scholarly journals Hitting Sets for Low-Degree Polynomials with Optimal Density

2014 ◽  
Author(s):  
Venkatesan Guruswami ◽  
Chaoping Xing
Keyword(s):  
Optimal Density ◽  
Hitting Sets ◽  
Low Degree ◽  
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 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.

Guest Column

2021 ◽  
Vol 51 (4) ◽  
pp. 48-72
Author(s):  
Mark Bun ◽  
Justin Thaler
Keyword(s):  
Boolean Function ◽  
Computer Science ◽  
Theoretical Computer Science ◽  
Theoretical Computer ◽  
Low Degree ◽  
Low Degree Polynomials

The approximate degree of a Boolean function f captures how well f can be approximated pointwise by low-degree polynomials. This article surveys what we know about approximate degree and illustrates some of its applications in theoretical computer science.

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