On the condition number of the Vandermonde matrix of the nth cyclotomic polynomial

Abstract

Abstract Recently, Blanco-Chacón proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number Cond(Vn ) of the Vandermonde matrix Vn associated to the nth cyclotomic polynomial. We prove some results on the singular values of Vn and, in particular, we determine Cond(Vn ) for n = 2 kpℓ , where k, ℓ ≥ 0 are integers and p is an odd prime number.