Communication-Efficient Search for an Approximate Closest Lattice Point

We consider the problem of finding the closest lattice point to a vector in n-dimensional Euclidean space when each component of the vector is available at a distinct node in a network. Our objectives are (i) minimize the communication cost and (ii) obtain the error probability. The approximate closest lattice point considered here is the one obtained using the nearest-plane (Babai) algorithm. Assuming a triangular special basis for the lattice, we develop communication-efficient protocols for computing the approximate lattice point and determine the communication cost for lattices of dimension n>1. Based on available parameterizations of reduced bases, we determine the error probability of the nearest plane algorithm for two dimensional lattices analytically, and present a computational error estimation algorithm in three dimensions. For dimensions 2 and 3, our results show that the error probability increases with the packing density of the lattice.

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08/30/2020

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I Introduction

Fig. 1: Centralized model
Fig. 2: Interactive model

We consider a network consisting of sensor-processor nodes (hereafter referred to as nodes) and possibly a central computing node (fusion center) interconnected by links with limited bandwidth. Node

observes real-valued random variable

. In the centralized model (Fig. 2), the objective is to compute a given function at the fusion center based on information communicated from each of the sensor nodes. In the interactive model (Fig. 2), the objective is to compute the function at each sensor node (the fusion center is absent). In general, since the random variables are real valued, these calculations would require that the system communicate an infinite number of bits in order to compute exactly. Since the network has finite bandwidth links, the information must be quantized in a suitable manner, but quantization affects the accuracy of the function that we are trying to compute. Thus, the main goal is to manage the tradeoff between communication cost and function computation accuracy. Here, computes the closest lattice point to a real vector in a given lattice .

The process of finding the closest lattice point is widely used for decoding lattice codes and for quantization. Lattice coding offers significant coding gains for noisy channel communication [8] and for quantization [5]. In a network, it may be necessary for a vector of measurements to be available at locations other than and possibly including nodes where the measurements are made. In order to reduce network bandwidth usage, it is logical to consider a vector quantized (VQ) representation of these measurements, subject to a fidelity criterion, for once a VQ representation is obtained, it can be forwarded in a bandwidth efficient manner to other parts of the network. However, there is a communication cost to obtaining the vector quantized representation. This paper is our attempt to understand the costs and tradeoffs involved. Bounds for the error probability in dimensions 2 and 3, when we considered the nearest plane (Babai) algorithm for lattices, are derived as well as the rate computation which underlies the decoding process in a distributed system. The communication cost/error tradeoff of refining the nearest-plane estimate in an interactive setting is addressed in a companion paper [25].

Example application settings include MIMO systems [23], and network management in wide area networks [14], to name a few. For prior work in the computer science community, see [28][15]. Information theory [References] has resulted in tight bounds, [20][17].

We observe here that algorithms for the closest lattice point problem have been studied in great detail, see [1] and the references therein, for a comprehensive survey and algorithms. However, in all these algorithms it is assumed that the vector components are available at the same location. In our work, the vector components are available at physically separated nodes and we are interested in the communication cost of exchanging this information in order to determine the closest lattice point. None of the previously proposed fast algorithms consider this communication cost.

The closest lattice point problem has also been proposed as a basis for lattice cryptography ([2],[18],[11][13],[21]), a topic of great interest in recent years, examples being the GGH and LWE cryptosystems. The idea is to require solution of the closest lattice point problem, which is known to be NP-hard [10], assuring security. The Babai algorithm is used in some of the proposed cryptosystems, thus computation of its error probability is of interest in this context.

The remainder of our paper is organized into five sections: Sec. II presents some basic definitions and facts about lattices. Sec. III establishes a framework for measuring the cost and error rate and presents an expression for the error probability of the distributed closest lattice point problem in an arbitrary two dimensional case, Sec. IV develops a computational error analysis procedure for lattices in dimension 3. Sec. V presents efficient protocols for computing the approximate closest lattice point along with rate estimates for both models for . Directions for future work and conclusions are in Sec. VI.

Ii Lattice Basics, Voronoi and Babai Partitions

Notation, properties, partitions and special bases for lattices are described in this section.

A lattice is a set of integer linear combinations of independent vectors which can be written as where is a matrix whose columns are the vectors and vectors are considered here in the column format. is a generator matrix of and is the associated Gram matrix. In this paper, we only consider full rank lattices ().

A set is called a fundamental region of a lattice if all its translations by elements of define a partition of . Examples of fundamental regions are the fundamental parallelepiped supported by a set of basis vectors and the Voronoi region or Voronoi cell of a lattice point defined as where denotes the Euclidean norm. Note that is congruent to .

The volume of a lattice is the volume of any of its fundamental regions. It is given by where is a generator matrix of

A vector is called Voronoi vector

if the hyperplane

has a non-empty intersection with A Voronoi vector is said to be relevant if this intersection is an dimensional face of

The packing radius of a lattice is half of the minimum distance between lattice points and the packing density is the fraction of space that is covered by balls of radius in centered at lattice points, i.e.,

The closest vector problem (CVP) in a lattice can be described as an integer least squares problem with the objective of determining such that where the norm considered is the standard Euclidean norm. The closest lattice point to is then given by . Observe that the mapping partitions into Voronoi cells.

The nearest plane (np) algorithm [4], an approach for approximating the closest lattice point, computes , an approximation to , given by , where is obtained as follows.

Let denote the subspace spanned by the vectors , . Let be the orthogonal projection of onto and let be the closest vector to in . Consider the decomposition and let . Start with and and compute , , for (here denotes the nearest integer to ).

We denote the vector as Babai point, which is an approximate solution to the closest lattice point problem. The mapping partitions into hyper-rectangular cells with volume and we refer to this partition as a Babai partition.

Example 1.

Fig. 3 represents the Babai partition (black lines) and the Voronoi partition (pink lines) for the hexagonal lattice generated by and illustrates geometrically the manner in which the np algorithm approximates the closest point problem.

Fig. 3: Cells of the Babai partition and of the Voronoi partition for the hexagonal lattice

Note that a Babai partition is basis dependent. In case the generator matrix is upper triangular with entry , each rectangular cell is axis-aligned and has sides of length . In this specific case, the vectors mentioned above are of type .

We remark that given a lattice with an arbitrary generator matrix

we can apply the QR decomposition,

and the matrix will generate a rotation of the original lattice. We will introduce next two types of special bases we will work closely in this paper: Minkowski-reduced basis and obtuse superbase.

A basis of a lattice in is said to be Minkowski-reduced if with is such that , for any for which can be extended to a basis of .

For two dimensional lattices, a Minkowski-reduced basis is also called Lagrange-Gauss reduced basis and there is a simple characterization [8] for it: a lattice basis is a Minkowski-reduced basis if only if and It follows that the angle between the subsequent minimum norm vectors and must satisfy

It is also possible to characterize a Minkowski-reduced basis for lattices in dimensions three or smaller [8] :

Proposition 1.

Consider a Gram matrix of a lattice and the conditions below:

(1)
(2)
(3)

Then, inequalities (1), (1)–(2) and (1)–(3) define a Minkowski-reduced basis for dimensions 1,2 and 3, respectively.

All lattices have a Minkowski-reduced basis, which roughly speaking, consists of short vectors that are as perpendicular as possible [8].

We describe next the concept of an obtuse superbase that will be used in the following sections.

Definition 1.

Let be a basis for a lattice . A superbase with is said to be obtuse if for . A lattice is said to be of Voronoi’s first kind if it has an obtuse superbase

The above parameters are called Selling parameters and if we say that the superbase is strictly obtuse.

Example 2.

Consider the standard basis for the body-centered cubic (BCC) lattice where We set and it is not hard to see that is a strictly obtuse superbase for BCC lattice. Indeed and Thus, BCC is of Voronoi’s first kind.

The existence of an obtuse superbase allows a characterization of the relevant Voronoi vectors for a lattice.

Theorem 1.

[9, Th.3, Sec. 2] Let be a lattice of Voronoi’s first kind with obtuse superbase . Vectors of the form where is a strict non-empty subset of are Voronoi vectors of .

It was demonstrated [9] that all lattices with dimension less or equal than three are Voronoi’s first kind. In three dimensions, considering an obtuse superbase, since all Voronoi vectors described in the above theorem can be written as one of the following seven vectors or their opposites [9]:

(4)

Given an obtuse superbase, we also characterize the norms where as vonorms and as conorms, for the superbase Precise definitions of conorms and vonorms for the general dimensional case can be found in [9].

The nonzero cosets of naturally form a discrete projective plane of order The vonorms are marked as the nodes of the projective plane and the corresponding conorms and at the nodes of the dual plane in the following Figure 4.

Fig. 4: Projective and dual planes labelled with vonorms and conorms respectively (based on [9], p. 61)
Remark 1.

[9] Two projective planes labeled with conorms represent the same lattice precisely when there is a conorm preserving collineation between them.

The existence of obtuse superbases for three dimensional lattices let us characterize the five parallelohedra that can be their Voronoi regions: truncated octahedron, hexa-rhombic dodecahedron, rhombic dodecahedron, hexagonal prism and cuboid.

Let be an arbitrary dimensional lattice, with obtuse superbase and conorms A vector can be specified by its inner products

(5)

since the determinant of the Gram matrix is non vanishing.

The most generic Voronoi region in three dimensions is the truncated octahedron, with 14 faces and 24 vertices. It is know [9] that the vertices of this Voronoi cell are all the 24 points where is any permutation of :

(6)

Using Equations (5) and (II) one can define all the points that generates a generic Voronoi region.

It is possible to guarantee that two lattices for which the correspondent conorms are zero have combinatorially equivalent Voronoi regions (since one can be continuously deformed into the other without any edges being lost). So, when we construct the dual projective planes to represent the conorms, there are five choices for zeros: one, two, three collinear zeros, three non-collinear zeros or four zeros. Each of these configuration produces a different Voronoi cell according to Figure 5.

Fig. 5: Dual plane labeled with conorms and its correspondent Voronoi cells (based on [9], p. 65). (Note: Authors have requested permission to reproduce from the publisher and will present only if permission is granted.)
Theorem 2.

In dimensions , if a lattice has a Minkowski-reduced basis with vectors , with , , then the superbase , with is an obtuse superbase for . Conversely, if has an obtuse superbase, then a Minkowski-reduced basis can be obtained from it.

Proof.

The case n=1 is trivial, hence we will start with n=2: Suppose that is a Minkowski-reduced basis, then, according to Proposition 1, and Moreover, by hypothesis, Define and to guarantee that is an obtuse superbase, we need to check that and Indeed,

(7)

Similarly we have that

If is an obtuse superbase, any permutation of it is also an obtuse superbase. So, we may consider one such that Then we have that and

From the last inequality, we have that

(8)

For n=3: Consider a Minkowski-reduced basis such that and To check if is an obtuse superbase, we need to verify that and

Observe that

(9)

With analogous arguments, we show that and

To prove the converse, up to a permutation, we may consider an obtuse superbase such that This basis will be Minkowski-reduced if we prove conditions (2) and (3) from Proposition 1, i.e.,

(10)

and

(11)

The inequalities in Equation (10) are shown similarly to the two dimensional case starting from and Starting from it follows the inequality in Equation (11) concluding the proof.

Iii Error Probability Analysis for Arbitrary Two Dimensional Lattices

We assume that node observes an independent identically distributed (iid) random process , where is the time index and that random processes observed at distinct nodes are mutually independent. The time index is suppressed in the sequel. The random vector is obtained by projecting a random process on the basis vectors of an underlying coordinate frame, which is assumed to be fixed.

Consider that the lattice is generated by the scaled generator matrix , where is the generator matrix of the unscaled lattice. Let and denote the Voronoi and Babai cells, respectively, associated with lattice vector . The error probability , is the probability of the event and .

As will be discussed further, the Babai partition is dependent on, and the Voronoi partition is invariant to, the choice of lattice basis. Thus the error probability depends on the choice of the lattice basis. We will assume here that a Minkowski-reduced lattice basis, which is also obtuse (Theorem 2) can be chosen by the designer of the lattice code and it can be transformed into an equivalent basis This can be accomplished by applying QR decomposition to the lattice generator matrix (which has the original chosen basis vectors on its columns) in addition to convenient scalar factor. The reason for working with a Minkowski-reduced basis is partly justified by Example 3 below and the fact that the Voronoi region is easily determined since the relevant vectors are known; see Lemma 1 below.

An example to demonstrate the dependence of the error probability on the lattice basis is now presented.

Example 3.

Consider a lattice with basis The error probability in this case is (Fig. 6), whereas if we start from the basis we achieve after the QR decomposition and since the Babai region associated with an orthogonal basis and the Voronoi region for rectangular lattices coincides.

Fig. 6: Voronoi region and Babai partition of the triangular basis

Example 3 illustrates the importance of working with a good basis and partially explains our choice to work with a Minkowski-reduced basis. As mentioned above, additional motivation come from the observation that for a Minkowski-reduced basis in two dimensions, the relevant vectors are known.

To see this, we first note that an equivalent condition for a basis to be Minkowski-reduced in dimension two is Thus, we can state the following result, which was derived from the two dimensional analysis proposed in [9].

Lemma 1.

If a Minkowski-reduced basis is given by then, besides the basis vectors, a third relevant vector is

(12)

where is the angle between and

Note that, if is a Minkowski basis then so is and hence any lattice has a Minkowski basis with . So, if we consider the Minkowski-reduced basis as with and it is possible to use Lemma 1 to describe the Voronoi region of and determine its intersection with the associated Babai partition. Observe that the area of both regions must be the same and in this specific case, equal to This means that the vertices that define the Babai rectangular partition are

In addition is an obtuse superbase for so the relevant vectors that defines the Voronoi region are and We will choose for the analysis proposed in Theorem 3 only the relevant vectors in the first quadrant, i.e., due to the symmetry that a Voronoi cell has. Therefore, we can state the following result

Theorem 3.

Consider a lattice with a triangular Minkowski-reduced basis such that the angle between and satisfies . The error probability for the Babai partition is given by

(13)
Proof.

To calculate for the lattice , we first obtain the vertices of the Voronoi region. This is done by calculating the points of intersection of the perpendicular bisectors of the three relevant vectors and (according to Lemma 1, Fig. 7). Thus the vertices of the Voronoi region are given by , and .

Fig. 7: Voronoi region, Babai partition and three relevant vectors

is then computed as the ratio between the area of the Babai region which is not overlapped by the Voronoi region and the area of the Babai region. From Fig. 7, we get the error as the sum the areas of four triangles, where two of them are defined respectively by the points and The remaining two triangles are symmetric to these two. Therefore, the error probability is the sum of the four areas, normalized by the area of the Voronoi region . The explicit formula for it is given by

We obtain the following Corollary, illustrated in Fig. 8, from the error probability obtained in Theorem 3 with and

Corollary 1.

For any two dimensional lattice and a Babai partition constructed from the QR decomposition associated with a Minkowski-reduced basis where we have

(14)

and

  • i.e., the lattice is orthogonal.

  • i.e., the lattice is equivalent to hexagonal lattice.

  • the level curves of are described as ellipsoidal arcs (Figure 8) in the region and

Fig. 8: Level curves of in right-left ordering, for and Notice that is represented in the horizontal axis and in vertical axis.

Iii-a Variations with Packing Density and Angle

The packing density for a lattice with basis in Minkowski-reduced form is given by and following the notation from Theorem 3. For a fixed density (fixed ) the error probability is decreasing with and for fixed it is increasing with (decreasing with ).

So, if we consider the error probability for a given density , we have that is minimized by , where

and maximized by for any

Figure 9 represents the minimum error probability function for

Fig. 9: Minimum error probability for given packing density assuming

Note also that if and is defined to be the angle between the basis vectors, then the result of Theorem 3 can be rewritten as

(15)

In this case, we can see that for a fixed the error probability increases with achieving its minimum in and maximum in

Iv Error Probability Analysis for Arbitrary Three Dimensional Lattices

To analyse the error in the three dimensional case, we developed and implemented an algorithm in the software Mathematica [27] which calculates the error probability of any three dimensional lattice, given an obtuse superbase. We assume, as we did in the two dimensional analysis, an initial upper triangular lattice basis given by where It can be accomplished by performing a QR decomposition and a multiplication by a scalar factor in the original basis.

It is important to remark that the error probability is, in the general case, dependent on the basis ordering. Our algorithm searches over all orderings and determines the best one. As an example, the performance of the BCC lattice is invariant over basis ordering, due to its symmetries. On the other hand, for the FCC lattice, depending on how the basis is ordered, we can find two different error probabilities, and but only is tabulated.

A detailed description of the algorithm is presented below.

Algorithm 1: Error probability of the closest lattice point problem in a distributed system (three dimensional case)

  • Voronoi region: Provided an obtuse superbase, the vertices and faces that define the Voronoi region of are determined by Equations (5) and (II), following the method proposed by Conway and Sloane [9]

    . In this stage, we determine, generate and classify the correspondent Voronoi region of

    into one of five possibilities described in Figure 5.

  • Babai partition: Determine the vertices of the Babai cell. Since we have assumed a generator matrix in upper triangular form, , the vertices are:

    (16)
  • Intersection: In this stage, using a function in Mathematica [27], we calculate the intersection between the Voronoi and Babai regions obtained previously. This function runs through all points that define both solids and select the coincident ones, providing in the end of the process, the vertices that determine the intersection region. We calculate then the volume of the intersection normalized by the volume of the lattice The algorithm determines first the format of each type of Voronoi cell (Figure 5) to simplify the calculations of the error probability. To be more specific, if all conorms are nonzero (truncated octahedron) or if only one conorm is zero (hexa-rhombic dodecahedron) or if two collinear conorms are zero (rhombic dodecahedron), we implement the general intersection algorithm, defined as: let be, respectively, the set of vertices, edges and faces that define the Babai region of and be, respectively, the set vertices, edges and faces that define the Voronoi region of Thus, we solve:

    Solve {Or and Or and }.

    The union of points resulting from the previous system will define the intersection of Voronoi and Babai regions of For the two remaining cases, i.e., when we have two non-collinear zeros (hexagonal prism) we only calculate the intersection between the hexagonal basis and the rectangular basis of both prisms and when we have four zeros, the error probability is zero.

  • Packing density: Finally, we calculate the packing density .

Iv-a Calculations for Known Lattices

We present results obtained by applying Algorithm 1 to some known lattices in this section.

In Fig. 10 we have

  • in red, the cubic lattice with basis

  • in green, the lattice with basis Voronoi region: hexagonal prism;

  • in blue, the body-centered cubic (BCC) lattice, with basis , Voronoi region: truncated octahedron;

  • in black, the face-centered cubic (FCC) lattice, with basis ; Voronoi region: rhombic dodecahedron;

  • in purple, lattice with basis Voronoi region: hexa-rhombic dodecahedron.

Fig. 10: Performance of known lattices

Table I below presents some lattice performances when we run Algorithm

Lattice/Voronoi cell Notation Table ,[8] Conorms
Cubic/ Cuboid
Hexa-rhombic dodecahedron
Hexagonal prism (corresp. lattice)
BCC/ Truncated octahedron
FCC/ Rhombic dodecahedron
TABLE I: Performance in Algorithm for known lattices

We remark that the error probability for the hexagonal prism is identical to the two dimensional case (see Theorem 3) and cuboids have a null error probability (when aligned to the coordinate axes). We also see that the face-centered cubic lattice, which results in the best packing density for lattices in three dimensions, is the worst case when one considers its error probability.

Iv-B Random Lattice Selection

In this section we applied Algorithm 1 to lattices whose basis was chosen randomly. Specifically, we start by considering a basis at random, with the format where are real numbers in the range Then, the program tests if this basis is both an obtuse superbase and Minkowski-reduced according to Theorem 2. If this condition is false, another random basis is selected, until a suitable one is found. At the end of this stage, we will have a randomly chosen obtuse, Minkowski-reduced superbase for the lattice

In Figure 11, we have plotted the known points already seen in Figure 10, together with orange points that are associated with lattices having a packing density greater than randomly chosen as above. Note that with overwhelming probability, a randomly chosen basis will have a truncated octahedron as a Voronoi region (the most general Voronoi region in three dimensions).

Fig. 11: Comparison between random and known performances

However, by considering conorms that are approximately zero, we can identify cases that are ‘almost’ like one of the degenerate polyhedra. These cases are presented in Table II, illustrated as square points in Figure 11, where the color characterizes the cell type, following the notation of Figure 10.

(Aproximate) Voronoi cell Conorms
Hexa-rhombic dodecahedron
Hexa-rhombic dodecahedron
Rhombic-dodecahedron
Rhombic-dedecahedron
Hexagonal prism
TABLE II: Performance in Algorithm for random lattices

Iv-C Remarks

We conjecture, after trials that:

Conjecture 1.

For any three dimensional lattice and a Babai partition constructed from the QR decomposition associated with an obtuse superbase which is also Minkowski-reduced,

(17)

Compared with the two dimensional case, we have an increase of in the conjectured bound for the error probability and we expect this number to grow more as the dimension increases.

We also conjecture, assuming a more ”spherical shape” for Voronoi regions of densest lattices the following

Conjecture 2.

The worst error probability for a lattice in dimension is achieved by the densest lattice and it tends to one when goes to infinity.

V Rate Computation for Constructing a Babai Partition for arbitrary

Communication protocols are presented for the centralized and interactive model along with associated rate calculations in the limit as .

V-a Centralized Model

We now describe the transmission protocol by which the nearest plane lattice point can be determined at the fusion center . Let where and are relatively prime. Note that we are assuming the generator matrix is such that the aforementioned ratios are rational, for Let , where denotes the least common multiple of its argument. By definition .

Protocol 1.

(Transmission, ). Let be the largest for which . Then node sends and to , (by definition ).

Let be the coefficients of , the Babai point.

Theorem 4.

The coefficients of the Babai point can be determined at the fusion center after running transmission protocol .

Proof.

Observe that each coefficient of is given by

(18)

which is written in terms of and , the fractional and integer parts of real number , resp., (, ) by

(19)

Since the fractional part in the above equation is of the form , , where is defined above, it follows that . Thus

(20)

can be computed in the fusion center in the order . ∎

Corollary 2.

The rate required to transmit , is no larger than bits.

Thus the total rate for computing the Babai point at the fusion center under the centralized model is no larger than bits, where is the differential entropy of random variable , and scale factor is small. Thus the incremental cost due to the ’s does not scale with . However when is small, this incremental cost can be considerable, if the lattice basis is not properly chosen as we will see in further examples.

This rate computation can be visualized geometrically and under the light of the decoding in orthogonal lattices. Consider a lattice generated by