A remark on flat ternary cyclotomic polynomials

Let \(\Phi_n(x)\) be the \(n\)-th cyclotomic polynomial. In this paper, for odd primes \(p\lt q \lt r\) with \(q\equiv \pm1\pmod p\) and \(8r\equiv \pm1\pmod {pq}\), we prove that the coefficients of \(\Phi_{pqr}(x)\) do not exceed \(1\) in modulus if and only if (i) \(p=3\), \(q\geq 19\) and \(q\equiv 1\pmod 3\) or (ii) \(p=7\), \(q\geq83\) and \(q\equiv -1\pmod 7\).

Cyclotomic numerical semigroup polynomials with at most two irreducible factors

A numerical semigroup S is cyclotomic if its semigroup polynomial $$\mathrm {P}_S$$ P S is a product of cyclotomic polynomials. The number of irreducible factors of $$\mathrm {P}_S$$ P S (with multiplicity) is the polynomial length $$\ell (S)$$ ℓ ( S ) of S. We show that a cyclotomic numerical semigroup is complete intersection if $$\ell (S)\le 2$$ ℓ ( S ) ≤ 2 . This establishes a particular case of a conjecture of Ciolan et al. (SIAM J Discrete Math 30(2):650–668, 2016) claiming that every cyclotomic numerical semigroup is complete intersection. In addition, we investigate the relation between $$\ell (S)$$ ℓ ( S ) and the embedding dimension of S.

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

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.