Monogenic Fields of Cryptographic Size

Abstract

For a monic irreducible $f\in \mathbb{Z}[x]$ of degree $n$ and an integer $c$, we study the palindromic transform $F(x)=x^{n}f(x+x^{-1}+c)$, which produces a polynomial of degree $2n$. We give a discriminant formula $\mathrm{disc}(F)=f(c+2)f(c-2)\mathrm{disc}(f{)}^2$, sufficient conditions for the irreducibility of $F$, and a criterion showing that $F$ is monogenic when $f$ is monogenic and $f(c+2)f(c-2)$ is squarefree, together with a matching non-monogenicity criterion. Iterating the transform yields monogenic number fields of degree $2^{k}n$ from a fixed base polynomial. As an explicit example, we construct a monogenic field of degree 512 from $x^4+x+1$.

arXiv comment17 pages