Improved Hardness of BDD and SVP Under Gap-(S)ETH

Our first result shows improved exponential hardness of $\mathrm{BDD}_{p,\alpha}$ for all sufficiently small values of $p$, including the important Euclidean case of $p=2$, assuming a variant of Gap-ETH (see the left plot in Figure 1). Indeed, it improves the smallest value of $\alpha$ for which exponential hardness of $\mathrm{BDD}_{2,\alpha}$ is known under a general-purpose complexity-theoretic assumption to $\alpha<0.98491$, showing such hardness for an explicit³³3Each of the quantities $\alpha_{\mathsf{kn}}$, $\alpha^{\ddagger}_{p}$, $\alpha^{\dagger}_{p,C}$, $\alpha_{p,C}^{*}$, $\alpha_{p}^{*}$, and $C_{p}$ described in this section is “explicit” in the sense that it is expressible via some (not necessarily closed-form) expression. These expressions are easily computed to high accuracy in practice as shown, e.g., in Figure 1. Additionally, we emphasize that these quantities are constants in that they do not depend on the rank $n$ of the lattice in the corresponding problem. constant less than the $\alpha=1$ threshold for the first time.⁴⁴4Using the ideas in this paper and [AS18], showing exponential hardness of $\mathrm{SVP}$ essentially corresponds to showing exponential hardness of $\mathrm{BDD}$ with $\alpha=1-\varepsilon$ for some constant $\varepsilon>0$. Additionally, using the $\mathrm{BDD}$-hardness framework in this paper, it would be relatively straightforward to show exponential hardness of $\mathrm{BDD}$ with $\alpha=1+\varepsilon$ for any constant $\varepsilon>0$. So, the $\alpha=1$ threshold is qualitatively quite natural, and trying to show hardness for an explicit constant $\alpha<1$ is a natural goal.

Our second result shows improved exponential hardness of $\mathrm{BDD}_{p,\alpha}$ in a different regime and under a somewhat weaker assumption.

In [AS18], Aggarwal and Stephens-Davidowitz showed SETH-hardness of $\mathrm{SVP}_{p,1}$ for all $p>p_{0}\approx 2.1397$. To partially overcome the “$p_{0}$ barrier,” they generalized their proof techniques to show Gap-ETH-hardness of $\mathrm{SVP}_{p,\gamma}$ for all $p>2$. The results in [BP20] adapted the former techniques of [AS18] to show SETH-hardness of $\mathrm{BDD}$, and similarly got stuck at the $p_{0}$ barrier in the sense that they could not prove hardness of $\mathrm{BDD}_{p,\alpha}$ with $p<p_{0}$ and $\alpha<1$ simultaneously. The proof of Theorem 1.2 can be thought of as adapting the latter, generalized techniques of [AS18] to $\mathrm{BDD}$, and analogously allows us to prove Gap-ETH-hardness of $\mathrm{BDD}_{p,\alpha}$ with $\alpha<1$ for all $p>2$.

Compared to related quantities, $\alpha^{\ddagger}_{p}$ is: at most “$\alpha_{p}^{*}$ with norm embeddings” for all $p\in[1,\infty)$; strictly less than $\alpha_{p}^{*}$ for all sufficiently small $p$; strictly less than $\alpha_{\mathsf{kn}}$ for all sufficiently large $p$; and strictly less than the minimum of $\alpha_{p}^{*}$ and $\alpha_{\mathsf{kn}}$ for intermediate values of $p$. That is, $\alpha^{\ddagger}_{p}$ improves on both $\alpha_{p}^{*}$ (even with norm embeddings) and $\alpha_{\mathsf{kn}}$ for intermediate values of $p$; again, see the left plot in Figure 1. (Recall that [BP20] shows exponential hardness of $\mathrm{BDD}_{p,\alpha}$ for $\alpha>\alpha_{p}^{*}$ assuming randomized ETH, and Theorem 1.1 above shows such hardness for $\alpha>\alpha_{\mathsf{kn}}$ assuming non-uniform Gap-ETH.) However, Theorem 1.2 relies on a somewhat stronger hardness assumption than the one used in [BP20], and a somewhat weaker hardness assumption than Theorem 1.1, so the prior and new results are formally incomparable.

Our third result shows $2^{n/C}$-hardness of $\mathrm{BDD}_{p,\alpha}$ for any $C>1$ and $\alpha>\alpha^{\dagger}_{p,C}$, where $\alpha^{\dagger}_{p,C}$ is an explicit constant. For all sufficiently small $p\geq 1$ and sufficiently large $C>1$, $\alpha^{\dagger}_{p,C}$ not only improves on the corresponding quantity $\alpha_{p,C}^{*}$ in [BP20], but also on $\alpha_{p}^{*}=\inf_{C>1}\alpha_{p,C}^{*}$. For example, we show that $\alpha^{\dagger}_{1.5,200}\approx 1.0247$ while [BP20] was only able to show $\alpha^{*}_{1.5,200}\approx 1.3624$ and $\alpha_{p}^{*}\approx 1.3554$ (see the right plot in Figure 1).

Theorems 1.3 and 1.1 both crucially rely on the recent breakthrough work of Vlăduţ [Vlă19] showing an explicit exponential lower bound on the $\ell_{2}$ lattice kissing number, i.e., on the maximum number of vectors $\boldsymbol{v}\in\mathcal{L}$ achieving $\lVert\boldsymbol{v}\rVert=\lambda_{1}(\mathcal{L})$. More specifically, [Vlă19] shows that there exists a lattice of every rank $n\in\mathbb{Z}^{+}$ with kissing number at least $2^{c_{\mathsf{kn}}n-o(n)}$, where $c_{\mathsf{kn}}\geq 0.02194$ (see Definition 3.5 and Theorem 3.6).

Vlăduţ’s specific lower bound on $c_{\mathsf{kn}}$ translates to the bounds on $\alpha_{\mathsf{kn}}$ and $\alpha^{\dagger}_{p,C}$ that we obtain. Additionally, our use of “non-uniform” rather than “randomized” Gap-(S)ETH in the preceding results arises from the fact that it is not clear whether Vlăduţ’s exponential kissing number lattices can be efficiently constructed uniformly (even using randomness); an affirmative answer would relax the assumptions accordingly. Indeed, our results are agnostic to Vlăduţ’s specific construction, and an improved lower bound on $c_{\mathsf{kn}}$ would immediately imply improved upper bounds on the values $\alpha_{\mathsf{kn}}$ and $\alpha^{\dagger}_{p,C}$. Additionally, in Section 3.4.1 we sketch an approach for removing the “gap” part of the assumption in Theorem 1.3, which would be a further relaxation.

Our final result shows the same strong runtime lower bounds for $\mathrm{SVP}_{p,\gamma}$ with some constant $\gamma>1$ under (randomized) Gap-SETH as [AS18] showed for $\mathrm{SVP}_{p,1}$ under (randomized) SETH. This answers a natural question left open by [ABGS21], which analogously showed the same runtime lower bounds for $\mathrm{CVP}_{p,\gamma}$ with some constant $\gamma>1$ under Gap-SETH as [BGS17] showed for the Closest Vector Problem ($\mathrm{CVP}$) under SETH.

The reduction used to prove Theorem 1.4 is itself a natural modification of the reduction used in [AS18] to prove SETH-hardness of exact $\mathrm{SVP}$, but its analysis is more nuanced. We emphasize that simply plugging an instance of $\mathrm{CVP}^{\prime}_{p,\gamma}$ with $\gamma>1$ rather than $\gamma=1$ into the reduction of [AS18] does not yield corresponding hardness of approximation for $\mathrm{SVP}_{p,\gamma^{\prime}}$ with $\gamma^{\prime}>1$; a modified reduction is necessary. Finally, we remark that the somewhat odd-looking $p\notin 2\mathbb{Z}$ requirement in Theorems 1.3 and 1.4 is an artifact of the “upstream” hardness results we employ for $\mathrm{CVP}^{\prime}_{p,\gamma}$; see Theorem 2.18.

As in nearly all prior work on the complexity of $\mathrm{BDD}$ and $\mathrm{SVP}$ (e.g., [Mic01, Kho05, LLM06, AS18, BP20]), a key component of our results is the construction of a family of “locally dense” lattices, which are specified by a lattice $\mathcal{L}^{\dagger}$ and corresponding target vector $\boldsymbol{t}^{\dagger}$. For our purposes, a locally dense lattice $\mathcal{L}^{\dagger}$ is one with few “short” vectors, many vectors “close” to $\boldsymbol{t}^{\dagger}$, but few vectors “too close” to $\boldsymbol{t}^{\dagger}$. (Other works such as [Mic14] define locally dense lattices in a closely related but different way, e.g., without the requirement of few “too close” vectors.)

For a discrete set $S$, which we will take to be a lattice or a subset of a lattice, define

Somewhat more formally, we define a locally dense lattice $\mathcal{L}^{\dagger}$, $\boldsymbol{t}^{\dagger}$ with relative distance $\alpha_{G}$ in the $\ell_{p}$ norm to be one for which

for some $\nu>1$. That is, $\mathcal{L}^{\dagger}$, $\boldsymbol{t}^{\dagger}$ is such that the number $G$ of “close vectors” (within distance $\alpha_{G}$ of $\boldsymbol{t}^{\dagger}$) is an exponential factor larger than the number of short vectors (of norm at most one). Similarly, we require there to be an exponential factor more close vectors than “too close” vectors, along with some other technical conditions. We defer discussing these issues until the main body of the paper, and for the remainder of the introduction focus on the constraint in Equation 1.

A crux in showing hardness of $\mathrm{BDD}_{p,\alpha}$ and $\mathrm{SVP}_{p,\gamma}$ is constructing good locally dense lattices, and their parameters govern the precise hardness results that we can obtain. A family of locally dense lattices with smaller relative distance $\alpha_{G}$ and larger $\nu$ leads to stronger hardness results. To obtain ETH-type hardness results, we simply need $\nu$ to be a constant greater than $1$, and then we can show $2^{\Omega(n)}$-hardness of $\mathrm{BDD}_{p,\alpha}$ for any constant $\alpha>\alpha_{G}$. For SETH-type hardness results, we get $2^{n/C}$-hardness of $\mathrm{BDD}_{p,\alpha}$ whenever our reduction to $\mathrm{BDD}$ has a multiplicative rank-increase factor of $C$. The value of $C$ depends on the gap factor $\nu$ in Equation 1, so to show such hardness for explicit $C>0$ we need an explicit lower bound on $\nu$. Our reductions also give a tradeoff between $C$ and $\alpha$, as shown in the right plot in Figure 1. The full situation is actually a bit more complicated when taking “too close” vectors into account, but we again defer discussing this for now. The situation for $\mathrm{SVP}$ is similar to the situation for $\mathrm{BDD}$.

An important technique in our work is randomized lattice sparsification, an efficient algorithm that essentially does the following. Given (a basis of) a lattice $\mathcal{L}$ and an index $q\in\mathbb{Z}^{+}$ as input, the algorithm randomly samples a sublattice $\mathcal{L}^{\prime}\subseteq\mathcal{L}$ such that for any fixed, finite set of lattice points $S\subseteq\mathcal{L}$ satisfying some mild conditions, $\left|{S\cap\mathcal{L}^{\prime}}\right|\approx\left|{S}\right|/q$ with probability near $1$. A variant of this algorithm, additionally given $\boldsymbol{t}\in\operatorname{span}(\mathcal{L})$ as input, randomly samples $\mathcal{L}^{\prime}\subseteq\mathcal{L}$ and $\boldsymbol{t}^{\prime}$ such that for any fixed, finite set of points $S\subseteq\mathcal{L}-\boldsymbol{t}$ satisfying some mild conditions, $\left|{S\cap(\mathcal{L}^{\prime}-\boldsymbol{t}^{\prime})}\right|\approx\left|{S}\right|/q$ with probability near $1$.

Intuitively, some mild caveats aside, sparsification says that a lattice with few short vectors (and few “too close” vectors) is just as good as a lattice with no short non-zero vectors (and no “too close” vectors), since the latter can be efficiently obtained from the former. Indeed, sparsifying with index $q\approx N_{p}^{o}(\mathcal{L},r,\boldsymbol{0})$ allows us to turn a lattice $\mathcal{L}$ and target $\boldsymbol{t}$ satisfying, say, $N_{p}(\mathcal{L},\alpha r,\boldsymbol{t})\geq 100\cdot N_{p}^{o}(\mathcal{L},r,\boldsymbol{0})$ into a lattice $\mathcal{L}^{\prime}$ and target $\boldsymbol{t}^{\prime}$ with $N_{p}(\mathcal{L}^{\prime},\alpha r,\boldsymbol{t}^{\prime})\geq 1$ and $N_{p}^{o}(\mathcal{L}^{\prime}\setminus\{{\boldsymbol{0}}\},r,\boldsymbol{0})=0$, so $\mathrm{dist}_{p}(\boldsymbol{t}^{\prime},\mathcal{L})\leq\alpha r$ and $\lambda^{(p)}_{1}(\mathcal{L}^{\prime})\geq r$. That is, the output $\mathcal{L}^{\prime}$, $\boldsymbol{t}^{\prime}$ satisfies the $\mathrm{BDD}$ promise $\mathrm{dist}_{p}(\boldsymbol{t}^{\prime},\mathcal{L})\leq\alpha\cdot\lambda^{(p)}_{1}(\mathcal{L}^{\prime})$. See Section 2.2 for a formal description of sparsification.

Define $\mathrm{CVP}^{\prime}_{p,\gamma}$ to be the following variant of the decision version of the $\gamma$-approximate Closest Vector Problem: given a basis $B$ of a rank-$n$ lattice $\mathcal{L}$ and a target vector $\boldsymbol{t}$ as input, decide whether there exists $\boldsymbol{x}\in\{{0,1}\}^{n}$ such that $\lVert B\boldsymbol{x}-\boldsymbol{t}\rVert_{p}\leq 1$ (i.e., the input is a YES instance), or whether $\mathrm{dist}_{p}(\boldsymbol{t},\mathcal{L})>\gamma$ (i.e., the input is a NO instance), under the promise that one of the two cases holds. In other words, $\mathrm{CVP}^{\prime}$ is the variant of $\mathrm{CVP}$ that asks whether there is a binary combination of basis vectors close to the target. Much is known about the (fine-grained) complexity of $\mathrm{CVP}^{\prime}$, which will be useful for us (see Theorems 2.17 and 2.18).

Our reductions from $\mathrm{CVP}^{\prime}$ to $\mathrm{BDD}$ and to $\mathrm{SVP}$ have the same basic form. Given a rank-$n^{\prime}$ instance $B^{\prime},\boldsymbol{t}^{\prime}$ of $\mathrm{CVP}^{\prime}_{p,\gamma}$ for some $\gamma>1$, we apply the following transformation with some scaling factors $s,\ell>0$ and some locally dense lattice $\mathcal{L}^{\dagger}=\mathcal{L}(B^{\dagger})$ of rank $n-n^{\prime}$ with target $\boldsymbol{t}^{\dagger}$ satisfying Equation 1:

Essentially the same transformation appears in both [AS18] and [BP20], and similar ideas appear in a number of works before that. Our work differs in its constructions of locally dense lattice gadgets $(\mathcal{L}^{\dagger}$, $\boldsymbol{t}^{\dagger})$, its more general reductions, and its improved analysis.

Here we give a rough analysis of the transformation using two observations. First, we observe that appending $I_{n^{\prime}}$ to $B^{\prime}$ allows us to upper bound the number of short lattice vectors in $\mathcal{L}(B)$ by

for any $r^{\prime}>0$. Second, suppose that $B^{\prime}$, $\boldsymbol{t}^{\prime}$ is a YES instance of $\mathrm{CVP}^{\prime}$. Then there exists $\boldsymbol{x}\in\{{0,1}\}^{n^{\prime}}$ such that $\lVert B^{\prime}\boldsymbol{x}-\boldsymbol{t}^{\prime}\rVert\leq 1$, and hence for each $\boldsymbol{y}\in\mathbb{Z}^{n-n^{\prime}}$ with $\lVert B^{\dagger}\boldsymbol{y}-\boldsymbol{t}^{\dagger}\rVert_{p}\leq\alpha_{G}$ we get that $\lVert B(\boldsymbol{x},\boldsymbol{y})-\boldsymbol{t}\rVert_{p}\leq r$, where $r:=(s^{p}+n^{\prime}/2^{p}+(\alpha_{G}\cdot\ell)^{p})^{1/p}$. So,

To transform a YES instance of $\mathrm{CVP}^{\prime}_{p,\gamma}$ to a valid instance of $\mathrm{BDD}_{p,\alpha}$ for some $\alpha>0$, it essentially suffices to set the parameters $r,s,\ell$ and use suitable $B^{\dagger}$, $\boldsymbol{t}^{\dagger}$ so that, say,

Indeed, if Equation 5 holds, then by Equations 3 and 4, $N_{p}(\mathcal{L}(B),r,\boldsymbol{t})\geq 100\cdot N_{p}^{o}(\mathcal{L}(B),r/\alpha,\boldsymbol{0})$. We can then sparsify $\mathcal{L}(B)$ to obtain a lattice with no non-zero vectors of norm less than $r/\alpha$, and at least one vector within distance $r$ of $\boldsymbol{t}$, as needed.

We recall that by assumption, $N_{p}(\mathcal{L}(B^{\dagger}),\alpha_{G},\boldsymbol{t}^{\dagger})\geq\nu^{n-n^{\prime}}$, which is important for satisfying Equation 5 since $N_{p}^{o}(\mathbb{Z}^{n^{\prime}}\oplus\mathcal{L}(\ell B^{\dagger}),r/\alpha,\boldsymbol{0})$ will typically be exponentially large in $n^{\prime}$. We also need that if the input $\mathrm{CVP}^{\prime}$ instance is a NO instance, then there will be few vectors in $\mathcal{L}(B)$ that are close to $\boldsymbol{t}$, which depends on $\mathcal{L}^{\dagger}$ having few vectors “too close” to $\boldsymbol{t}^{\dagger}$, but again we defer discussing this. See Lemma 3.2 for a precise description of the useful properties of the transformation given in Equation 2.

When reducing to $\mathrm{SVP}_{p,\gamma^{\prime}}$ instead of $\mathrm{BDD}_{p,\alpha}$ we apply a further transformation to $B,\boldsymbol{t}$ before sparsifying. Namely, we apply Kannan’s embedding, which appends the vector $(\boldsymbol{t},u)$, for some value $u>0$, to $B$ to obtain a new basis:

The analysis in this case is a bit more subtle as well—we need to upper bound quantities of the form $N_{p}(\mathcal{L}(B),(r^{p}-(wu)^{p})^{1/p},w\cdot(\boldsymbol{t},u))$ not just for $w=0,1$ (corresponding to short and “too close” vectors in the $\mathrm{BDD}$ case, respectively) but for all integers $w\geq 2$ too—but the idea is similar. In fact, we use a result from [AS18] (presented in Theorem 4.2) that analyzes the combination of Kannan’s embedding and sparsification already, essentially reducing our task to bounding the quantities $N_{p}(\mathcal{L}(B),(r^{p}-(wu)^{p})^{1/p},w\cdot(\boldsymbol{t},u))$.

We conclude this summary of techniques by describing the specific locally dense lattices $\mathcal{L}^{\dagger},\boldsymbol{t}^{\dagger}$ that we use to instantiate Equation 2. We use two main families of locally dense lattices for our results.