I am trying to understand the Kannan embedding technique. But I am confused about the formation of the B' and the finding of the short vector inside that basis. How does this basis matrix in the algorithm produce a lattice? Or more specifically why a uSVP instance consist of a vector(e^T) and an integer(M)?
Thanks.
The columns of matrix $B'$ generate a lattice $\Lambda'=\{\mathbf v'\in\mathbb Z^{n+1}\}$ where the first $n$ entries form a vector which differs from a vector of $\Lambda$ by some multiple of $\mathbf e$ and the last entry is the same multiple of $M$. We also need some guarantee that $\mathbf e$ is shorter than the shortest vector of $\Lambda$ by some amount and that $\mathbf {As}$ is the closest vector to $\mathbf b$.
$M$ is not necessarily an integer, but is some value such that e.g. $M=||\mathbf e||$ when we know $||\mathbf e||<\lambda_1/2$ where $\lambda_1$ is the length of the shortest vector of $\Lambda$. It follows that the vector $(\mathbf e, M)^T$ has length at most $\sqrt 2M<\lambda_1/\sqrt 2$. This is shorter than any vector of the form $(\mathbf x, 0)^T$ which corresponds to a vector $\mathbf x$ of $\Lambda$ and so is of length at least $\lambda_1$. It is shorter than any other vector of the form $(\mathbf y, M)^T$ by the definition of the closest vector. It is also short than any vector of the form $(\mathbf z, kM)^T$ for any $|k|\ge 2$ as such a vector has length at least $|k|M>\sqrt 2 M$ based on the final component. It follows that $(\mathbf e, M)$ is the unique shortest vector of $\Lambda'$.
