Table 1. Notations.

N	:	Dimension
$R_q=\frac{{\mathbb{Z}}_q[x]}{x^N-1}$	:	Polynomial ring.
$q,q_1,q_2$	:	Modulo values.
$\oplus$	:	XOR operation.
$\otimes$	:	Multiplication in a polynomial ring.
$x{\leftarrow }^{r}X$	:	$x$ is chosen uniformly random from distribution X.
$||x||{}_{\mathrm{\infty }}$	:	For $x\in R$, $||x||{}_{\mathrm{\infty }}=\mathrm{m}\mathrm{a}{\mathrm{x}}_{1\le i\le N}\{x_i\}-\mathrm{m}\mathrm{i}{\mathrm{n}}_{1\le i\le N}\{x_i\}$.
$||x||{}_2$	:	For $x\in R$, $||x||{}_2=({\sum }_{i=1}^N{(x_i-\overline{x})}^2{)}^{1/2}$, where $\overline{x}=\frac{1}{N}{\sum }_{i=1}^N{x}_i$.
$\kappa \in {\mathbb{Z}}_{\ge 0}$	:	The main security parameter.
$\mathrm{\perp }$	:	Error message.
$d\in {\mathbb{Z}}^{+}$	:	The parameter of polynomial spaces.
		For D-NTRU, $d=d_f=d_g$.
$\delta ,\varphi$	:	The failure parameters.
$x_p^{-1}$	:	The inverse of the polynomial x in mod p.
		Let $x=f$. Then, $f\otimes f_p^{-1}\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}p$.
$\mathrm{\neg }Z$	:	The complement of Z.
$L(d,d)$	:	Ternary polynomials. If $x{\leftarrow }^{r}L(d,d)$, then $d$ coefficient of $x$ is equal to $1$, $d$ coefficient of $x$ is equal to $-1$, and the others are equal to $0$.
Kdf	:	Key derivation function such that $R_{q_2}\to \{0,1{\}}^N$.
H	:	Hash function such that $R_{q_1}\to \{0,1{\}}^N$.