(Posting this again since the last one was an image and people told me to write it down).
Found this from this paper (Page 4; Section 4.1). Image version.
I wanted help in understand what $\theta_{i}$ and $x = ( x_{1},\dots,x_{k} ) _{\equiv}$ mean here.
Is $\theta$ a 'sequence' of zero's and then a one at the i'th position? If so, how does one calculate something like $\alpha_{i}^{\delta{i}} \ \theta_{i} = \mathbb{mod} \ \mathit{\Pi}$ where $\alpha_{i} \in \{1, 0\}^{t}$ and $\{0, 1\}^{t}$ is a set of t-bit numbers. (Page 2; Section 1)
The definitions of the notation:
Let $\mathit{\Pi} = \prod_{i=1}^{k}{p_{i}^{\delta_{i}}}$ be the $n$-bit product of the first $k$ primes with some small exponents $\delta_{i}$, and let $\Delta = \textrm{max}_{i}\delta_{i}$. We denote by $x = ( x_{1},\dots,x_{k} ) _{\equiv}$ the modular representation of $x \in \mathbb{Z}_{\mathit{\Pi}}$, i.e., $x_{i} = x \bmod \ p_{i}^{\delta_{i}}$. For $i = 1, \dots, k$, one then defines $\theta_{i} = (0, \dots, 1, \dots, 0)_{\equiv}$ where the "$1$" stands for the $i^{\text{th}}$ position. It is obvious to see that we always have: $$\forall x \in \mathbb{Z}_{\mathit{\Pi}}\qquad x = \sum_{i=1}^{k}x_{i} \theta_{i} \bmod{\mathit{\Pi}}$$ [some proof/reasoning here for the calculations below] $$\begin{align}\forall x \in \mathbb{Z}_{\mathit{\Pi}} \qquad \forall x \in \mathbb{Z}^{*}_{\mathit{\Pi}} &\Longleftrightarrow x_{i} \in \mathbb{Z}^{*}_{p_{i}^{\delta_{i}}} \\ &\Longleftrightarrow x_{i}^{\delta_{i}} \not\equiv 0 \pmod{ p_{i}^{\delta_{i}}} \\ &\Longleftrightarrow x_{i}^{\delta_{i}} \theta_{i} \not\equiv 0 \pmod{\mathit{\Pi}} \;\text{for $i = 1, \dots, k$} \end{align}$$