How to find the value of a vector modulo a basis in lattice-based cryptography In Gentry's paper on fully homomorphic encryption using ideal lattices, he finds the values of vectors modulo a certain basis. For instance:

$\psi \leftarrow \psi' \mod B$ 

Taken from page 69 of Gentry's thesis.

What does it mean to evaluate a vector modulo a basis?
Examining the definition provided by A. Sakzad, I now have the intuition for what this means which I'll explain through an example.

In the diagram below, the red dots are a lattice, and the two red arrows represent a basis, $B$, for this lattice. We have a green vector, $ψ'$ = [2, 2]. We might ask which point on the lattice $ψ'$ is closest to. If we are just looking at this in the grid reference frame, $ψ'$ is equidistant from [3, 1] and [3, 3]. But within the reference frame of basis $B$, which is shown by the red parallelograms, $ψ'$ is closest to (3, 3). The value of $B⋅⌈B^{−1}ψ′⌋$ is the point in the lattice to which $ψ'$ is closest in the reference frame of $B$. $ψ=ψ′(\mod B)=ψ′−B⋅⌈B^{−1}ψ′⌋$ is the vector from $B⋅⌈B^{−1}ψ′⌋$ to the original vector. It could be viewed as the error or remainder when $ψ′$ is approximated by a point on the lattice. In the diagram below, $ψ$ is the blue arrow.



Since the basis vectors are [3, 1] and [0, 2] the basis matrix is: 
$$
B = \left( \begin{array}{ccc}
3 &amp; 0\\
1 &amp; 2 \end{array} \right) $$
Doing the math we find that (3, 3) is indeed the value of $B⋅⌈B^{−1}ψ′⌋$:
$$
B⋅⌈B^{−1}ψ′⌋ 
=B⋅⌈1/6 \left( \begin{array}{ccc}
2 &amp; 0\\
-1 &amp; 3 \end{array} \right)
\left( \begin{array}{ccc}
2\\
2 \end{array} \right)⌋ $$
$$
= B⋅⌈\left( \begin{array}{ccc}
2/3\\
2/3 \end{array} \right)⌋
= \left( \begin{array}{ccc}
3 &amp; 0\\
1 &amp; 2 \end{array} \right)
\left( \begin{array}{ccc}
1\\
1\end{array} \right)
= \left( \begin{array} {ccc}
3\\
3\end{array} \right)
$$
Therefore the remainder of $ψ′$ modulo the basis is:
$$ ψ=ψ′(modB)=ψ′−B⋅⌈B^{−1}ψ′⌋ = \left( \begin{array} {ccc}
-1 \\
-1 \end{array} \right)
$$