In modulo $q$ (i.e., in the world of remainders where all numbers have been divided by $q$), for each $a\in \{0,1,2,\cdots ,q-1\}$:

• Additive Inverse of $a$ is denoted as $a_{+}^{-1}$ which satisfies $a+a_{+}^{-1}\equiv 0modq$. For example, in modulo 11, $3_{+}^{-1}=8$, because $3+8\equiv 0mod11$.
• Multiplicative Inverse of $a$ is denoted as $a_{\ast }^{-1}$ which satisfies $a\cdot a_{\ast }^{-1}\equiv 1modq$. Such an inverse exists if and only if $\gcd <!--\; FUNCTION\; APPLICATION\; -->(a,q)=1$. For example, modulo 11, $3_{\ast }^{-1}=4$, because $3\cdot 4\equiv 1mod11$.