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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?

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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.

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  • $\begingroup$ Thank you for your answer. I get the point. $\endgroup$
    – Alex Ideal
    Mar 9 '20 at 13:43

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