A survey on coefficients of cyclotomic polynomials

Sanna, Carlo

doi:10.1016/j.exmath.2022.03.002

Cited by 10 publications

(6 citation statements)

References 85 publications

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“…The study of the coefficients of the cyclotomic polynomials has a very long history, which goes back at least to Gauss. For a survey, see [20]. Let A(n) be the maximum of the absolute values of a n (0), .…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

RLWE and PLWE over cyclotomic fields are not equivalent

Scala¹,

Sanna²,

Signorini³

2022

Preprint

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We prove that the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems over the cyclotomic field Q(ζn) are not equivalent. Precisely, we show that reducing one problem to the other increases the noise by a factor that is more than polynomial in n. We do so by providing a lower bound, holding for infinitely many positive integers n, for the condition number of the Vandermonde matrix of the nth cyclotomic polynomial.

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Section: Proof Of Theorem 11mentioning

confidence: 99%

RLWE and PLWE over cyclotomic fields are not equivalent

Scala¹,

Sanna²,

Signorini³

2022

Preprint

Self Cite

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“…The coefficients of cyclotomic polynomials have been intensively studied; we refer to [20] for a recent survey. The following lemma lists some basic properties of cyclotomic polynomials.…”

Section: Cyclotomic Polynomialsmentioning

confidence: 99%

“…(2) Let G = Z n and let k be a divisor of n such that 5,6,7,8,9,12,13,14,15,16,17,20,22,24,26,28,31,32,33,34,35,36,39,40,41,42,43,46,47…”

mentioning

confidence: 99%

Intersective sets over abelian groups

Xu¹,

Yip²

2022

Preprint

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Given a finite abelian group G and a subset J ⊂ G with 0 ∈ J, let D G (J, N ) be the maximum size of A ⊂ G N such that the difference set A − A and J N have no non-trivial intersection. Recently, this extremal problem has been widely studied for different groups G and subsets J. In this paper, we generalize and improve the relevant results by Alon and by Hegedűs by building a bridge between this problem and cyclotomic polynomials. In particular, we construct infinitely many non-trivial families of G and J for which the upper bounds on D G (J, N ) obtained by them (via linear algebra method) can be improved exponentially. We also obtain a new upper bound D Fp ({0, 1}, N ) ≤ ( 1 2 + o( 1))(p − 1) N , which improves the previously best known result by Huang, Klurman and Pohoata. Our main tools are from algebra, number theory, and probability.

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“…Due to its importance in many branches of mathematics, there have been extensive investigation on its properties. Recently, Sanna in [20], write a concise survey and attempts to collect the main results regarding the coefficients of the cyclotomic polynomials and to provide all the relevant references to their proofs.…”

Section: Introductionmentioning

confidence: 99%