scholarly journals Small Sample Spaces Cannot Fool Low Degree Polynomials

2008 ◽  
pp. 266-275 ◽  
Author(s):  
Noga Alon ◽  
Ido Ben-Eliezer ◽  
Michael Krivelevich
Keyword(s):  
Small Sample ◽  
Low Degree ◽  
Low Degree Polynomials

Testing Low-Degree Polynomials over Prime Fields

2004 ◽  
Author(s):  
C.S. Jutla ◽  
A.C. Patthak ◽  
A. Rudra ◽  
D. Zuckerman
Keyword(s):  
Low Degree ◽  
Prime Fields ◽  
Low Degree Polynomials

scholarly journals Irreducible skew polynomials over domains

2021 ◽  
Vol 29 (3) ◽  
pp. 75-89
Author(s):  
C. Brown ◽  
S. Pumplün
Keyword(s):  
Polynomial Ring ◽  
Skew Polynomial Ring ◽  
Weyl Algebras ◽  
Skew Polynomials ◽  
Low Degree ◽  
Skew Polynomial ◽  
Low Degree Polynomials ◽  
Injective Endomorphism

Abstract Let S be a domain and R = S[t; σ, δ] a skew polynomial ring, where σ is an injective endomorphism of S and δ a left σ -derivation. We give criteria for skew polynomials f ∈ R of degree less or equal to four to be irreducible. We apply them to low degree polynomials in quantized Weyl algebras and the quantum planes. We also consider f(t) = tm − a ∈ R.

scholarly journals Infiltrating CD16+ Are Associated with a Reduction in Peripheral CD14+CD16++ Monocytes and Severe Forms of Lupus Nephritis

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Anabel Barrera García ◽  
José A. Gómez-Puerta ◽  
Luis F. Arias ◽  
Catalina Burbano ◽  
Mauricio Restrepo ◽  
...  
Keyword(s):  
Lupus Nephritis ◽  
Small Sample ◽  
Class Iii ◽  
Recruitment Process ◽  
Diffuse Infiltration ◽  
Low Degree ◽  
The Mean ◽  
Immunohistochemistry Stain ◽  
High Degree ◽  
Number Of Cells

Our aim was to characterize glomerular monocytes (Mo) infiltration and to correlate them with peripheral circulating Mo subsets and severity of lupus nephritis (LN). Methods. We evaluated 48 LN biopsy samples from a referral hospital. Recognition of Mo cells was done using microscopic view and immunohistochemistry stain with CD14 and CD16. Based on the number of cells, we classified LN samples as low degree of diffuse infiltration (<5 cells) and high degree of diffuse infiltration (≥5 cells). Immunophenotyping of peripheral Mo subsets was done using flow cytometry. Results. Mean age was 34.0±11.7 years and the mean SLEDAI was 17.5±6.9. The most common SLE manifestations were proteinuria (91%) and hypocomplementemia (75%). Severe LN was found in 70% of patients (Class III, 27%; Class IV, 43%). Severe LN patients and patients with higher grade of CD16+ infiltration had lower levels of nonclassical (CD14+CD16++) Mo in peripheral blood. Conclusions. Our results might suggest that those patients with more severe forms of LN had a higher grade of CD14+CD16+ infiltration and lower peripheral levels of nonclassical (CD14+CD16++) Mo and might reflect a recruitment process in renal tissues. However, given the small sample, our results must be interpreted carefully.

scholarly journals Revisiting Multivariate Ring Learning with Errors and Its Applications on Lattice-Based Cryptography

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 858
Author(s):  
Alberto Pedrouzo-Ulloa ◽  
Juan Ramón Troncoso-Pastoriza ◽  
Nicolas Gama ◽  
Mariya Georgieva ◽  
Fernando Pérez-González
Keyword(s):  
Cyclotomic Polynomials ◽  
Time Efficiency ◽  
Space And Time ◽  
Low Degree ◽  
Learning With Errors ◽  
Lattice Based Cryptography ◽  
Low Degree Polynomials ◽  
Learning With Errors Problem ◽  
Recent Attack

The “Multivariate Ring Learning with Errors” problem was presented as a generalization of Ring Learning with Errors (RLWE), introducing efficiency improvements with respect to the RLWE counterpart thanks to its multivariate structure. Nevertheless, the recent attack presented by Bootland, Castryck and Vercauteren has some important consequences on the security of the multivariate RLWE problem with “non-coprime” cyclotomics; this attack transforms instances of m-RLWE with power-of-two cyclotomic polynomials of degree n=∏ini into a set of RLWE samples with dimension maxi{ni}. This is especially devastating for low-degree cyclotomics (e.g., Φ4(x)=1+x2). In this work, we revisit the security of multivariate RLWE and propose new alternative instantiations of the problem that avoid the attack while still preserving the advantages of the multivariate structure, especially when using low-degree polynomials. Additionally, we show how to parameterize these instances in a secure and practical way, therefore enabling constructions and strategies based on m-RLWE that bring notable space and time efficiency improvements over current RLWE-based constructions.

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