The Learning With Errors (LWE) problem is one of the most important hardness assumptions latticebased constructions base their security on. In 2015, Albrecht, Player and Scott presented the software tool LWEEstimator to estimate the hardness of concrete LWE instances, making the choice of parameters for latticebased primitives easier and better comparable. To give lower bounds on the hardness, it is assumed that each algorithm has given the corresponding optimal number of samples. However, this is not the case for many cryptographic applications. In this work we first analyze the hardness of LWE instances given a restricted number of samples. For this, we describe LWE solvers from the literature and estimate their runtime considering a limited number of samples. Based on our theoretical results we extend the LWEEstimator. Furthermore, we evaluate LWE instances proposed for cryptographic schemes and show the impact of restricting the number of available samples.
The Learning With Errors (LWE) problem is used in the construction of many cryptographic latticebased primitives [20, 30, 31]. It became popular due to its flexibility for instantiating very different cryptographic solutions and its (presumed) hardness against quantum algorithms. Moreover, LWE can be instantiated such that it is provably as hard as worstcase lattice problems [31].
In general, an instance of LWE is characterized by parameters
To ease the hardness estimation of concrete instances of LWE, the LWEEstimator [3, 4] was introduced. In particular, the LWEEstimator is a very useful software tool to choose and compare concrete parameters for latticebased primitives. To this end, the LWEEstimator summarizes and combines existing attacks to solve LWE from the literature. The effectiveness of LWE solvers often depend on the number of given LWE samples. To give conservative bounds on the hardness of LWE, the LWEEstimator assumes that the optimal number of samples is given for each algorithm, i.e., the number of samples for which the algorithm runs in minimal time. However, in cryptographic applications the optimal number of samples is often not available. In such cases the hardness of used LWE instances estimated by the LWEEstimator might be overly conservative. Hence, also the system parameters of cryptographic primitives based on those hardness assumptions are more conservative than necessary from the viewpoint of stateoftheart cryptanalysis. A more precise hardness estimation is to take the restricted number of samples given by cryptographic applications into account.
In this work we close this gap. We extend the theoretical analysis and the LWEEstimator such that the hardness of an LWE instance is computed when only a restricted number of samples is given. As in [4], our analysis is based on the following algorithms: exhaustive search, the Blum–Kalai–Wassermann (BKW) algorithm, the distinguishing attack, the decoding attack, and the standard embedding approach. In contrast to the existing LWEEstimator we do not adapt the algorithm proposed by Arora and Ge [8] due to its high costs and consequential insignificant practical use. Additionally, we also analyze the dual embedding attack. This variant of the standard embedding approach is very suitable for instances with a small number of samples since the embedding lattice is of dimension
Moreover, we evaluate our implementation to show that the hardness of most of the considered algorithms are influenced significantly by limiting the number of available samples. Furthermore, we show how the impact of reducing the number of samples differs depending on the model the hardness is estimated in.
Our implementation is already integrated into the existing LWEEstimator at https://bitbucket.org/malb/lweestimator (from commitid eb45a74 on). In our implementation, we always use the existing estimations with optimal number of samples, if the given restricted number of samples exceeds the optimal number. If not enough samples are given, we calculate the computational costs using the estimations presented in this work.
Figure 1 shows the categorization by strategies used to solve LWE: One approach reduces LWE to finding a short vector in the dual lattice formed by the given samples, also known as Short Integer Solution (SIS) problem. Another strategy solves LWE by considering it as a Bounded Distance Decoding (BDD) problem. The direct strategy solves for the secret directly. In Figure 1, we dashframe algorithms that make use of basis reduction methods. The algorithms considered in this work are written in bold.
In Section 2 we introduce notations and definitions required for the subsequent sections. In Section 3 we describe basis reduction and its runtime estimations. In Section 4 we give our analyses of the considered LWE solvers. In Section 5 we describe and evaluate our implementation. In Section 6 we explain how restricting the number of samples impacts the bithardness in different models. In Section 7 we summarize our work.
We follow the notation used by Albrecht, Player and Scott [4]. Logarithms are base 2 if not indicated otherwise. We write
With
For definitions of a lattice L, its rank, its bases, and its determinant
The distance between a lattice L and a vector
Furthermore, the ith successive minimum
and the Hermite factor of a basis is given as
where
At last we define the fundamental parallelepiped as follows. Let
In the following we recall the definition of LWE.
Let n and
Let
In Regev’s original definition of LWE, the attacker has access to arbitrary many LWE samples, which means that
often written as matrix
In the original definition,
Two characterizations of LWE are considered in this work: (1) the generic characterization by
In the following, let
The transformed samples can be constructed such that
and
The result is an LWE instance with errors having standard deviation
The two main hardness assumptions leading to the basic strategies of solving LWE are the Short Integer Solutions (SIS) problem and the Bounded Distance Decoding (BDD) problem. We describe both of them in the following.
The Short Integer Solutions (SIS) problem is defined as follows: Given a matrix
Solving the SIS problem with appropriate parameters solves DecisionLWE. Given m samples written as
The BDD problem is defined as follows. Given a lattice L, a target vector
An LWE instance
Basis reduction is a very important building block of most of the algorithms to solve LWE considered in this paper. It is applied to a lattice L to find a basis
Following the convention of Albrecht, Player and Scott [4], we assume that the first nonzero vector
Let L be a lattice with basis
Let
The BKZ algorithm employs an algorithm to solve several SVP instances of smaller dimension, which can be seen as an SVP oracle. The SVP oracle can be implemented by computing the Voronoi cells of the lattice, by sieving, or by enumeration. During BKZ several BKZ rounds are done. In each BKZ round an SVP oracle is called several times to receive a better basis after each round. The algorithm terminates when the quality of the basis remains unchanged after another BKZ round. The difference between BKZ and BKZ 2.0 are the usage of extreme pruning [19], early termination, limiting the enumeration radius to the Gaussian Heuristic, and local block preprocessing [14].
There exist several practical estimations of the runtime
in clock cycles, called LP model. This result should be used carefully, since applying this estimation implies the existence of a subexponential algorithm for solving LWE [4]. The estimation – shown by Albrecht, Cid, Faugère, Fitzpatrick and Perret [1] –
called deltasquared model, is nonlinear in
where ρ is the number of BKZ rounds and
where
The estimation
Under the Gaussian heuristic and geometric series assumption, the following correspondence between the block size k and
where
In this section we describe the algorithms used to estimate the hardness of LWE and analyze them regarding their computational cost. If there exists a small secret variant of an algorithm, the corresponding section is divided into general and small secret variant.
Since the goal of this paper is to investigate how the number of samples m influences the hardness of LWE, we restrict our attention to attacks that are practical for restricted m. This excludes Arora and Ge’s algorithm and BKW, which require at least subexponential m. Furthermore, we do not include purely combinatorial attacks like exhaustive search or meetinthemiddle, since there runtime is not influenced by m.
The distinguishing attack solves decisional LWE via the SIS strategy using basis reduction. For this, the dual lattice
The success probability ϵ is the advantage of distinguishing
In order to achieve a fixed success probability ϵ, a vector
is needed. Let
The logarithm of
where m is the given number of LWE samples. To estimate the runtime of the distinguishing attack, it is sufficient to determine
Model  Logarithmic runtime 
LP 

deltasquared 

Relation

Block size k in







On the one hand, the runtime of BKZ decreases exponentially with the length of
The distinguishing attack for small secrets works similar to the general case, but it exploits the smallness of the secret
Using the same reasoning as in the standard case, the required
where p can be estimated by equation (2.1). The rest of the algorithm remains the same as in the standard case. Table 3 gives the run times estimations of in the LP and the deltasquared model described in Section 3. Table 4 gives the block size k of BKZ derived in Section 3 following the second approach to estimate run times of the distinguishing attack with small secret. Combining this algorithm with exhaustive search as described in Section 2.3.1 may improve the runtime.
Model  Logarithmic runtime 
LP 

deltasquared 

Relation

Block size k in







The decoding approach solves LWE via the BDD strategy described in Section 2. The procedure considers the lattice
In the following let the target success probability be the overall success probability of the attack, chosen by the attacker (usually close to 1). In contrast, the success probability refers to the success probability of a single run of the algorithm. The target success probability is achieved by running the algorithm potentially multiple times with a certain success probability for each single run.
To solve BDD, and therefore LWE, the most basic algorithm is Babai’s Nearest Plane algorithm [9]. Given a BDD instance
The result of the algorithm is the lattice point
Hence, an attacker can adjust his overall runtime according to the tradeoff between the quality of the basis reduction and the success probability.
Lindner and Peikert [26] present a modification of the Nearest Plane algorithm named Nearest Planes. They introduce additional parameters
The success probability of the Nearest Planes algorithm is the probability of
To choose values
where
The runtime of the basis reduction is determined by
The runtime of the decoding step for Lindner and Peikert’s Nearest Planes algorithm is determined by the number of points
Since no closed formula is known to calculate the values
Since this contemplation only considers a fixed success probability, the best tradeoff between success probability and the running time of a single execution described above must be found by repeating the process above with varying values of the fixed success probability.
The decoding approach for small secrets works the same as in the general case, but it exploits the smallness of the secret
The standard embedding attack solves LWE via reduction to uSVP. The reduction is done by creating an
Let
be the qary lattice defined by the matrix
If
see [27, 16]. Therefore,
To determine the success probability and the runtime, we distinguish between two cases:
Based on Albrecht, Fitzpatrick and Göpfert [2], Göpfert shows [21, Section 3.1.3] that the standard embedding attack succeeds with nonnegligible probability if
where m is the number of LWE samples. The value τ is experimentally determined to be
In Table 5, we put all together and state the runtime for the cases from Section 3 of the standard embedding attack in the LP and the deltasquared model. Table 6 gives the block size k of BKZ derived in Section 3 following the second approach to estimate run times of the standard embedding attack.
Model  Logarithmic runtime 
LP 

deltasquared 

Relation

Block size k in







As discussed above, the success probability ϵ of a single run depends on τ and thus does not necessarily yield the desired target success probability
Consequently, it has to be considered that ρ executions of this algorithm have to be done, i.e., the runtime has to multiplied by ρ. As before we assume that the samples may be reused in each run.
To solve a small secret LWE instance based on embedding, the strategy described in Section 2.3.1 can be applied: First, modulus switching is used and afterwards the algorithm is combined with exhaustive search. The standard embedding attack on LWE with small secret using modulus switching works the same as standard embedding in the nonsmall secret case, except that it operates on instances characterized by n,
where p can be estimated by equation (2.1). As stated in the description of the standard case, the overall runtime of the algorithm is determined depending on
In Table 7, we state the runtime of the standard embedding attack in the LP and the deltasquared model. Table 8 gives the block size k of BKZ derived in Section 3 following the second approach to estimate runtime of the standard embedding attack with small secret. The success probability remains the same.
Model  Logarithmic runtime 
LP 

deltasquared 

Relation

Block size k in







Dual embedding is very similar to standard embedding shown in Section 4.3. However, since the embedding is into a different lattice, the dual embedding algorithm runs in dimension
For an LWE instance
with
to be the lattice in which uSVP is solved. Considering
and therefore
Since this attack is similar to standard embedding, the estimations of the success probability and the running time is the same except for adjustments with respect to the dimension and determinant. Hence, the dual embedding attack is successful if the rootHermite delta fulfills
while the number of LWE samples is m.
In Table 9, we state the runtime of the dual embedding attack in the LP and the deltasquared model. Table 10 gives the block size k of BKZ derived in Section 3 following the second approach to estimate runtime of the dual embedding attack.
Model  Logarithmic runtime 
LP 

deltasquared 

Relation

Block size k in







Since this algorithm is not mentioned in [4], we explain the analysis for an unlimited number of samples in the following. The case where the number of samples is not limited, and thus the optimal number of samples can be used, is a special case of the discussion above. To be more precise, to find the optimal number of samples
There are two small secret variant of the dual embedding attack: One is similar to the small secret variant of the standard embedding, the other is better known as the embedding attack by Bai and Galbraith. Both are described in the following.
As before, the strategy described in Section 2.3.1 can be applied: First, modulus switching is used and afterwards the algorithm is combined with exhaustive search. This variant works the same as dual embedding in the nonsmall secret case, except that it operates on instances characterized by n,
where p can be estimated by equation (2.1).
In Table 11, we state the runtime of the dual embedding attack with small secret in the LP and the deltasquared model. Table 12 gives the block size k of BKZ derived in Section 3 following the second approach to estimate runtime of the dual embedding attack with small secret. The success probability remains the same.
Model  Logarithmic runtime 
LP 

deltasquared 

Relation

Block size k in







The embedding attack by Bai and Galbraith [10] solves LWE with a small secret vector
for the matrix
in order to recover the short vector
To tackle this, the lattice should be scaled such that it is more balanced, i.e., the first n rows of the lattice basis are multiplied with a factor depending on σ (see [10]). Hence, the determinant of the lattice is increased by a factor of
where
In Table 13, we state the runtime of the BaiGalbraith embedding attack in the LP and the deltasquared model. Table 14 gives the block size k of BKZ derived in Section 3 following the second approach to estimate runtime of the BaiGalbraith embedding attack with small secret.
Model  Logarithmic runtime 
LP 

deltasquared 

Rel.

Block size k in







The success probability is determined similar to the standard embedding, see equation (4.3) in Section 4.3.
Similar to the other algorithms for LWE with small secret, the runtime of Bai and Galbraith’s attack can be combined with exhaustive search guessing parts of the secret. However, in contrast to the other algorithms using basis reduction, Bai and Galbraith state that applying modulus switching to their algorithm does not improve the result. The reason for this is, that modulus switching reduces q by a larger factor than it reduces the size of the error.
In this section, we describe our implementation of the results presented in Section 4 as an extension of the LWEEstimator introduced in [3, 4]. Furthermore, we compare results of our implementation focusing on the behavior when limiting the number of available LWE samples.
Our extension is also written in sage and it is already merged with the original LWEEstimator (from commitid eb45a74 on) in March 2017. In the following we used the version of the LWEEstimator from June 2017 (commitid: e0638ac) for our experiments.
Except for Arora and Ge’s algorithm based on Gröbner bases, we adapt each algorithm the LWEEstimator implements to take a fixed number of samples into account if a number of samples is given by the user. If not, each of the implemented algorithms assumes unlimited number of samples (and hence assumes the optimal number of samples is available). Our implementation also extends the algorithms codedBKW, decisionalBKW, searchBKW, and meetinthemiddle attacks (for a description of these algorithms see [4]) although we omitted the theoretical description of these algorithms in Section 4.
Following the notation in [4], we assign an abbreviation to each algorithm to refer:
dual  distinguishing attack, Section 4.1, 
dec  decoding attack, Section 4.2, 
usvpprimal  standard embedding, Section 4.3, 
usvpdual  dual embedding, Section 4.4, 
usvpbaigal  BaiGalbraith embedding, Section 4.4.2, 
usvp  minimum of usvpprimal, usvpdual, and usvpbaigal, 
mitm  exhaustive search, 
bkw  codedBKW, 
aroragb  Arora and Ge’s algorithm based on Gröbner bases. 
The shorthand symbol bkw solely refers to codedBKW and its small secret variant. DecisionBKW and SearchBKW are not assigned an abbreviation and are not used by the main method estimate_lwe, because codedBKW is the latest and most efficient BKW algorithm. Nevertheless, the other two BKW algorithms can be called separately via the function bkw, which is a convenience method for the functions bkw_search and bkw_decision, and its corresponding small secret variant bkw_small_secret.
In the LWEEstimator the three different embedding approaches usvpprimal, usvpdual, and usvpbaigal (in case of LWE with small secret is called) are summarized as the attack usvp and the minimum of the three embedding algorithms is returned. In our experiments we show the different impacts of those algorithms and hence we display the results of the three embedding approaches separately.
Let the LWE instance be defined by n,
The main function to call the LWEEstimator is called estimate_lwe. Listing 1 shows how to call the LWEEstimator on the given LWE instance (with Gaussian distributed error and secret) including the following attacks: distinguishing attack, decoding, and embedding attacks. The first two lines of Listing 1 define the parameters
Basic example of calling the LWEEstimator of the LWE instance
Listing 2 shows the estimations of the LWE instance with
Example of calling the hardness estimations of the small secret LWE instance
In the following, we give interesting insights earned during the implementation.
One problem arises in the decoding attack dec when very strictly limiting the number of samples. It uses enum_cost to calculate the computational cost of the decoding step. For this, amongst other things, the stretching factors
The original LWEEstimator routine to find the block size k for BKZ, called k_chen, iterates through possible values of k, starting at 40, until the resulting
Iteration to find k in method k_chen of the previous implementation used in the LWEEstimator.
Implementation of method k_chen to find k using the secantmethod.
In the following, we present hardness estimations of LWE with and without taking a restricted number of samples into account. The presented experiments are done for the following LWE instance:
We show the base2 logarithm of the estimated hardness of the LWE instance under all implemented attacks (except for Arora and Ge’s algorithm) in Table 15. According to the experiments, the hardness decreases with increasing the number of samples and remains the same after reaching the optimal number of samples. If our software could not find a solution the entry is filled with NaN. This is mostly due to too few samples provided to apply the respective algorithm.
Samples  mitm  dual  dec  usvpprimal  usvpdual 
100  326.5  127.3  92.3  NaN  95.7 
150  326.5  87.1  65.0  NaN  55.7 
200  326.5  77.2  57.7  263.4  49.2 
250  326.5  74.7  56.8  68.8  48.9 
300  326.5  74.7  56.8  51.4  48.9 
350  326.5  74.7  56.8  48.9  48.9 
400  326.5  74.7  56.8  48.9  48.9 
450  326.5  74.7  56.8  48.9  48.9 
Samples  bkw 

NaN 

NaN 

NaN 

NaN 

NaN 

NaN 

85.1 

85.1 

85.1 
In Table 16, we show the logarithmic hardness and the corresponding optimal number of samples estimated for unlimited number of samples. It should be noted that some algorithms rely on multiple executions, e.g., to amplify a low success probability of a single run to a target success probability. In such a case, the previous implementation of the LWEEstimator assumed new samples for each run of the algorithm. In our implementation, we assume that samples may be reused in repeated runs of the same algorithm, giving a lower bound on the hardness estimations. Hence, sometimes the optimal number of samples computed by the original LWEEstimator and the optimal number of samples computed by our method differ a lot in Table 16, e.g., decoding attack. To compensate this and to provide better comparability, we recalculate the optimal number of samples.
Optimal number of samples  
Algorithm  Original calculation  Recalculation  Hardness [bit] 
mitm  181  181  395.9 
sis  192795128  376  74.7 
dec  53436  366  58.1 
usvp  16412  373  48.9 
bkw 


85.1 
Comparing Table 15 and Table 16 shows that for a number of samples lower than the optimal number of samples, the estimated hardness is either (much) larger than the estimation using optimal number of samples or does not exist. In contrast, for a number of samples greater than or equal to the optimal number of samples, the hardness is exactly the same as in the optimal case, since the implementation falls back on the optimal number of samples when enough samples are given. Without this the hardness would increase again as can be seen for the dual embedding attack in Figure 2. For the results presented in Figure 2 we manually disabled the function to fall back to the optimal number of samples.
In Figure 3 we show the effect of limiting the available number of samples on the considered algorithms. We do not include codedBKW in this plot, since the number of required samples to apply the attack is very large (about



LP  
LWE solver 








mitm  407.0  407.0  407.0  407.0  407.0  407.0  407.0  407.0 
usvp  97.7  102.0  104.2  108.9  144.6  157.0  149.9  159.9 
dec  106.1  111.5  111.5  117.2  138.0  143.4  144.3  148.7 
dual  106.2  132.5  112.3  133.1  166.0  189.1  158.0  165.2 
bkw  212.8  –  212.8  –  212.8  –  212.8  – 
We tested and compared various proposed parameters of different primitives such as signature schemes [10, 5], encryption schemes [26, 18], and key exchange protocols [6, 13, 12]. In this section we explain our findings using an instantiation of the encryption scheme by Linder and Peikert [26] as an example. It aims at “medium security” (about 128 bits) and provides
Except bkw and mitm, all attacks considered use basis reduction as a subroutine. As explained in Section 3, several ways to predict the performance of basis reduction exist. Assuming that sieving scales as predicted to higher dimension leads to the smallest runtime estimates for BKZ on quantum (called
Our results are summarized in Table 17. We write “–” if the corresponding algorithm was not applicable for the tested instance of the Linder–Peikert scheme. Since bkw and mitm do not use basis reduction as subroutine, their runtimes are independent of the used BKZ prediction.
For the LWE instance considered, the best attack with arbitrary many samples always remains the best attack after restricting the number of samples. Restricting the samples always leads to an increased runtime for every attack, up to a factor of
In this work, we present an analysis of the hardness of LWE for the case of a restricted number of samples. For this, we describe the approaches distinguishing attack, decoding, standard embedding, and dual embedding shortly and analyze them with regard to a restricted number of samples. Also, we analyze the small secret variants of the mentioned algorithms under the same restriction of samples.
We adapt the existing software tool LWEEstimator to take the results of our analysis into account. Moreover, we also adapt the algorithms BKW and meetinthemiddle that are omitted in the theoretical description. Finally, we present examples, compare hardness estimations with optimal and restricted numbers of samples, and discuss our results.
The usage of a restricted set of samples has its limitations, e.g., if given too few samples, attacks are not applicable as in the case of BKW. On the other hand, it is possible to construct LWE instances from a given set of samples. For example, in [17] ideas how to generate additional samples (at cost of having higher noise) are presented. An integration in the LWEEstimator and comparison of those methods would give an interesting insight, since it may lead to improvements of the estimation, especially for the algorithms exhaustive search and BKW.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: CRC 1119 CROSSING
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