RLWE and PLWE over cyclotomic fields are not equivalent

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