A problem about Gaussian distribution in paper GPV08 These are contents from the paper Trapdoors for Hard Lattices
and New Cryptographic Constructions(GPV08). I do not know the reason about the last sentence. Why these two distributions D_{\Lambda, s, c}(x) and D_{\Lambda, s, c^{'}}(x) are identical?
The Gaussian function $\rho(x)=\exp(-\pi \| x \|^2)$ satisfies the property that if $x,y$ are orthogonal vectors, then $\rho(x+y)=\rho(x) \cdot \rho(y)$. This follows directly from the definition and the Pythagorean theorem. So, we can cancel out the contribution due to $c-c’$, which is orthogonal to every element in $\Lambda$, from both the numerator and the denominator.