Research Vignette: Generalization and Interpolation

by Daniel Hsu

The practice of deep learning has attracted new attention to several basic questions in statistical machine learning. One such question is how fitting machine learning models to relatively small “training” data sets can lead to accurate predictions on new data. In machine learning jargon, this is the question of generalization.

The conventional wisdom in machine learning offers the following about generalization:

  1. A model that is too simple will underfit the true patterns in the training data, and thus, it will predict poorly on new data.
  2. A model that is too complicated will overfit spurious patterns in the training data; such a model will also predict poorly on new data.

Consequently, one should choose a model that balances these concerns of underfitting and overfitting [30]. A textbook corollary is that a model that exactly fits — i.e., interpolates — the training data “is overfit to the training data and will typically generalize poorly” [17].

Recent machine learning practice appears to eschew this conventional wisdom in a dramatic fashion. For instance, a common starting point for training a neural network is to find a model that exactly fits the training data [27]. (Typically, the model is subsequently fine-tuned using different criteria, but the starting point is already nontrivial.) While this may be difficult or impossible with small neural networks, it becomes substantially easier after making the network very large. It is remarkable that this practice is compatible with generalization, despite the concerns about overfitting.

This apparent contradiction of conventional wisdom is not new. In the last century, machine learning practitioners were already aware of the efficacy (and advantages) of using “oversized” neural networks where the number of model parameters exceeds the number of training data [9, 20]. Similar observations were made about large ensembles of decision trees: it was found that increasing the number of decision trees in an ensemble could improve accuracy, even beyond the point at which the ensemble exactly fit the training data [28]. These empirical observations motivated the development of a new theory that greatly sharpens the existing body of knowledge in statistical machine learning [1, 28], and this theory has been recently extended to deep neural networks [2, 15, 26].

Figure 1: Leo Breiman’s list of important questions regarding neural networks (reproduced from [9])

However, the new theory does not fully explain recent observations about interpolating neural networks. First, in more recent experiments, neural networks are trained to interpolate noisy data [34]. By noisy data, we mean data in which the prediction targets (labels) may be erroneous. In fact, experiments have been conducted in which noise is deliberately injected in the data by assigning random labels to subsets of the training data. Neural networks that interpolate these data were nevertheless found to have nontrivial accuracy on new data. Second, similar phenomena have been observed in the context of predicting real numerical targets. (The aforementioned theoretical and empirical studies mostly focused on classification targets.) When predicting real numbers, it is almost always a given that training data will have noisy labels, so the thought of interpolating training data seems even more preposterous. But again, in recent experimental studies, interpolating (or nearly interpolating) models were found to have nontrivially accurate predictions on new data [4, 7].

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Research Vignette: The Many Dimensions of High-Dimensional Expanders

by Prahladh Harsha

An emerging theme in several disciplines of science over the last few decades is the study of local-to-global phenomena. Let me elucidate with a few examples. In biology, one tries to understand the global properties of an organism by studying local interactions at a cellular level. The Internet graph is impossible to predict or control at a global level; what one can at best do is understand its behavior by making changes at a very local level. Moving to examples closer to theoretical computer science, the pioneering works in computation theory due to Turing and others define the computation of any global function as one that can be broken down into a sequence of local computations. The seminal work on NP-completeness due to Cook, Levin, and Karp demonstrates that any verification task can be in fact reduced to a conjunction of verifying very local objects.

Given these examples, an intriguing question that arises in multiple disciplines is to identify which objects support such local-to-global phenomena. This question as stated is an ill-formed one, but one can attempt to make it well-defined in a variety of contexts. Let me illustrate by giving three specific instances.

Mixing time in graphs: Which graphs have the property of having a global mixing property that can be inferred by understanding the mixing properties of several related smaller graphs?

Local testing and decoding (in coding theory): Which error-correcting codes have the property of being able to be tested or decoded by looking at the code word or received word very locally (i.e., at very few locations)?

Topologically expanding graphs: Which graphs or hypergraphs have the property that their global topological property can be inferred by their local topological behavior?

These questions, seemingly disparate and arising in very different contexts and communities (in particular, approximate sampling, coding theory, and topology), surprisingly all led to the investigation of a similar expanding object: the high-dimensional expander. The Simons Institute 2019 summer cluster on Error-Correcting Codes and High-Dimensional Expansion brought together researchers from these diverse communities to study high-dimensional expanders and their potential applications to mathematics and theoretical computer science.

What is a high-dimensional expander?

High-dimensional expanders (HDXs) are a high-dimensional analogue of expander graphs. An expander graph, loosely speaking, is an extremely well-connected graph. Analytically, this is best captured via the second-largest eigenvalue (in absolute value) of the normalized adjacency matrix of the graph. More precisely, a graph G is said to be a λ-expander (for λ ∈ [0,1]) if the second-largest eigenvalue (in absolute value) of the normalized adjacency matrix AG of G is bounded above by λ. The smaller the λ, the better the expander. An equivalent definition is in terms of how well the random walk on the vertices induced by the graph mixes: G is a λ-expander if the spectral norm of the difference of the normalized adjacency matrix AG and the normalized all-ones matrix J (the normalized adjacency matrix of the complete graph) is bounded above by λ (i.e., \[\left\| A_G – \frac1n J \right\| \leq \lambda,\]

where n is the number of vertices of the graph G).

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Lattice Blog Reduction – Part II: Slide Reduction

This is the second entry in a series of posts about lattice block reduction. See here for the first part. In this post I will assume you have read the first one, so if you haven’t, continue at your own risk. (I suggest reading at least the first part for context, notations and disclaimers.)

Last time we focused on BKZ which applies SVP reduction to successive projected subblocks. In this post we consider slide reduction, which allows for a much cleaner and nicer analysis. But before we can do that, we need a little more background.

A New Tool: Dual SVP Reduction

As you hopefully know, duality is a very useful concept in lattice theory (and in mathematics more generally). It allows to pair up lattices, which are related in a well defined way. Similarly, we can pair up the different bases of two dual lattices to obtain dual bases. I’ll skip the definition of these two concepts since we will not need them. It is sufficient to know that we can compute the dual basis from the primal basis efficiently. One very cool feature of dual bases is that the last vector in the dual basis has a length that is inverse to the length of the last GSO vector of the primal basis. In math: if \({\mathbf{B}}\) and \({\mathbf{D}}\) are dual bases, then \(\| {\mathbf{b}}_n^* \| = \| {\mathbf{d}}_n \|^{-1}\). (If you want to see why this is true at least for full rank lattices, use the fact that in this case \({\mathbf{D}} = {\mathbf{B}}^{-T}\), and the QR-factorization.) It follows that if \({\mathbf{d}}_n\) happens to be the shortest vector in the dual lattice, then \({\mathbf{b}}_n^*\) is as long as possible, since the existence of any basis \(\bar{{\mathbf{B}}}\) where \(\|\bar{{\mathbf{b}}}_n^*\| > \|{\mathbf{b}}_n^*\|\) would imply that there exists a dual basis \(\bar {{\mathbf{D}}}\) such that \(\| \bar{{\mathbf{d}}}_n\| < \| {\mathbf{d}}_n\| \). By analogy to SVP reduction, we call a basis, where \(\| {\mathbf{b}}^*_n \|\) is maximized, dual SVP reduced (DSVP reduced). This gives us a new tool to control the size of the GSO vectors: we can apply an SVP algorithm to the dual basis of a projected subblock \({\mathbf{B}}_{[i,j]}\). This will yield a shortest vector in the dual of this projected sublattice. Then we can compute a dual basis, which contains this shortest vector in the last position and finally compute a new primal basis for this projected subblock, which now locally maximizes \(\|{\mathbf{b}}_j^* \|\). As we did for primal SVP reduction in the last post, we will assume access to an algorithm that, given a basis \({\mathbf{B}}\) and indices \(i,j\), will return a basis such that \({\mathbf{B}}_{[i,j]}\) is DSVP reduced and the rest of the basis is unchanged. We will call such an algorithm a DSVP oracle. It may sound like this should be somewhat less efficient than SVP reduction, since we have to switch between the dual and the primal bases (which, when done explicitly, requires matrix inversion), but this is not actually the case. In fact, one can implement a DSVP reduction entirely without having to explicitly compute (any part of) the dual basis as shown in [GN08,MW16].

Effect of a call to the DSVP oracle. GSO log norms of the input in black, of the output in blue. Note that the sum of the GSO log norms is a constant, so increasing the length of the last vector, decreases the (average of the) remaining vectors.

I hope this figure provides some intuition that such an oracle can be useful. Now let us quantify how much this DSVP oracle helps us. Recall that in the primal SVP reduction we used Minkowski’s theorem to bound the length of the first vector. Since we are now applying the SVP algorithm to the dual, it should come as no surprise that we will use Minkowski’s theorem on the dual lattice, which tells us that \[\lambda_1(\widehat{\Lambda}) \leq \sqrt{\gamma_n} \det(\widehat{\Lambda})^{1/n} = \sqrt{\gamma_n} \det(\Lambda)^{-1/n}\] where \(\widehat{\Lambda}\) is the dual lattice, i.e. the lattice generated by the dual basis. Furthermore, by exploiting above fact that for basis \({\mathbf{B}}\) and its dual \({\mathbf{D}}\) we have \(\| {\mathbf{b}}_n^* \| = \|{\mathbf{d}}_n \|^{-1}\), this shows that if \({\mathbf{B}}\) is DSVP reduced, i.e. \({\mathbf{d}}_n\) is a shortest vector in the dual lattice, then \[\| {\mathbf{b}}_n^* \| = \|{\mathbf{d}}_n \|^{-1} = \lambda_1(\widehat{\Lambda})^{-1} \geq \frac{\det(\Lambda)^{1/n}}{\sqrt{\gamma_n}}.\] So after we’ve applied the DSVP oracle to a projected block \({\mathbf{B}}_{[i-k+1,i]}\), we have \[\|{\mathbf{b}}^*_i \| \geq \frac{\left(\prod_{j = i-k+1}^{i} \|{\mathbf{b}}_j^* \| \right)^{1/k}}{\sqrt{\gamma_{k}}}.\]

Slide Reduction

Now we have all the tools we need to describe slide reduction [GN08]. One of the major hurdles to apply an LLL-style running time analysis to BKZ seems to be that the projected subblocks considered in that algorithm are maximally overlapping. So slide reduction takes a different route: it applies primal and dual SVP reduction to minimally overlapping subblocks, which allows to still prove nice bounds on the output quality (in fact, even better than BKZ), but also on the running time via a generalization of the LLL analysis. More specifically, let \({\mathbf{B}}\) be the given lattice basis of an \(n\)-dimensional lattice and \(k\) be the blocksize. We require that \(k\) divides \(n\). (We’ll come back to that restriction later.) Instead of applying our given SVP oracle to successive projected subblocks, we apply it to disjoint projected subblocks, i.e. to the blocks \({\mathbf{B}}_{[1,k]}\), \({\mathbf{B}}_{[k+1,2k]}\), etc. So we locally minimize the GSO vectors \({\mathbf{b}}^*_{ik + 1}\) for \(i \in \{0,\cdots,n/k – 1\}\). (Technically, we iterate this step with a subsequent LLL reduction until there is no more change, which is important for the runtime analysis, but let’s ignore this for now). So now we have a basis where these disjoint projected subblocks are SVP-reduced. In the next step we shift the blocks by 1 and apply our DSVP oracle to them. (Note that the last block now extends beyond the basis, so we ignore this block.) This has the effect of locally maximizing the vectors \({\mathbf{b}}^*_{ik + 1}\) for \(i \in \{1,\cdots,n/k – 1\}\). This might seem counter-intuitive at first, but note that the optimization context for \({\mathbf{b}}^*_{ik + 1}\) changes between the SVP reduction and the DSVP reduction: \({\mathbf{b}}^*_{ik + 1}\) is first minimized with respect to the block \({\mathbf{B}}_{[ik+1,(i+1)k]}\) and then maximized with respect to the block \({\mathbf{B}}_{[(i-1)k+1,ik+1]}\). So one can view this as using the block \({\mathbf{B}}_{[ik+2, (i+1)k]}\) as a pivot to lower the ratio between the lengths of the GSO vectors \({\mathbf{b}}^*_{ik+1}\) and \({\mathbf{b}}^*_{(i+1)k+1}\). This view is reminiscent of the proof of Mordell’s inequality \(\gamma_n^{\frac{1}{n-1}} \leq \gamma_{n-1}^{\frac{1}{n-2}}\), which explains the title of the paper [GN08]. The idea of slide reduction is to simply iterate these two steps until there is no more change.

Slide reduction in one picture: apply the SVP oracle to the disjoint projected blocks in parallel, then shift the blocks by 1 and apply the DSVP oracle. Repeat.

Let’s dive into the analysis.

The Good

When the algorithm terminates, we are guaranteed that the following conditions hold simultaneously:

  1. The blocks \({\mathbf{B}}_{[ik+1, (i+1)k]}\) are SVP reduced for all \(i \in \{0,\cdots,n/k – 1\}\) (the primal conditions), which implies \[\|{\mathbf{b}}^*_{ik+1} \|^{k-1} \leq \gamma_k^{k/2} \prod_{j=ik+2}^{(i+1)k} \|{\mathbf{b}}^*_j \|\] (Note that we raised Minowski’s bound to the \(k\)-th power and canceled one the \(\|{\mathbf{b}}^*_{ik+1} \|\) on both sides.)

  2. The blocks \({\mathbf{B}}_{[ik+2, (i+1)k+1]}\) are DSVP reduced for all \(i \in \{0,\cdots,n/k – 2\}\) (the dual conditions), which implies \[\gamma_{k}^{k/2} \|{\mathbf{b}}^*_{(i+1)k+1} \|^{k-1} \geq \prod_{j = ik+2}^{(i+1)k} \|{\mathbf{b}}_j^* \|\]

(Technically, there is a constant slack factor \(>1\) involved, which can be set arbitrarly close to 1, but is important for running time. We’ll sweep under the rug for simplicity.)

Just by staring at the two inequalities, you will notice that they can easily be combined to yield: \[\|{\mathbf{b}}^*_{ik+1} \| \leq \gamma_k^{\frac{k}{k-1}} \|{\mathbf{b}}^*_{(i+1)k+1} \|\] for all \(i \in \{0,\dots,n/k-2\}\) and in particular \[\|{\mathbf{b}}^*_{1} \| \leq \gamma_k^{\frac{k}{k-1} (\frac{n}{k}-1)} \|{\mathbf{b}}^*_{n-k+1} \| = \gamma_k^{\frac{n-k}{k-1}} \|{\mathbf{b}}^*_{n-k+1} \|\] By a similar trick as last time we can assume that \(\lambda_1({\mathbf{B}}) \geq \|{\mathbf{b}}^*_{n-k+1} \|\), because the last block is SVP-reduced, which shows that slide reduction achieves an approximation factor \[\|{\mathbf{b}}^*_{1} \| \leq \gamma_k^{\frac{n-k}{k-1}} \lambda_1({\mathbf{B}}).\] Done! Yes, it is really that simple. With (very) little more work one can similarly show a bound on the Hermite factor \[\|{\mathbf{b}}^*_{1} \| \leq \gamma_k^{\frac{n-1}{2(k-1)}} \det({\mathbf{B}})^{\frac1n}.\] Simply reuse the bounds on the ratios of \(\| {\mathbf{b}}^*_1 \|\) and \(\| {\mathbf{b}}^*_{ik+1} \|\) in combination with Minkowski’s bound for each block. (You guessed it: Homework!) Note that both of them are better than what we were able to obtain for BKZ in our last blog post. And in contrast to BKZ one can easily bound the number of calls to the SVP oracle by a polynomial in \(n, k\) and the bit size of the original basis. The analysis is similar to the one of LLL with a modified potential function: we let \(P({\mathbf{B}}) = \prod_{i=0}^{n/k-2} \det({\mathbf{B}}_{[1,ik]})^2\). If the basis \({\mathbf{B}}\) consists of integer coefficients only, this potential is also integral. Furthermore, one can show that if an iteration of slide reduction modifies the basis, it will decrease this potential by at least a constant factor (by using the slack factor we brushed over). This shows that while the basis is modified, the potential decreases exponentially, which results in a polynomial number of calls to the (D)SVP oracle.

The Bad

We just sketched a complete and elegant analysis of the entire algorithm and it checks all the boxes: best known approximation factor, best known Hermite factor, a polynomial number of calls to its (D)SVP oracle. So what could possibly be bad about it? Remember that we required that the blocksize \(k\) divides the dimension \(n\). It seems like it should be easy to get rid of this restriction, for example one could artificially increase the dimension of the lattice to assure that the blocksize divides it. Unfortunately, this and similar approaches will degrade the bound on the output quality – there will be a rounding-up operator in the exponent [LW13]. For small \(k\) this might not be too much of an issue, but as \(k\) grows, this results in a significant performance hit. Luckily, a recent work [ALNS19] shows that one can avoid this degradation by combining slide reduction with yet another block reduction algorithm: SDBKZ, which will be the topic of the next post.

The Ugly

Slide reduction is beautiful and there is little one can find ugly about it in theory. Unfortunately, experimental studies so far concluded that this algorithm is significantly inferior to BKZ, which (at least to me) is puzzling. This is often attributed to the fact that BKZ uses maximally overlapping blocks, which seems to allow it to obtain stronger reduction notions (even though we cannot prove it). So, one could wonder if there is an algorithm that uses maximally overlapping blocks (and is thus hopefully competetive in practice), but allows for a clean analysis. It turns out that the topic of the next post (SDBKZ) is such an algorithm.

  • Gama, Nguyen. Finding short lattice vectors within Mordell’s inequality. STOC 2008

  • Li, Wei. Slide reduction, successive minima and several applications. Bulletin of the Australian Mathematical Society 2013

  • Micciancio, Walter. Practical, predictable lattice basis reduction. EUROCRYPT 2016

  • Aggarwal, Li, Nguyen, Stephens-Davidowitz. Slide Reduction, Revisited—Filling the Gaps in SVP Approximation. https://arxiv.org/abs/1908.03724

Research Vignette: Geometry of Polynomials

by Shayan Oveis Gharan

Classical theorems in theoretical computer science typically heavily exploit combinatorial or probabilistic tools. Later on, linear algebraic and geometric tools proved to be fundamental elements in numerous algorithmic or hardness results. In this research vignette, I will explain a new tool known as the “polynomial paradigm,” which can be seen as a bridge between linear algebraic and probabilistic methods; and I will discuss some unexpected applications in computer science. 

Given a polynomial \(p \in R\lbrack z_{1},\ldots,z_{n}\rbrack\), we can think of it in three ways: (i) coefficients of p, (ii) zeros of p, or (iii) a function that maps n-dimensional complex vectors to complex numbers. Having these three different representations, one can study what happens to one representation when we change another one, or what happens when we apply a mathematical operation such as differentiation, integration, or specialization. For many years, mathematicians studied these questions for many classes of polynomials.

Over the last two decades, a new theme that we call the “polynomial paradigm” emerged. It was observed that if the roots of the (multivariate) polynomial \(p\) have a “nice” structure in the n-dimensional complex plane, then one can study the interaction of zeros, coefficients, and function values rigorously. In several pioneering works, Borcea, Brändén, Choe, Liggett, Oxley, Sokal, and Wagner studied the class of real stable polynomials, which can be seen as the right multivariate generalization of real-rooted polynomials. A polynomial \(p\) is real stable if it has no root in the upper half of the n-dimensional complex plane. They observed that this class is closed under many operations, such as specialization, differentiation, inversion, etc.

Over the last decade, the polynomial paradigm proved to be a fruitful approach in many areas of science and mathematics. Several applications of this paradigm in theoretical computer science, combinatorics, and linear algebra were discovered, such as improved approximation algorithms for the traveling salesman problem, a purely algebraic proof of the van der Waerden conjecture, and a resolution of the notorious Kadison-Singer conjecture. The general recipe in almost all these applications is as follows: Given a discrete phenomenon, encode it in a complex multivariate polynomial with a nice zero-free region, such as a real stable polynomial. Then study this polynomial via the interplay of coefficients, zeros, and function values. Finally, translate your findings to the domain of the original problem.

In Spring 2019, with Nikhil Srivastava, we organized a semester-long program on Geometry of Polynomials at the Simons Institute, where we invited researchers across several areas of science, mathematics, and computer science to explore this rapidly evolving area. A large group tried to better understand these polynomials and their properties, while many others tried to explore further applications. In the rest of this research vignette, I will explain the journey toward the discovery of a new class of multivariate polynomials that we call completely log-concave polynomials. This discovery and their application motivated several researchers within the program to start collaborating to better understand this class of polynomials and its properties.

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Fine-grained hardness of lattice problems: Open questions

1 Introduction

1.1 Lattices and lattice-based cryptography

Lattices are classically-studied geometric objects that in the past few decades have found a multitude of applications in computer science. The most important application area is lattice-based cryptography, the design of cryptosystems whose security is based on the apparent intractability of computational problems on lattices, even for quantum computers. Indeed, lattice-based cryptography has revolutionized the field because of its apparent quantum resistance and its other attractive security, functionality, and efficiency properties.

Intuitively, a lattice is a regular ordering of points in some (typically high-dimensional) space. More precisely, a lattice \( {{\cal{L}}}\) of rank \( {n}\) is the set of all integer linear combinations of some linearly independent vectors \( {\mathbf{b}_1, \ldots, \mathbf{b}_n}\), which are called a basis of \( {{\cal{L}}}\). We will be primarily interested in analyzing the running times of lattice algorithms as functions of the lattice’s rank \( {n}\).

1.2. Computational lattice problems

The two most important computational problems on lattices are the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP). SVP asks, given a basis of a lattice \( {{\cal{L}}}\) as input, to find a shortest non-zero vector in \( {{\cal{L}}}\). CVP, which can be viewed as an inhomogeneous version of SVP, asks, given a basis of a lattice \( {{\cal{L}}}\) and a target point \( {\mathbf{t}}\) as input, to find a closest vector in \( {{\cal{L}}}\) to \( {\mathbf{t}}\).

Algorithms for solving SVP form the core of the best known attacks on lattice-based cryptography both in theory and in practice. Accordingly, it is critical to understand the precise complexity of SVP as well as possible. The best provably correct algorithms for both SVP and CVP run in \( {2^{n + o(n)}}\)-time [ADRS15, ADS15, AS18a]. The best heuristic algorithms for SVP run in \( {2^{cn + o(n)}}\)-time for \( {c = 0.292}\) classically [BDGL16] and \( {c = 0.265}\) using quantum speedups [Laa15] (see also [KMPR19]), and most real-world lattice-based cryptosystems assume that these algorithms are close to optimal. Indeed, many of these cryptosystems assume what Bos et al. [B+16] call a “paranoid” worst-case estimate of \( {c = 0.2075}\) (based on the kissing number and assuming that sieving algorithms are optimal) as the fastest hypothetical running time for SVP algorithms when choosing parameters. (See also Albrecht et al. [A+18], which surveys the security assumptions made in a wide range of lattice-based cryptosystems.) Accordingly, the difference in being able to solve SVP in \( {2^{0.2075n}}\) versus \( {2^{n/20}}\) versus \( {2^{\sqrt{n}}}\) time may mean the difference between lattice-based cryptosystems being secure, insecure with current parameters, or effectively broken in practice.

There is a rank-preserving reduction from SVP to CVP [GMSS99], so any algorithm for CVP immediately gives an essentially equally fast algorithm for SVP. In other words, CVP is at least as hard as SVP (and probably a bit harder). Indeed, historically, almost all lower bounds for SVP are proven via reduction from CVP (and nearly all algorithmic progress on CVP uses ideas originally developed for SVP).

1.3. Fine-grained hardness

The field of fine-grained complexity works to give strong, quantitative lower bounds on computational problems assuming standard complexity-theoretic assumptions. Proving such a (conditional) lower bound for an \( {{\mathsf{NP}}}\)-hard problem generally works by (1) assuming a stronger hardness assumption than \( {{\mathsf{P}} \neq {\mathsf{NP}}}\) about the complexity of \( {k}\)-SAT (such as ETH or SETH, defined below), and (2) giving a highly efficient reduction from \( {k}\)-SAT to the problem. The most important hardness assumptions for giving lower bounds on \( {{\mathsf{NP}}}\)-hard problems are the Exponential Time Hypothesis (ETH) and the Strong Exponential Time Hypothesis (SETH) of Impagliazzo and Paturi [IP01]. ETH asserts that there is no \( {2^{o(n)}}\)-time algorithm for \( {3}\)-SAT, and SETH asserts that for every \( {\epsilon > 0}\) there exists \( {k \in {\mathbb Z}^+}\) such that there is no \( {2^{(1 – \epsilon)n}}\)-time algorithm for \( {k}\)-SAT, where \( {n}\) denotes the number of variables in the SAT instance.

Here by “highly efficient” reductions we mean linear ones, i.e., reductions that map a \( {3}\)-SAT or \( {k}\)-SAT formula on \( {n}\) variables to an SVP or CVP instance of rank \( {C n + o(n)}\) for some absolute constant \( {C > 0}\). Indeed, by giving a reduction from \( {3}\)-SAT (respectively, \( {k}\)-SAT for any \( {k \in {\mathbb Z}^+}\)) instances on \( {n}\) variables to SVP or CVP instances of rank \( {C n + o(n)}\), we can conclude that there is no \( {2^{o(n)}}\)-time (resp., \( {2^{(1-\epsilon)n/C}}\)-time for any \( {\epsilon > 0}\)) algorithm for the corresponding problem assuming ETH (resp., SETH). Note that the smaller the value of \( {C}\) for which one can show such a reduction, the stronger the conclusion. In particular, a reduction mapping \( {k}\)-SAT instances on \( {n}\) variables to SVP or CVP instances of rank \( {n + o(n)}\) would imply an essentially tight lower bound on the corresponding problem assuming SETH — as mentioned above, the best provably correct algorithms for both SVP and CVP run in time \( {2^{n + o(n)}}\).

1.4. Fine-grained hardness of CVP (and SVP)

It is relatively easy to show that CVP is “ETH-hard,” i.e., to show that a \( {2^{o(n)}}\)-time algorithm for CVP would imply a \( {2^{o(n)}}\)-time algorithm for \( {3}\)-SAT instances with \( {n}\) variables. This would falsify ETH. (It’s a nice exercise to show that the Subset Sum problem on a set of size \( {n}\) reduces to CVP on a lattice of rank \( {n}\), which implies the result.)

With some work, Divesh Aggarwal and Noah extended this to SVP [AS18b]. In particular, we showed a reduction from CVP to SVP that only increases the rank of the lattice by some constant multiplicative factor. (Formally, the reduction only works with certain minor constraints on the CVP instance. The reduction originally relied on a geometric conjecture, which was open for decades. But, Serge Vlăduţ proved the conjecture [Vlă19] shortly after we published!)

So, unless ETH is false, there is no \( {2^{o(n)}}\)-time algorithm for CVP or SVP. But, for cryptographic applications, even, say, a \( {2^{n/20}}\)-time algorithm would be completely devastating. If such an algorithm were found, cryptographic schemes that we currently think are secure against absurdly powerful attackers straight out of science fiction (say, one with a computer the size of the sun running until the heat death of the universe) would turn out to be easily broken (e.g., in seconds on our laptops).

In [BGS17, ABGS20], we almost showed that CVP is “SETH-hard,” i.e., that a \( {2^{(1-\epsilon)n}}\)-time algorithm for CVP would imply such an algorithm for \( {k}\)-SAT for any constant \( {k}\). This would falsify SETH. So, we almost showed that the [ADS15] algorithm is optimal. The “almost” is because our proof works with \( {\ell_p}\) norms, that is, we show hardness for the version of CVP in which the distance from the target to a lattice vector is defined in terms of the \( {\ell_p}\) norm,

\( \displaystyle \|\mathbf{x}\|_p := (|x_1|^p + \cdots + |x_d|^p)^{1/p} \; . \)

We call the corresponding problem \( {{\mathrm{CVP}}_p}\). In fact, our proof works for all \( {\ell_p}\) norms except when \( {p}\) is an even integer. (To see why this might happen, notice \( {\|\mathbf{x}\|_p^p}\) is a polynomial in the \( {x_i}\) if and only if \( {p}\) is an even integer. In fact, there’s some sense in which “\( {\ell_2}\) is the easiest norm,” because for any \( {p}\), there is a linear map \( {A \in {\mathbb R}^{d \times m}}\) such that \( {m}\) is not too large and \( {\|\mathbf{x}\|_2 \approx \|A \mathbf{x}\|_p}\).) Of course, we are most interested in the case \( {p= 2}\) (the only case for which the [ADS15] algorithm works), which is an even integer! Indeed, for all \( {p \neq 2}\), the fastest known algorithm for CVP is still Ravi Kannan’s \( {n^{O(n)}}\)-time algorithm from 1987 [Kan87]. (For SVP and for constant-factor approximate CVP, \(2^{O(n)}\)-time algorithms are known [DPV11].)

In fact, we showed that for \( {p = 2}\), no “natural” reduction can rule out a \( {2^{3n/4}}\)-time algorithm for CVP under SETH. A “natural” reduction is one with a fixed bijection between witnesses. In particular, any “natural” reduction from \( {3}\)-SAT to CVP must reduce to a lattice with rank at least roughly \( {4n/3}\). So, new ideas will be needed to prove stronger hardness of CVP in the \( {\ell_2}\) norm.

2. Open problems

We now discuss some of the problems that we left open in [BGS17, ABGS20]. For simplicity, we ask for specific results (e.g., “prove that problem \( {A}\) is \( {T}\)-hard under hypothesis \( {B}\)“), but of course any similar results would be very interesting (e.g., “\( {A}\) is \( {T’}\)-hard under hypothesis \( {B’}\)“).

2.1. Hardness in the \(\ell_2\) norm

The most obvious question that we left open is, of course, to prove similar \( {2^n}\)-time hardness results for \( {{\mathrm{CVP}}_2}\) (and more generally for \( {{\mathrm{CVP}}_p}\) for even integers \( {p}\)).

Open problem 1. Show that there is no \( {2^{0.99 n}}\)-time algorithm for \( {{\mathrm{CVP}}_2}\) assuming SETH.

Remember that we showed that any proof of such a strong result would have to use an “unnatural” reduction. So, a fundamentally different approach is needed. One potentially promising direction would be to find a Cook reduction, as our limitations only apply to Karp reductions.

Alternatively, one might try for a different result that gets around this “natural” reduction limitations. E.g., even the following much weaker result would be very interesting.

Open problem 2. Show an efficient reduction from \( {3}\)-SAT on \( {n}\) variables to \( {{\mathrm{CVP}}_2}\) on a lattice of rank \( {\approx 10n}\).

Such a reduction to \( {{\mathrm{CVP}}_2}\) on a lattice of rank \( {Cn}\) for some large constant \( {C}\) is known by applying the Sparsification Lemma [IPZ01] to \( {3}\)-SAT, but showing such a reduction for any reasonably small \( {C}\) or even any explicit \( {C}\) using a different proof technique would be interesting.

Also, our limitations only apply to reductions that map satisfying assignments to exact closest vectors. So, one might try to get around our limitation by working directly with approximate versions of \( {3}\)-SAT and \( {{\mathrm{CVP}}_2}\). (In [ABGS20], we show such reductions from Gap-\( {k}\)-SAT to constant-factor approximate \( {{\mathrm{CVP}}_p}\) for all \( {p \notin 2{\mathbb Z}}\) as well as all \( {k \leq p}\). We also show reductions from Gap-\( {k}\)-Parity that achieve relatively large approximation factors.)

Open problem 3. Show an efficient reduction from Gap-\( {3}\)-SAT on \( {n}\) variables to approximate \( {{\mathrm{CVP}}_2}\) on a lattice of rank \( {n}\).

2.2. Hardness in \(\ell_p\) norms

Intuitively, one reason that we are able to prove such strong results for \( {\ell_p}\) norms for \( {p \neq 2}\) is because we can use lattices with large ambient dimension \( {d}\) but low rank \( {n}\). In other words, while our reductions produce lattices \( {{\cal{L}}}\) that live in some \( {n}\)-dimensional subspace of \( {\ell_p}\)-space, the ambient space itself has large dimension \( {d}\) relative to \( {n}\). Of course, any subspace of the \( {\ell_2}\) norm is an \( {\ell_2}\) subspace (i.e., every slice of a ball is a lower-dimensional ball), so in the \( {\ell_2}\) norm, one can assume without loss of generality that \( {d = n}\). In particular, if we were able to prove \( {2^n}\)-hardness for the \( {\ell_2}\) norm, then we would actually prove \( {2^d}\)-hardness for free. However, a potentially easier problem would be to improve the \( {2^n}\)-hardness of \( {{\mathrm{CVP}}_p}\) shown in [BGS17, ABGS20]  to \( {2^d}\)-hardness for some \( p \neq 2 \).

Open problem 4. Show that there is no \( {2^{0.99 d}}\)-time algorithm for \( {{\mathrm{CVP}}_p}\) (for some \( {p}\)) assuming SETH.

More generally, it would be very interesting to settle the fine-grained complexity of \( {{\mathrm{CVP}}_p}\) for some \( {p \neq 2}\) (either in terms of rank \( {n} \) or dimension \( {d} \)). This could take the form either of showing improved algorithms (currently the fastest algorithms for \( {{\mathrm{CVP}}_p}\) for general \( {p}\) run in \( {n^{O(n)}}\)-time [Kan87], and \( {2^{O(n)}}\)-time for a constant approximation factor [DPV11]), or showing super-\( {2^n}\) hardness, or both.

Open problem 5. Show matching upper bounds and lower bounds (under SETH) for \( {{\mathrm{CVP}}_p}\) for some \( {p}\) (possibly with a constant approximation factor).

The case where \( {p = \infty}\) is especially interesting. Indeed, because the kissing number in the \( {\ell_\infty}\) norm is \( {3^n-1}\), one might guess that the fastest algorithms for \( {{\mathrm{CVP}}_\infty}\) and \( {{\mathrm{SVP}}_\infty}\) actually run in time \( {3^{n + o(n)}}\) or perhaps \( {3^{d + o(d)}}\). (See [AM18], which essentially achieves this.) We therefore ask whether stronger lower bounds can be proven in this special case.

Open problem 6. Show that \( {{\mathrm{CVP}}_\infty}\) cannot be solved in time \( {3^{0.99n}}\) (under SETH).

2.3. Hardness closer to crypto

The most relevant problem to cryptography is approximate \( {{\mathrm{SVP}}_2}\) with an approximation factor that is polynomial in the rank \( {n}\). Our fastest algorithms to solve this problem work via a reduction to exact (or near exact) \( {{\mathrm{SVP}}_2}\) with some lower rank \( {n’ = \Theta(n)}\), so that even for these polynomial approximation factors, our fastest algorithms run in time \( {2^{\Omega(n)}}\) (where the hidden constant depends on the polynomial; see Michael’s post for more on this topic). And, hardness results for exact SVP rule out attacks on cryptography that use such reductions. We currently only know how to rule out \( {2^{o(n)}}\)-time algorithms for \( {{\mathrm{SVP}}_2}\) (under the Gap-ETH assumption). We ask whether we can do better. (In [AS18b], we proved the stronger result below for \( {\ell_p}\) norms for large enough \( {p \notin 2{\mathbb Z}}\).)

Open problem 7. Prove that there is no \( {2^{n/10}}\)-time algorithm for \( {{\mathrm{SVP}}_2}\) (under SETH).

Of course, we would ideally like to directly rule out faster algorithms for approximate \( {{\mathrm{SVP}}_2}\) with the approximation factors that are most directly relevant to cryptography. There are serious complexity-theoretic barriers to overcome to get all the way there (e.g., \( {{\mathrm{CVP}}_p}\) and \( {{\mathrm{SVP}}_p}\) are known to be in \( {{\mathsf{NP}}} \cap {{\mathsf{coNP}}}\) for large enough polynomial approximation factors. But, we can still hope to get as close as possible, by proving stronger hardness results for approximate \( {{\mathrm{CVP}}_p}\) and approximate \( {{\mathrm{SVP}}_p}\). Indeed, a beautiful sequence of works showed hardness for approximation factors up to \( {n^{c/\log \log n}}\) (so “nearly polynomial) [DKRS03, HR12], but these results are not fine grained.

The best fine-grained hardness of approximation results known rule out algorithms for small constant-factor approximations for \( {{\mathrm{CVP}}_p}\) with \( {p \notin 2{\mathbb Z}}\) in time \( {2^{0.99n}}\) for \( {{\mathrm{CVP}}_p}\) and \( {{\mathrm{SVP}}_p}\) for any \( {p}\) in time \( {2^{o(n)}}\). We ask whether we can do better.

Open problem 8. Prove that there is no \( {2^{0.99 n}}\)-time algorithm for \( {2}\)-approximate \( {{\mathrm{CVP}}_p}\) (under some form of Gap-SETH, see below).

Open problem 9. Prove that there is no \( {2^{o(n)}}\)-time algorithm for \( {\gamma}\)-approximate \( {{\mathrm{CVP}}_p}\) for superconstant \( {\gamma = \omega(1)}\) (under Gap-ETH).

2.4. Gap-SETH?

One issue that arose in our attempts to prove fine-grained hardness of approximation results is that we don’t even know the “right” complexity-theoretic assumption about approximate CSPs to use as a starting point. For fine-grained hardness of exact problems, ETH and SETH are very well established hypotheses, and they are in some sense “the weakest possible” assumptions of their form. E.g., it is easy to see that \( {k}\)-SAT is \( {2^{Cn}}\) hard if any \( {k}\)-CSP is. But, for hardness of approximation, the situation is less clear.

The analogue of ETH in the regime of hardness of approximation is the beautiful Gap-ETH assumption, which was defined independently by Irit Dinur [Din16] and Pasin Manurangsi and Prasad Raghavendra [MR17]. This assumption says that there exists some constant approximation factor \( {\delta \neq 1}\) such that \( {\delta}\)-Gap-\( {3}\)-SAT cannot be solved in time \( {2^{o(n)}}\). (Formally, both Dinur and Manurangsi and Raghavendra say that there is no \( {2^{o(n)}}\)-time algorithm that distinguishes a satisfiable formula from a formula for which no assignment satisfies more than a \( {(1-\epsilon)}\) fraction of the clauses, but we ignore this requirement of perfect completeness here.) It is easy to see that this hypothesis is equivalent to a similar hypothesis about any \( {3}\)-CSP (or, indeed, any \( {k}\)-CSP for any constant\( {k}\)).

However, to prove hardness of approximation with the finest of grains, we need some “gap” analogue of SETH, i.e., we would like to assume that for large enough \( {k}\), some Gap-\( {k}\)-CSP is hard to approximate up to some constant factor \( {\delta \neq 1}\) in better than \( {2^{0.99n}}\)-time. (Formally, we should add an additional variable \( {\epsilon > 0}\) and have such a hypothesis for every running time \( {2^{(1-\epsilon)n}}\), but we set \( {\epsilon = 0.01}\) here to keep things relatively simple.)

An issue arises here concerning the dependence of the approximation factor \( {\delta}\) on the arity \( {k}\). In particular, recall that \( {k}\)-SAT can be trivially approximated up to a factor of \( {1-2^{-k}}\) (since a random assignment satisfies a \( {1-2^{-k}}\) fraction of the clauses in expectation). So, if we define Gap-SETH in terms of Gap-\( {k}\)-SAT, then we must choose \( {\delta = \delta(k) \geq 1-2^{-k}}\) that converges to one as \(k\) increases. Manurangsi proposed such a version of Gap-SETH in his thesis [Man19, Conjecture 12.1], specifically that for every large enough constant \( {k}\) there exists a constant \( {\delta = \delta(k) \neq 1}\) such that Gap-\( {k}\)-SAT cannot be approximated up to a factor of \( {\delta}\) in time \( {2^{0.99n}}\). (Again, we are leaving out an additional variable, \( {\epsilon}\).)

If we rely on this version of Gap-SETH, then our current techniques seem to get stuck at proving hardness of approximation for, say, \( {\gamma}\)-approximate \( {{\mathrm{CVP}}_p}\) for some non-explicit constant \( {\gamma_p > 1}\) (and, if one works out the numbers, one can see immediately that \( {\gamma_p}\) must be really quite close to one). However, other Gap-\(k\)-CSPs are known to be (\(\mathsf{NP}\)-)hard to approximate up to much better approximation factors. E.g., for any \( {k}\), Gap-\(k\)-Parity is \( {{\mathsf{NP}}}\)-hard to approximate up to any constant approximation factor \( {1/2 < \delta \leq 1}\) [Hås01], and Gap-\( {k}\)-AND is \( {{\mathsf{NP}}}\)-hard to approximate for any constant approximation factor \( {\Omega(k/2^k) \leq \delta \leq 1}\) [Cha16]. Indeed, Gap-\( {k}\)-AND is a quite natural problem to consider in this context since there is a fine-grained, approximation-factor preserving reduction from any Gap-\( {k}\)-CSP to Gap-\( {k}\)-AND. This generality motivates understanding the precise complexity of Gap-\( {k}\)-AND.

Open problem 10. What is the fine-grained complexity of the \( {\delta}\)-Gap-\( {k}\)-AND problem in terms of \( {n}\), \( {k}\), and \( {\delta}\)? In particular, if

\( \displaystyle C_{k,\delta} := \inf \{ C > 0 \ : \ \text{there is a $2^{C_{k,\delta}}$-time algorithm for algorithm for $\delta$-Gap-$k$-AND}\}\)

then what is the behavior of \( {C_{k,\delta}}\) as \( {k \rightarrow \infty}\) (for various functions \( {\delta = \delta(k)}\) of \( {k}\))?

In particular, if one were to hypothesize sufficiently strong hardness of \( {\delta}\)-Gap-\( {k}\)-AND — i.e., to define an appropriate variant of Gap-SETH based on Gap-\( {k}\)-AND — then one might be able to use this hypothesis to prove very strong fine-grained hardness of approximation results. There is a fine-grained (but non-approximation preserving) reduction from Gap-\( {k}\)-AND to Gap-\( {k}\)-SAT, and so Manurangsi’s Gap-SETH is equivalent to the conjecture that there exists some non-explicit \( {\delta(k)}\) such that \( {\lim_{k \rightarrow \infty} C_{k,\delta} = 1}\).

  • [ABGS20] Aggarwal, Bennett, Golovnev, Stephens-Davidowitz. Fine-grained hardness of CVP(P)— Everything that we can prove (and nothing else)
  • [A+18] Albrecht, Curtis, Deo, Davidson, Player, Postlethwaite, Virdia, Wunderer. Estimate all the {LWE, NTRU} schemes! SCN, 2019.
  • [ADRS15] Aggarwal, Dadush, Regev, Stephens-Davidowitz. Solving the Shortest Vector Problem in \(2^n\) time via discrete Gaussian sampling. STOC, 2015.
  • [ADS15] Aggarwal, Dadush, Stephens-Davidowitz. Solving the Closest Vector Problem in \(2^n\) time–The discrete Gaussian strikes again! FOCS, 2015.
  • [AM18] Aggarwal, Mukhopadhyay. Faster algorithms for SVP and CVP in the \(\ell_\infty\) norm. ISAAC, 2018.
  • [AS18a] Aggarwal, Stephens-Davidowitz. Just take the average! An embarrassingly simple \(2^n\)-time algorithm for SVP (and CVP). SOSA, 2018.
  • [AS18b] Aggarwal, Stephens-Davidowitz. (Gap/S)ETH hardness of SVP. STOC, 2018.
  • [B+16] Bos, Costello, Ducas, Mironov, Naehrig, Nikolaenko, Raghunathan, Stebila. Frodo: Take off the ring! Practical, Quantum-Secure Key Exchange from LWE. CCS, 2016.
  • [BDGL16] Becker, Ducas, Gama, Laarhoven. New directions in nearest neighbor searching with applications to lattice sieving. SODA, 2016.
  • [BGS17] Bennett, Golovnev, Stephens-Davidowitz. On the quantitative hardness of CVP. FOCS, 2017.
  • [Cha16] Chan. Approximation resistance from pairwise-independent subgroups. J. ACM, 2016.
  • [Din16] Dinur. Mildly exponential reduction from gap 3SAT to polynomial-gap label-cover.
  • [DKRS03] Dinur, Kindler, Raz, Safra. Approximating CVP to within almost-polynomial factors is NP-hard. Combinatorica, 2003.
  • [DPV11] Dadush, Peikert, Vempala. Enumerative lattice algorithms in any norm via \(M\)-ellipsoid coverings. FOCS, 2011.
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  • [HR12] Haviv, Regev. Tensor-based hardness of the Shortest Vector Problem to within almost polynomial factors. TOC, 2012.
  • [IP01] Impagliazzo, Paturi. On the complexity of \(k\)-SAT. JCSS, 2001.
  • [IPZ01] Impagliazzo, Paturi, Zane. Which problems have strongly exponential complexity? JCSS, 2001.
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  • [KMPR19] Kirshanova, Mårtensson, Postlethwaite, Roy Moulik. Quantum algorithms for the approximate \(k\)-list problem and their application to lattice sieving. Asiacrypt, 2019.
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  • [MR17] Manurangsi, Raghavendra. A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs. ICALP, 17.
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Lattice Blog Reduction – Part I: BKZ

This is the first entry in a (planned) series of at least three, potentially four or five, posts about lattice block reduction. The purpose of this series is to give a high level introduction to the most popular algorithms and their analysis, with pointers to the literature for more details. The idea is to start with the obvious – the classic BKZ algorithm. In the next two posts we will look at two lesser known algorithm, which allow to highlight useful tools in lattice reduction. These three posts will focus on provable results. I have not decided how to proceed from there, but I could see the series being extended to topics involving heuristic analyses, practical considerations, and/or a survey of more exotic algorithms that have been considered in the literature.

Target Audience

I will assume that readers of this series are already familiar with basic concepts of lattices, e.g. bases, determinants, successive minima, Minkowski’s bound, Gram-Schmidt orthogonalization, dual lattices and dual bases, etc. If any of these concepts seem new to you, there are great resources to familiarize yourself with them first (see e.g. lecture notes by Daniele, Oded, Daniel/Léo). It will probably help if you are familiar with the LLL algorithm (also covered in aforementioned notes), but I’ll try to phrase everything so it is understandable even if if you aren’t.

Ok, so let’s get started. Before we look at BKZ in particular, first some comments about lattice block reduction in general.

The Basics

The Goal

Why would anyone use block reduction? There are (at least) two reasons.

1) Block reduction allows you to find short vectors in a lattice. Recall that finding the shortest vector in a lattice (i.e. solving SVP) is really hard (as far as we know, this takes at least \(2^{\Omega(n)}\) time or even \(n^{\Omega(n)}\) if you are not willing to also spend exponential amounts of memory). On the other hand, finding somewhat short vectors that are longer than the shortest vector by “only” an exponential factor is really easy (see LLL). So what do you do if you need something that is shorter than what LLL gives you, but you don’t have enough time to actually find the shortest vector? (This situation arises practically every time you use lattice reduction for cryptanalysis.) You can try to find something in between and hope that it doesn’t take as long. This is where lattice reduction comes in: it gives you a smooth trade-off between the two settings. It is worth mentioning that when it comes to approximation algorithms, block reduction is essentially the only game in town, i.e. there are, as far as I know, no non-trivial approximation algorithms that cannot be viewed as block reduction. (In fact, this is related to an open problem that Noah stated during the program: to come up with a non-trivial approximation algorithm that does not rely on a subroutine to find the shortest lattice vector in smaller dimensions.) The only exception to this are quantum algorithms that are able to find subexponential approximations in polynomial time in lattices with certain (cryptographically highly relevant) structure (see [CDPR16] and follow up work).

2) Block reduction actually gives you more than just short vectors. It gives you guarantees on the “quality” of the basis. What do we mean by the quality of the basis? Consider the Gram-Schmidt vectors \({\mathbf{b}}_i^*\) (GSO vectors) associated to a lattice basis \({\mathbf{B}}\). What we want is that the length of these Gram-Schmidt vectors (the GSO norms) does not drop off too quickly. The reason why this is a useful measure of quality for lattice bases is that it gives a sense of how orthogonal the basis vectors are: conditioned on being bases of the same lattice, the less accentuated the drop off in the GSO vectors, the more orthogonal the basis, and the more useful this basis is to solve several problems in a lattice. In fact, recall that the product of the GSO norms is equal to the determinant of the lattice and thus remains constant. Accordingly, if the GSO norms do not drop off too quickly, the first vector can be shown to be relatively short. So by analyzing the quality of the basis that block reduction achieves, a guarantee on the length of the first vector comes for free (see goal 1)). If you are familiar with the analysis of LLL, this should not come as a surprise to you.

Tools

In order to ensure that the GSO norms do not drop off to quickly, it seems useful to be able to reduce them locally. To this end, we will work with projected lattice blocks (this is where the term “block” in block reduction comes from). More formally, given a basis \({\mathbf{B}}\) we will consider the block \({\mathbf{B}}_{[i,j]}\) for \(i < j\) as the basis formed by the basis vectors \({\mathbf{b}}_i, {\mathbf{b}}_{i+1}, \dots, {\mathbf{b}}_{j}\) projected orthogonally to the first \(i-1\) basis vectors. So \({\mathbf{B}}_{[i,j]}\) is a basis for the lattice given by the sublattice formed by \({\mathbf{b}}_1, {\mathbf{b}}_{2}, \dots, {\mathbf{b}}_{j}\) projected onto the orthogonal subspace of the vectors \({\mathbf{b}}_1, {\mathbf{b}}_{2}, \dots, {\mathbf{b}}_{i-1}\). Notice that the first vector of \({\mathbf{B}}_{[i,j]}\) is exactly \({\mathbf{b}}^*_i\) – the \(i\)-th GSO vector. Another way to view this is to consider the QR-factorization of \({\mathbf{B}} = {\mathbf{Q}} {\mathbf{R}}\), where \({\mathbf{B}}\) is the matrix whose columns are the basis vectors \({\mathbf{b}}_i\). Since \({\mathbf{Q}}\) is orthonormal, it represents a rotation of the lattice and we can consider the lattice generated by the columns of \({\mathbf{R}}\) instead, which is an upper triangular matrix. For an upper triangular basis, the projection of a basis vector orthogonal to the previous basis vectors simply results in dropping the first entries from the vector. So considering a projected block \({\mathbf{R}}_{i,j}\) is simply to consider the square submatrix of \({\mathbf{R}}\) consisting of the rows and columns with index \(k\) between \(i \leq k \leq j\).

Now we need a tool that allows us to control these GSO vectors, which we view as the first basis vectors in projected sublattices. For this, we will fall back to algorithms that solve SVP. Recall that this is very expensive, so we will not call this on the basis \({\mathbf{B}}\) but rather on the projected blocks \({\mathbf{B}}_{[i,j]}\), where we ensure that the dimension \(k = j-i+1\) of the lattice generated by this projected block is not too large. In fact, the maximum dimension \(k\) that we call the SVP algorithm on will control the time/quality trade-off achieved by our block reduction algorithms and is usually denoted by the block size. So we will assume that we have access to such an SVP algorithm. Actually, we will assume something slightly stronger: we will assume access to a subroutine that takes as input the basis \({\mathbf{B}}\) and indices \(i,j\) and outputs a basis \({\mathbf{C}}\) such that

  • the lattice generated by the basis remains the same

  • the first \(i-1\) and the last vectors starting from \(j+1\) remain unchanged

  • the projected block \({\mathbf{C}}_{[i,j]}\) is SVP reduced, meaning that \({\mathbf{c}}^*_i\) is the shortest vector in the lattice generated by \({\mathbf{C}}_{[i,j]}\). Additionally, if \({\mathbf{B}}_{[i,j]}\) is already SVP reduced, we assume that the basis \({\mathbf{B}}\) is left unchanged.

We will call an algorithm that achieves this an SVP oracle. Such an oracle can be implemented given any algorithm that solves SVP (for arbitrary lattices). The technical detail of filling in the gap is left as homework to the reader.

Effect of a call to the SVP oracle. GSO log norms of the input in black, of the output in red. Note that the sum of the GSO log norms is a constant, so reducing the first vector, increases the (average of the) remaining vectors.

For the analysis we need to know what such an SVP oracle buys us. This is where Minkowski’s theorem comes in: we know that for any \(n\)-dimensional lattice \(\Lambda\) we have \(\lambda_1(\Lambda) \leq \sqrt{\gamma_n} \det(\Lambda)^{1/n}\) (where \(\lambda_1(\Lambda)\) is the length of the shortest vector in \(\Lambda\) and \(\gamma_n = \Theta(n)\) is Hermite’s constant). This tells us that after we’ve applied the SVP oracle to a projected block \({\mathbf{B}}_{[i,i+k-1]}\), we have \[\|{\mathbf{b}}^*_i \| \leq \sqrt{\gamma_{k}} \left(\prod_{j = i}^{i+k-1} \|{\mathbf{b}}_j^* \| \right)^{1/k}.\] Almost all of the analyses of block reduction algorithms, at least in terms of their output quality, rely on this single inequality.

Disclaimer

Before we finally get to talk about BKZ, I want to remark that throughout this series I will punt on a technical (but very important) topic: the number of arithmetic operations (outside of the oracle calls) and the size of the numbers. The number of arithmetic operations is usually not a problem, since it will be dominated by the calls to the SVP oracle. We will only compute projections of sublattices corresponding to projected blocks as described above to pass them to the oracle, which can be done efficiently using the Gram-Schmidt orthogonalization. The size of the numbers is a more delicate issue. We need to ensure that the required precision for these projections does not explode somehow. This is usually addressed by interleaving the calls to the SVP oracle with calls to LLL. If you are familiar with the LLL algorithm, it should be intuitive that this allows to control the size of the number. For a clean example of how this can be handled, we refer to e.g. [GN08a]. So, in summary, we will measure the running time of our algorithms thoughout simply in the number of calls to the SVP oracle.

BKZ

Schnorr [S87] introduced the concept of BKZ reduction in the 80’s as a generalization of LLL. The first version of the BKZ algorithm as we consider it today was proposed by Schnorr and Euchner [SE94] a few years later. With our setup above, the algorithm can be described in a very simple way. Let \({\mathbf{B}}\) be a lattice basis of an \(n\)-dimensional lattice and \(k\) be the block size. Recall that this is a parameter that will determine the time/quality trade-off as we shall see in the analysis. We start by calling the SVP oracle on the first block \({\mathbf{B}}_{[1,k]}\) of size \(k\). Once this block is SVP reduced, we shift our attention to the next block \({\mathbf{B}}_{[2,k+1]}\) and call the oracle on that. Notice that SVP reduction of \({\mathbf{B}}_{[2,k+1]}\) may change the lattice generated by \({\mathbf{B}}_{[1,k]}\) and \({\mathbf{b}}_1\) may not be the shortest vector in the first block anymore, i.e. it can potentially be reduced even further. However, instead of going back and fixing that, we will simply leave this as a problem to “future us”. For now, we continue in this fashion until we reach the end of the basis, i.e. until we called the oracle on \({\mathbf{B}}_{n-k,n}\). Note that so far this can be viewed as considering a constant sized window moving from the start of the basis to the end and reducing the first vector of the projected block in this window as much as possible using the oracle. Once we have reached the end of the basis, we start reducing the window size, i.e. we call the oracle on \({\mathbf{B}}_{n-k+1,n}\), then on \({\mathbf{B}}_{n-k+2,n}\), etc. This whole process is called a BKZ tour.

Now that we have finished a tour, it is time to go back and fix the blocks that are not SVP reduced anymore. We do this simply by running another tour. Again, if the second tour modified the basis, there is no guarantee that all the blocks are SVP redcued. So we simply repeat, and repeat, and … you get the idea. We run as many tours as required until the basis does not change anymore. That’s it. If this looks familiar to you, that’s not a coincidence: if we plug in \(k=2\) as our block size, we obtain (a version of) LLL! So BKZ is a proper generalization of LLL.

BKZ in one picture: apply the SVP oracle to the projected blocks from start to finish and when you reach the end, repeat.

The obvious questions now are: what can we expect from the output? And how long does it take?

The Good

We will now take a closer look at the approximation factor achieved by BKZ. If you want to follow this analysis along, you might want to get out pen and paper. Otherwise, feel free to trust me on the calculations (I wouldn’t!) and/or jump ahead to the end of this section for the result (no spoilers!). Let’s assume for now that the BKZ algorithm terminates. If it does, we know that the projected block \({\mathbf{B}}_{[i, i+k-1]}\) is SVP reduced for every \(i \in [1,\dots,n-k+1]\). This means that we have \[\|{\mathbf{b}}^*_i \|^k \leq \gamma_{k}^{k/2} \prod_{j = i}^{i+k-1} \|{\mathbf{b}}_j^* \|\] for all these \(n-k+1\) values of \(i\). Multiplying all of these inequalities and canceling terms gives the inequality \[\|{\mathbf{b}}^*_1 \|^{k-1}\|{\mathbf{b}}^*_2 \|^{k-2} \dots \|{\mathbf{b}}^*_{k-1} \| \leq \gamma_{k}^{\frac{(n-k+1)k}{2}} \|{\mathbf{b}}_{n-k+2}^* \|^{k-1} \|{\mathbf{b}}_{n-k+3}^* \|^{k-2} \dots \|{\mathbf{b}}_{n}^* \|.\] Now we make two more observations: 1) not only is \({\mathbf{B}}_{[1, k]}\) SVP reduced, but so is \({\mathbf{B}}_{[1, i]}\) for every \(i < k\). (Why? Think about it for 2 seconds!) This means we can multiply the inequalities \[\|{\mathbf{b}}^*_1 \|^i \leq \gamma_{i}^{i/2} \prod_{j = 1}^{i} \|{\mathbf{b}}_j^* \|\] for all \(i \in [2,k-1]\) together with the trivial inequality \(\|{\mathbf{b}}^*_1 \| \leq \|{\mathbf{b}}^*_1 \|\), which gives \[\|{\mathbf{b}}^*_1 \|^{\frac{k(k-1)}{2}} \leq \left(\prod_{i = 2}^{k-1} \gamma_{i}^{i/2} \right) \prod_{i = 1}^{k-1} \|{\mathbf{b}}_i^* \|^{k-1}\] Now we use the fact that \(\gamma_k^k \geq \gamma_i^i\) for all \(i \leq k\) (Why? Homework!) and combine with our long inequality above to get \[\|{\mathbf{b}}^*_1 \|^{\frac{k(k-1)}{2}} \leq \gamma_k^{\frac{k(n-1)}{2}} \|{\mathbf{b}}_{n-k+2}^* \|^{k-1} \|{\mathbf{b}}_{n-k+3}^* \|^{k-2} \dots \|{\mathbf{b}}_{n}^* \|.\] (I’m aware that this is a lengthy calculation for a blog post, but we’re almost there, so bear with me. It’s worth it!)

We now use one final observation, which is a pretty common trick in lattice algorithms: w.l.o.g. assume that for some shortest vector \({\mathbf{v}}\) in our lattice its projection orthogonal to the first \(n-1\) basis vectors is non-zero (if it is zero for all of the shortest vectors, simply drop the last vector from the basis, the result is still BKZ reduced, so use induction). Then we must have that \(\lambda_1 = \| {\mathbf{v}} \| \geq \|{\mathbf{b}}_i^* \|\) for all \(i \in [n-k+2, \dots, n]\), since otherwise the projected block \({\mathbf{B}}_{i,n}\) would not be SVP reduced. This means, we have \(\lambda_1 \geq \max_{i \in [n-k+2, \dots, n]} \|{\mathbf{b}}_i^* \|\). This is the final puzzle piece to get our approximation bound: \[\|{\mathbf{b}}^*_1 \| \leq \gamma_{k}^{\frac{n-1}{k-1}} \lambda_1.\] Note that this analysis (dating back to Schnorr [S94]) is reminiscent of the analysis of LLL and if we plug in \(k=2\), we get exactly what we’d expect from LLL. Though we do note a gap in the other extreme: if we plug in \(k=n\), we know that the approximation factor is \(1\) (we are solving SVP in the entire lattice), but the bound above yields a factor \(\gamma_n = \Theta(n)\).

The Bad

Now that we’ve looked at the output quality of the basis, let’s see what we can say about the running time (recall that our focus is on the number of calls to the SVP oracle). The short answer is: not much and that’s very unfortunate. Ideally, we’d want a bound on the number of SVP calls that is polynomial in \(n\) and \(k\). This would mean that the overall running time for large \(k\) is dominated by the running time of the SVP oracle in dimension \(k\) and the block size would give us exactly the expected trade-off. However, an LLL style analysis has so far only yielded a bound on the number of tours which is \(O(k^n)\) [HPS11, Appendix]. This is quite bad – for large \(k\) the number of calls will be the dominating factor in the running time.

The Ugly

Recall that the analysis of LLL does not only provide a bound on the approximation factor, but also on the Hermite factor, i.e. on the ratio of \(\| {\mathbf{b}}_1\|/\det(\Lambda)^{1/n}\). Since an LLL-style analysis worked out nicely for the approximation factor of BKZ, it stands to reason that a similar analysis should yield a similar bound for BKZ. By extrapolating from LLL, one could expect a bound along the lines of \(\| {\mathbf{b}}_1\|/\det(\Lambda)^{1/n} \leq \gamma_{k}^{n/2k}\) (note the square root improvement w.r.t. the trivial bound obtained from the approximation factor). And, in fact, a bound of \(\gamma_{k}^{\frac{n-1}{2(k-1)} + 1}\) has been claimed in [GN08b] but without proof (as pointed out in [HPS11]) and it is not clear, how one would prove this. ([GN08b] claims that one can use a similar argument as we did for the approximation factor, but I don’t see it.)

The Rescue

So it seems different techniques are necessary to complete the analysis of BKZ. The work of [HPS11] introduced such a new technique based on the analysis of dynamical systems. This work applied the technique successfully to BKZ, but the analysis is quite involved. What it shows is that one can terminate BKZ after a polynomial number of tours and still get a guarantee on the output quality, which is very close to the conjectured bound on the Hermite factor above. (Caveat: Technically, [HPS11] only showed this result for a slight variant of BKZ, but the difference to the standard BKZ algorithm only lies in the scope of the interleaving LLL applications, which is something that we glossed over above.) This is in line with experimental studies [SE94,GN08b,MW16], which show that BKZ produces high quality bases after a few tours already.

We will revisit this approach when considering a different block reduction variant, SDBKZ, where the analysis is much cleaner. As a teaser for the next post though, recall that BKZ can be viewed as a generalization of LLL (which corresponds to BKZ with block size \(k=2\)). Since the analysis of LLL did not carry entirely to BKZ, one could wonder if there is a different generalization of LLL such that an LLL-style analysis also generalizes naturally. The answer to this is yes, and we will consider such an algorithm in the next post.

  • [CDPR16] Cramer, Ducas, Peikert, Regev. Recovering short generators of principal ideals in cyclotomic rings. EUROCRYPT 2016
  • [GN08a] Gama, Nguyen. Finding short lattice vectors within Mordell’s inequality. STOC 2008
  • [GN08b] Gama, Nguyen. Predicting lattice reduction. EUROCRYPT 2008
  • [HPS11] Hanrot, Pujol, Stehlé. Analyzing blockwise lattice algorithms using dynamical systems. CRYPTO 2011
  • [MW16] Micciancio, Walter. Practical, predictable lattice basis reduction. EUROCRYPT 2016
  • [SE94] Schnorr, Euchner. Lattice basis reduction: Improved practical algorithms and solving subset sum problems. Mathematical Programming 1994
  • [S87] Schnorr. A hierarchy of polynomial time lattice basis reduction algorithms. Theoretical Computer Science 1987
  • [S94] Schnorr. Block reduced lattice bases and successive minima. Combinatorics, Probability and Computing 1994

Workshop “Lattices: New Cryptographic Capabilities”

On the behalf of the organizers, I am excited to announce that the next Simons workshop Lattices: New Cryptographic Capabilities will take place next week Mar 23-27, 2020 over Zoom!

The workshop will cover advanced lattice-based cryptographic constructions, while also highlighting some of the recurring themes and techniques, reiterated through a game of Bingo! The rest of this post provides a sneak preview along with the Bingo puzzle.

Looking forward to seeing everyone at the workshop!

Hoeteck, together with Shweta, Zvika and Vinod


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A Sneak Preview

Let A1, A2 be square matrices and t a row vector such that

tA1 = x1t, tA2 = x2t
Using high-school algebra lingo, we would refer to t as the eigenvector of A1, A2. It is easy to see that

t ⋅ (A1 + A2) = (x1 + x2)t, t ⋅ A1A2 = x1x2t
This extends readily to any polynomial p(x1, …, xn), namely: if tAi = xit, then

t ⋅ f(A1, …, An) = f(x1, …, xn)t
As in turns out, much of advanced lattice-based crypto boils down to a generalization of this statement! The generalization is along two orthogonal dimensions:

  1. arbitrary matrices A1, …, An that may not share the same eigenvector t, and
  2. a relaxation to “approximate” equality, namely tAi ≈ xit.

The generalization underlies fully homomorphic encryption, homomorphic signatures, attribute-based encryption schemes and many more!

Bingo!

Here’s the 4×4 bingo puzzle:

GGH15 Bonsai AR+G noise growth
G − 1 LWE Vinod LHL
Gaussian Af FHE Dec linear noise flooding
homomorphic trapdoor smoothing parameter Hf, x

Research Vignette: Foundations of Data Science

by Ilias Diakonikolas (University of Southern California), Santosh Vempala (Georgia Institute of Technology), and David P. Woodruff (Carnegie Mellon University)

Algorithmic High-Dimensional Robust Statistics
Fitting a model to a collection of observations is one of the quintessential goals of statistics and machine learning. A major recent advance in theoretical machine learning is the development of efficient learning algorithms for various high-dimensional models, including Gaussian mixture models, independent component analysis, and topic models. The Achilles’ heel of these algorithms is the assumption that data is precisely generated from a model of the given type.

This assumption is crucial for the performance of these algorithms: even a very small fraction of outliers can completely compromise the algorithm’s behavior. For example, k adversarially placed points can completely alter the k-dimensional principal component analysis, a commonly used primitive for many algorithms. However, this assumption is only approximately valid, as real data sets are typically exposed to some source of contamination. Moreover, the data corruption is often systematic, and random models do not accurately capture the nature of the corruption. Hence, it is desirable that any estimator that is to be used in practice is stable in the presence of arbitrarily and adversarially corrupted data.

Indeed, the problem of designing outlier-robust estimators is natural enough that it is studied by a classical body of work, namely robust statistics, whose prototypical question is the design of estimators that perform well in the presence of corrupted data. This area of statistics was initiated by the pioneering works of Tukey and Huber in the 1960s and addresses a conceptual gap in Fisher’s theory of exact parametric inference — since parametric models are typically only approximately valid, robust statistics is essential to complete the theory. From a practical perspective, the question of how to make good inferences from data sets in which pieces of information are corrupted has become a pressing challenge. Specifically, the need for robust statistics is motivated by data poisoning attacks, in the context of adversarial machine learning, as well as by automatic outlier removal for high-dimensional data sets from a variety of applications.

Classical work in robust statistics pinned down the fundamental information-theoretic aspects of high-dimensional robust estimation, establishing the existence of computable and information-theoretically optimal robust estimators for fundamental problems. In contrast, until very recently, the computational complexity aspects of robust estimators were poorly understood. In particular, even for the basic problem of robustly estimating the mean of a high-dimensional data set, all known robust estimators were hard to compute (i.e., computationally intractable). In addition, the accuracy of the known efficient heuristics degrades quickly as the dimension increases. This state of affairs prompted the following natural question: can we reconcile robustness and computational efficiency in high-dimensional estimation?

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Research Vignette: Lower Bounds in Computational Complexity

by Rahul Santhanam (University of Oxford)

Computational complexity theory studies the possibilities and limitations of algorithms. Over the past several decades, we have learned a lot about the possibilities of algorithms. We live today in an algorithmic world, where the ability to reliably and quickly process vast amounts of data is crucial. Along with this social and technological transformation, there have been significant theoretical advances, including the discovery of efficient algorithms for fundamental problems such as linear programming and primality.

We know far less about the limitations of algorithms. From an empirical point of view, certain important problems seem inherently hard to solve. These include the satisfiability problem in logic, the Traveling Salesman Problem in graph theory, the integer linear programming problem in optimization, the equilibrium computation problem in game theory, and the protein folding problem in computational biology. What all these problems have in common is that it is easy to verify if a given solution is correct, but it seems hard to compute solutions. This phenomenon is encapsulated in the celebrated NP vs. P question, which asks if all problems with solutions verifiable in polynomial time (NP) can also be solved in polynomial time (P). The NP vs. P problem is important both mathematically, as evidenced by its inclusion in the list of Millennium Prize Problems by the Clay Mathematics Institute, and scientifically, since natural problems that arise in a variety of scientific contexts are in NP but not known to be in P. To make progress on NP vs. P and related questions, we need to show complexity lower bounds (i.e., prove that a given computational problem cannot be solved efficiently).

Complexity lower bounds are interesting for many reasons. First, from a pragmatic point of view, they map the boundaries of what is possible and, hence, save us from wasting our efforts on finding efficient solutions to problems beyond those boundaries. Second, lower bounds can be exploited algorithmically to design secure cryptographic protocols and efficiently convert randomized algorithms into deterministic ones. Thus, intriguingly, proving new limits on the power of computation also opens up new possibilities for computation! Third, and perhaps most importantly, lower bounds provide a deeper understanding of the nature of computation. An efficient solution to an algorithmic problem tells us something new about that specific problem, while a lower bound often tells us something new about the computational model and hence about algorithms in general.

The Fall 2018 Simons Institute program on Lower Bounds in Computational Complexity gathered together researchers working on different aspects of complexity lower bounds, in Boolean, algebraic, and interactive settings, with the goal of making progress on the major open problems concerning lower bounds. In some of these cases, such as the setting of interactive computation, good lower bounds are known for various problems. In other cases, such as the case of general Boolean circuits, very little is known. In the remainder of this article, I will briefly survey what is known about Boolean circuit lower bounds and describe a phenomenon called hardness magnification that provides some new insights. This line of work was developed partly during the course of the Lower Bounds program with my collaborators Igor Oliveira and Ján Pich.

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Research Vignette: Real-Time Decision Making in Energy (RTDM-E)

by Xinbo Geng (Texas A&M University), Swati Gupta (Massachusetts Institute of Technology)Tong Huang (Texas A&M University), and Le Xie (Texas A&M University)

The electric grid serves as one of the backbone infrastructure systems that support the well-being of billions of citizens in modern society. There has been a significant transformation of energy systems over the past decade, even though these changes have largely gone unnoticed due to affordable and reliable grid services for end-users in most developed nations. The transformation on the supply side mainly manifests itself through the decarbonization of the power supply, i.e. greater use of low-carbon energy sources. As an example, a recent National Bureau of Economic Research (NBER) study released in December 2018 suggested that US power-sector emissions have decreased by 45% since 2008. This dramatic change is fundamentally driven by two factors: (1) a sharp increase in the renewable-energy portfolio; and (2) the substantial replacement of coal-fired power plants with natural gas and other resources.

Due to the supply-side transformation, in order to maintain or improve the quality of electricity services to end-users, it is necessary to make use of real-time information that is becoming readily available thanks to the proliferation of information and communication technology (ICT) infrastructures.

Decision making in the electric grid enabled by real-time information occurs at multiple time scales, which makes the problem even more interesting. At the time scale of 15 minutes or longer, a central challenge lies in how to best allocate different resources that will meet the balance of the fluctuating demand at the lowest cost. The underlying mathematical problem is typically formulated as a nonlinear multi-stage optimization problem. Another big area of investment in real-time decision making (RTDM) is in the proliferation of globally synchronized sensors called synchrophasors. Compared with conventional SCADA systems, synchrophasors offer a two-orders-of-magnitude faster sampling rate, and synchronized time stamps for electrical variables. Therefore, they offer new opportunities to detect, locate, and classify anomalies that would have not been possible to capture by conventional means, as well as opportunities to avoid blackouts.

In what follows, we elaborate on the real-time decision-making challenges and opportunities at two different time scales and two different spatial scales. The first one deals with day-ahead decisions on phase balancing in the distribution grid. The second one studies the possibility of real-time localization of anomalies such as forced oscillation in large transmission systems.

A focal area of the Spring 2018 Simons Institute program on Real-Time Decision Making was formalizing and studying various issues in the modern energy grid pertaining to efficiency, reliability and security. Among the research themes, a good amount of effort was devoted to modernizing the optimization core of grid operations [4]. In addition, by tapping into expertise in the program in combinatorial algorithms, concrete collaboration occurred at the interface between combinatorial algorithms and grid optimization. For example, electricity is typically distributed radially to address security concerns, and this translates to selecting a spanning tree in a network with desirable properties. The project below on the distribution grid grew out of frequent conversations that the RTDM program facilitated for researchers with backgrounds in optimization and in power-systems engineering.

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