# Recovering dilithium secret keys implies solving MSIS

I'm currently looking at different ways to break Dilithium and found this post: What is the effect of solving short integer solution problem in Dilithium or any other post quantum signature scheme?

However I can't seem to wrap my head around the answer given by Daniel S., he states that:

If one can solve MSIS, powerful attacks are possible: consider the key generation algorithm on page 4 of the specification:

Gen

01 $$A←R^{k×l}_q$$

02 $$(\mathbf s_1,\mathbf s_2)\leftarrow S_\eta^l \times S_\eta^k$$

03 $$\mathbf t := A\mathbf s_1 +\mathbf s_2$$

04 return(pk=($$A,\mathbf t$$),sk=($$A,\mathbf t,\mathbf s_1,\mathbf s_2$$))

We are given the values $$A$$ and $$\mathbf t$$ and wish to recover $$\mathbf s_1$$ and $$\mathbf s_2$$ for a key recovery attack (top of the attack outcome hierarchy).

Note that the matrix $$\begin{matrix}(A|I|-T)\end{matrix}$$ where $$I$$ is the $$k\times k$$ identity matrix and $$T$$ is the $$k\times k$$ diagonal matrix with entries taken from $$\mathbf t$$ has the short solution $$(\mathbf s_1^T,\mathbf s_2^T,1,\cdots 1)^T$$.

Since $$A|I|$$ and $$T$$ are of different dimensions, how can we subtract the two?

Furthermore, can anyone explain why $$(\mathbf s_1^T,\mathbf s_2^T,1,\cdots 1)^T$$ would be a small solution or give a simple example of how recovering dilithium secret keys implies solving MSIS?