The Splitting Field of $Y^n-2$, Two-Variable NTT and Lattice-Based Cryptography

ePrint Report: The Splitting Field of $Y^n-2$, Two-Variable NTT and Lattice-Based Cryptography

Wenzhe Yang

The splitting field $F$ of the polynomial $Y^n-2$ is an extension over $\mathbb{Q}$ generated by $\zeta_n=\exp(2 \pi \sqrt{-1} /n)$ and $\sqrt[n]{2}$. When $n$ ($\geq 8$) is a power-of-two integer, the degree of $F$ over $\mathbb{Q}$ is $n^2/4$. In this paper, we lay the foundation for applying the Order-LWE in $\mathcal{R}=\mathbb{Z}[\zeta_n, \sqrt[n]{2}]$ to cryptographic uses. More specifically, We will compute the Galois group $\text{Gal}\left(F/\mathbb{Q} \right)$ and the …

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