermite-Pad polynomials) of Type II with $l$ measures. We also find the ratio between the determinant of the moment matrix corresponding to the multiple orthogonal polynomials and the determinant of the moment ...
Seung-Yeop Lee, Meng Yang
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Transactions on Cryptographic Hardware and Embedded Systems, 2020 In this paper, we present an instruction set coprocessor architecture for lattice-based cryptography and implement the module lattice-based post-quantum key encapsulation mechanism (KEM) Saber as a case study.
Sujoy Sinha Roy, Andrea Basso
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Polynomial degree reduction in the L2-norm on a symmetric interval for the canonical basis
Results in Applied Mathematics, 2021 In this paper, we develop a direct formula for determining the coefficients in the canonical basis of the best polynomial of degree M that approximates a polynomial of degree N>Mon a symmetric interval for the L2-norm.
Habib Ben Abdallah +2 more
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Multiple big q-Jacobi polynomials
Bulletin of Mathematical Sciences, 2020 Here, we investigate type II multiple big [Formula: see text]-Jacobi orthogonal polynomials. We provide their explicit formulae in terms of basic hypergeometric series, raising and lowering operators, Rodrigues formulae, third-order [Formula: see text]-difference equation, and we obtain recurrence relations.
Fethi Bouzeffour, Mubariz Garayev
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Ehrhart polynomial and multiplicity Tutte polynomial
, 2011 6 pages, 1 ...
D'Adderio, Michele, Moci, Luca
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Negacyclic Polynomial Multiplication
, 2022 In this article I’ll cover three techniques to compute special types of polynomial products that show up in lattice cryptography and fully homomorphic encryption. Namely, the negacyclic polynomial product, which is the product of two polynomials in the quotient ring $\mathbb{Z}[x] / (x^N + 1)$.
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Karatsuba-Ofman Multiplier with Integrated Modular Reduction for GF(2m)
Advances in Electrical and Computer Engineering, 2013 In this paper a novel GF(2m) multiplier based on Karatsuba-Ofman Algorithm is presented. A binary field multiplication in polynomial basis is typically viewed as a two steps process, a polynomial multiplication followed by a modular reduction step ...
CUEVAS-FARFAN, E. +6 more
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Simple Multivariate Polynomial Multiplication
Journal of Symbolic Computation, 1994 Let \(u= u(x_1, \dots, x_m)\) and \(v= v(x_1, \dots, x_m)\) be \(m\)-variate polynomials over a field \(F\) and let \(N= c^m\), where \(c\) is chosen so that \(c\geq 2\deg_{x_i} (u)+ 1,2 \deg_{x_i} (v)+ 1\) \((1\leq i\leq m)\). It is known that the product \(uv\) can be computed in time \(O(N \log N\log \log N)\).
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Asymptotics for multiple Meixner polynomials
Journal of Mathematical Analysis and Applications, 2014 The n-root asymptotic behavior of multiple Meixner polynomials is studied. A method based on an algebraic function formulation in connection with some available techniques from logarithmic potential theory has been developed. It represents an alternative to the use of Riemann-Hilbert techniques and the steepest descent method for oscillatory RH ...
Aptekarev, A. I. +1 more
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