Gaussian Difference Dynamics: Breaking the $2^{n/2}$ Barrier for the Shortest Vector Problem and Discrete Gaussian Sampling
We give classical randomized algorithms in a word RAM model that break the $2^{n/2}$ barrier for both discrete Gaussian sampling (DGS) and the Shortest Vector Problem (SVP). For every fixed $α>1$, shifted DGS on arbitrary lattices at width $σ\geαη_{1/2}(L)$ takes $$ (3/2)^{n/2+o(n)}\text{ time} \qquad\text{and}\qquad (4/3)^{n/2+o(n)}\text{ space}. $$ These bounds match the Becker--Ducas--Gama--Laarhoven (BDGL) heuristic sieving exponents. At widths at least $3η_{1/2}(L)/2$, the same total cost suffices for an entire batch of $\lfloor(4/3)^{n/2}\rfloor$ samples jointly close to independent target Gaussians. For SVP on arbitrary lattices, our algorithm returns a shortest nonzero vector with high probability in $$ 2^{0.4130n+o(n)}\text{ time} \qquad\text{and}\qquad 2^{0.3138n+o(n)}\text{ space}. $$ As a corollary, we obtain centered DGS at every width with the same exponents, answering a question of Stephens-Davidowitz (SODA 2016). We introduce Gaussian difference dynamics (GDD), a framework for tracking expected counts and pairwise clustering of vectors through repeated differencing. A closed recursion for the first two counting moments controls these quantities relative to a Gaussian reference, without requiring independence within intermediate lists. Accurate moment bounds let us extract a Gaussian sample by weighted selection, while the weaker bounds needed to establish a shortest vector representation allow us to retain more differences. We perform these transitions on thin spherical shells, where the Gaussian rejection weights can be renormalized at subexponential cost. For batch sampling, we combine this geometry with the double rejection technique of Doulgerakis and Laarhoven to control the joint distribution of outputs while reusing vectors.