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In Peikert et al.'s most recent work (STOC 2017) a direct reduction of worst-case lattice problems to decision R-LWE is achieved for $\alpha q \ge 2 \cdot \omega(1)$ (Theorem 6.2), where $\alpha q$ is the Gaussian standard deviation.

However, it is unclear to me how this parameter constraint translates to the spherical error case (Corollary 7.3). Does the hardness guarantee in this case require a potentially wider standard deviation or does it remain the same?

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