# 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 $$D_{\Lambda, s, c^\prime}(x)$$ 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.