# GGH Encryption Scheme - Example

Example

Let be a lattice with the basis and its inverse

$B= \begin{pmatrix} 7 & 0 \\ 0 & 3 \\ \end{pmatrix}$ and $B^{-1}= \begin{pmatrix} \frac{1}{7} & 0 \\ 0 & \frac{1}{3} \\ \end{pmatrix}$

With

$U = \begin{pmatrix} 2 & 3 \\ 3 &5\\ \end{pmatrix}$ and
$U^{-1} = \begin{pmatrix} 5 & -3 \\ -3 &2\\ \end{pmatrix}$

this gives

$B' = U B = \begin{pmatrix} 14 & 9 \\ 21 & 15 \\ \end{pmatrix}$

Let the message be and the error vector . Then the ciphertext is

To decrypt one must compute

This is rounded to and the message is recovered with

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