1 Introduction
The recent years have seen a boom in the number of security incidents in computer networks. In order to alleviate the impact of network attacks and intrusions, Internet providers want to detect malicious traffic at their network’s entry points and on the backbones between subnetworks. Softwarebased network intrusion detection systems (NIDSes), such as the popular opensource system Snort [1], are capable of detecting suspicious network traffic by testing (among others) whether a packet payload matches a regular expression (regex) describing known patterns of malicious traffic. NIDSes collect and maintain vast databases of such regexes that are typically divided into groups according to types of the attacks and target protocols.
Regex matching is the most computationally demanding task of a NIDS as its cost grows with the speed of the network traffic as well as with the number and complexity of the regexes being matched. The current softwarebased NIDSes cannot perform the regex matching on networks beyond 1 Gbps [2, 3], so they cannot handle the current speed of backbone networks ranging between tens and hundreds of Gbps. A promising approach to speed up NIDSes is to (partially) offload regex matching into hardware [3, 4, 5]. The hardware then serves as a prefilter of the network traffic, discarding the majority of the packets from further processing. Such prefiltering can easily reduce the traffic the NIDS needs to handle by two or three orders of magnitude [3].
Fieldprogrammable gate arrays (FPGAs) are the leading technology in highthroughput regex matching. Due to their inherent parallelism, FPGAs provide an efficient way of implementing nondeterministic finite automata (NFAs), which naturally arise from the input regexes. Although the amount of available resources in FPGAs is continually increasing, the speed of networks grows even faster. Working with multigigabit networks requires the hardware to use many parallel packet processing branches in a single FPGA [5]; each of them implementing a separate copy of the concerned NFA, and so reducing the size of the NFAs is of the utmost importance. Various languagepreserving automata reduction approaches exist, mainly based on computing (bi)simulation relations on automata states (cf. the related work). The reductions they offer, however, do not satisfy the needs of highspeed hardwareaccelerated NIDSes.
Our answer to the problem is approximate reduction of NFAs, allowing for a tradeoff between the achieved reduction and the precision of the regex matching. To formalise the intuitive notion of precision, we propose a novel probabilistic distance of automata. It captures the probability that a packet of the input network traffic is incorrectly accepted or rejected by the approximated NFA. The distance assumes a probabilistic model of the network traffic (we show later how such a model can be obtained).
Having formalised the notion of precision, we specify the target of our reductions as two variants of an optimization problem: (1) minimizing the NFA size given the maximum allowed error (distance from the original), or (2) minimizing the error given the maximum allowed NFA size. Finding such optimal approximations is, however, computationally hard (PSPACEcomplete, the same as precise NFA minimization).
Consequently, we sacrifice the optimality and, motivated by the typical structure of NFAs that emerge from a set of regexes used by NIDSes (a union of many long “tentacles” with occasional small stronglyconnected components), we limit the space of possible reductions by restricting the set of operations they can apply to the original automaton. Namely, we consider two reduction operations: (i) collapsing the future of a state into a selfloop (this reduction overapproximates the language), or (ii) removing states(such a reduction is underapproximating).
The problem of identifying the optimal sets of states on which these operations should be applied is still PSPACEcomplete. The restricted problem is, however, more amenable to an approximation by a greedy algorithm. The algorithm applies the reductions statebystate in an order determined by a precomputed error labelling of the states. The process is stoppped once the given optimization goal in terms of the size or error is reached. The labelling is based on the probability of packets that may be accepted through a given state and hence overapproximates the error that may be caused by applying the reduction at a given state. As our experiments show, this approach can give us highquality reductions while ensuring formal error bounds.
Finally, it turns out that even the precomputation of the error labelling of the states is costly (again PSPACEcomplete). Therefore, we propose several ways to cheaply overapproximate it such that the strong error bound guarantees are still preserved. Particularly, we are able to exploit the typical structure of the “union of tentacles” of the hardware NFA in an algorithm that is exponential in the size of the largest “tentacle” only, which is indeed much faster in practice.
We have implemented our approach and evaluated it on regexes used to classify malicious traffic in Snort. We obtain quite encouraging experimental results demonstrating that our approach provides a much better reduction than languagepreserving techniques with an almost negligible error. In particular, our experiments, going down to the level of an actual implementation of NFAs in FPGAs, confirm that we can squeeze into an uptodate FPGA chip reallife regexes encoding malicious traffic, allowing them to be used with a negligible error for filtering at speeds of 100 Gbps (and even 400 Gbps). This is far beyond what one can achieve with current exact reduction approaches.
Related Work
Hardware acceleration for regex matching at the line rate is an intensively studied technology that uses generalpurpose hardware [6, 7, 8, 9, 10, 11, 12, 13, 14] as well as FPGAs [15, 16, 17, 18, 19, 20, 3, 4, 5]. Most of the works focus on DFA implementation and optimization techniques. NFAs can be exponentially smaller than DFAs but need, in the worst case, memory accesses to process each byte of the payload where is the number of states. In most cases, this incurs an unacceptable slowdown. Several works alleviate this disadvantage of NFAs by exploiting reconfigurability and finegrained parallelism of FPGAs, allowing one to process one character per clock cycle (e.g. [15, 16, 19, 20, 3, 4, 5]).
In [14], which is probably the closest work to ours, the authors consider a set of regexes describing network attacks. They replace a potentially prohibitively large DFA by a tree of smaller DFAs, an alternative to using NFAs that minimizes the latency occurring in a nonFPGAbased implementation. The language of every DFAnode in the tree overapproximates the languages of its children. Packets are filtered through the tree from the root downwards until they belong to the language of the encountered nodes, and may be finally accepted at the leaves, or are rejected otherwise. The overapproximating DFAs are constructed using a similar notion of probability of an occurrence of a state as in our approach. The main differences from our work are that (1) the approach targets approximation of DFAs (not NFAs), (2) the overapproximation is based on a given traffic sample only (it cannot benefit from a probabilistic model), and (3) no probabilistic guarantees on the approximation error are provided.
Approximation of DFAs was considered in various other contexts. Hyperminimization is an approach that is allowed to alter language membership of a finite set of words [21, 22]. A DFA with a given maximum number of states is constructed in [23]
, minimizing the error defined either by (i) counting prefixes of misjudged words up to some length, or (ii) the sum of the probabilities of the misjudged words wrt the Poisson distribution over
. Neither of these approaches considers reduction of NFAs nor allows to control the expected error with respect to the real traffic.In addition to the metrics mentioned above when discussing the works [23, 21, 22], the following metrics should also be mentioned. The CesaroJaccard distance studied in [24] is, in spirit, similar to [23] and does also not reflect the probability of individual words. The edit distance of weighted automata from [25] depends on the minimum edit distance between pairs of words from the two compared languages, again regardless of their statistical significance. None of these notions is suitable for our needs.
Languagepreserving minimization of NFAs is a PSPACEcomplete problem [26, 27]. More feasible (polynomialtime) is languagepreserving size reduction of NFAs based on (bi)simulations [28, 29, 30, 31], which does not aim for a truly minimal NFA. A number of advanced variants exist, based on multipebble or lookahead simulations, or on combinations of forward and backward simulations [32, 33, 34]. The practical efficiency of these techniques is, however, often insufficient to allow them to handle the large NFAs that occur in practice and/or they do not manage to reduce the NFAs enough. Finally, even a minimal NFA for the given set of regexes is often too big to be implemented in the given FPGA operating on the required speed (as shown even in our experiments). Our approach is capable of a much better reduction for the price of a small change of the accepted language.
2 Preliminaries
We use to denote the set and to denote the set . Given a pair of sets and , we use to denote their symmetric difference, i.e., the set . We use the notation
to denote a vector of
elements, to denote the all 1’s vector , to denote a matrix, and for its transpose, andfor the identity matrix.
In the following, we fix a finite nonempty alphabet . A nondeterministic finite automaton (NFA) is a quadruple where is a finite set of states, is a transition function, is a set of initial states, and is a set of accepting states. We use , and to denote , and , respectively, and to denote that . A sequence of states is a run of over a word from a state to a state , denoted as , if , , and . Sometimes, we use in set operations where it behaves as the set of states it contains. We also use to denote that and to denote that . The language of a state is defined as and its banguage (backlanguage) is defined as . Both notions can be naturally extended to a set : and . We drop the subscript when the context is obvious. accepts the language defined as . is called deterministic (DFA) if and and , and unambiguous (UFA) if .
The restriction of to is an NFA given as . We define the trim operation as where . For a set of states , we use to denote the set of states reachable from , formally, . We use the number of states as the measurement of the size of , i.e., .
A (discrete probability) distribution over a set is a mapping such that . An state probabilistic automaton (PA) over is a triple where is a vector of initial weights, is a vector of final weights, and for every , is a transition matrix for symbol . We abuse notation and use to denote the set of states . Moreover, the following two properties need to hold: (i) (the initial probability is 1) and (ii) for every state it holds that (the probability of accepting or leaving a state is 1). We define the support of as the NFA s.t.
Let us assume that every PA is such that . For a word , we use to denote the matrix . It can be easily shown that represents a distribution over words defined as . We call the probability of in . Given a language , we define the probability of in as .
If Conditions (i) and (ii) from the definition of PAs are dropped, we speak about a pseudoprobabilistic automaton (PPA), which may assign a word from its support a quantity that is not necessarily in the range , denoted as the significance of the word below. PPAs may arise during some of our operations performed on PAs.
3 Approximate Reduction of NFAs
In this section, we first introduce the key notion of our approach: a probabilistic distance of a pair of finite automata wrt a given probabilistic automaton that, intuitively, represents the significance of particular words. We discuss the complexity of computing the probabilistic distance. Finally, we formulate two problems of approximate automata reduction via probabilistic distance.
3.1 Probabilistic Distance
We start by defining our notion of a probabilistic distance of two NFAs. Assume NFAs and and a probabilistic automaton specifying the distribution . The probabilistic distance between and wrt is defined as
Intuitively, the distance captures the significance of the words accepted by one of the automata only. We use the distance to drive the reduction process towards automata with small errors and to assess the quality of the resulting automata.
The value of can be computed as follows. Using the fact that (1) and (2) , we get
Hence, the key step is to compute for an NFA and a PA . Problems similar to computing such a probability have been extensively studied in several contexts including verification of probabilistic systems [35, 36, 37]. The below lemma summarises the complexity of this step.
Lemma 1 ()
Let be a PA and an NFA. The problem of computing is PSPACEcomplete. For a UFA , can be computed in PTIME.
In our approach, we apply the method of [37] and compute in the following way. We first check whether the NFA is unambiguous. This can be done by using the standard product construction (denoted as ) for computing the intersection of the NFA with itself and trimming the result, formally , followed by a check whether there is some state s.t. [38]. If is ambiguous, we either determinise it or disambiguate it [38], leading to a DFA/UFA , respectively.^{1}^{1}1In theory, disambiguation can produce smaller automata, but, in our experiments, determinisation proved to work better. Then, we construct the trimmed product of and (this can be seen as computing while keeping the probabilities from on the edges of the result), yielding a PPA .^{2}^{2}2 is not necessarily a PA since there might be transitions in that are either removed or copied several times in the product construction. Intuitively, represents not only the words of but also their probability in . Now, let be the matrix that expresses, for any , the significance of getting from to via any . Further, it can be shown (cf. the proof of Lemma 1 in the Appendix) that the matrix , representing the significance of going from to via any , can be computed as . Then, to get , it suffices to take . Note that, due to the determinisation/disambiguation step, the obtained value indeed is despite being a PPA.
3.2 Automata Reduction using Probabilistic Distance
We now exploit the above introduced probabilistic distance to formulate the task of approximate reduction of NFAs as the following two optimisation problems. Given an NFA and a PA specifying the distribution , we define

sizedriven reduction: for , find an NFA such that and the distance is minimal,

errordriven reduction: for , find an NFA such that and the size is minimal.
The following lemma shows that the natural decision problem underlying both of the above optimization problems is PSPACEcomplete, which matches the complexity of computing the probabilistic distance as well as that of the exact reduction of NFAs [26].
Lemma 2 ()
Consider an NFA , a PA , a bound on the number of states , and an error bound . It is PSPACEcomplete to determine whether there exists an NFA with states s.t. .
The notions defined above do not distinguish between introducing a false positive ( accepts a word ) or a false negative ( does not accept a word ) answers. To this end, we define overapproximating and underapproximating reductions as reductions for which the additional conditions and hold, respectively.
A naïve solution to the reductions would enumerate all NFAs of sizes from 0 up to (resp. ), for each of them compute , and take an automaton with the smallest probabilistic distance (resp. a smallest one satisfying the restriction on ). Obviously, this approach is computationally infeasible.
4 A Heuristic Approach to Approximate Reduction
In this section, we introduce two techniques for approximate reduction of NFAs that avoid the need to iterate over all automata of a certain size. The first approach underapproximates the automata by removing states—we call it the pruning reduction—while the second approach overapproximates the automata by adding selfloops to states and removing redundant states—we call it the selfloop reduction. Finding an optimal automaton using these reductions is also PSPACE
complete, but more amenable to heuristics like greedy algorithms. We start with introducing two highlevel greedy algorithms, one for the size and one for the errordriven reduction, and follow by showing their instantiations for the pruning and the selfloop reduction. A crucial role in the algorithms is played by a function that labels states of the automata by an estimate of the error that will be caused when some of the reductions is applied at a given state.
4.1 A General Algorithm for SizeDriven Reduction
Algorithm 1 shows a general greedy method for performing the sizedriven reduction. In order to use the same highlevel algorithm in both directions of reduction (over/underapproximating), it is parameterized with three functions: , and . The real intricacy of the procedure is hidden inside these three functions. Intuitively, assigns every state of an NFA an approximation of the error that will be caused wrt the PA when a reduction is applied at this state, while the purpose of is to create a new NFA obtained from by introducing some error at states from .^{3}^{3}3We emphasize that this does not mean that states from will be simply removed from —the performed operation depends on the particular reduction. Further, estimates the error introduced by the application of , possibly in a more precise (and costly) way than by just summing the concerned error labels: Such a computation is possible outside of the main computation loop. We show instantiations of these functions later, when discussing the reductions used. Moreover, the algorithm is also parameterized with a total order that defines which states of are processed first and which are processed later. The ordering may take into account the precomputed labelling. The algorithm accepts an NFA , a PA , and and outputs a pair consisting of an NFA of the size and an error bound such that .
The main idea of the algorithm is that it creates a set of states where an error is to be introduced. is constructed by starting from an empty set and adding states to it in the order given by , until the size of the result of has reached the desired bound (in our setting, is always antitone, i.e., for , it holds that ). We now define the necessary condition for , and that makes Algorithm 1 correct.
Condition C1 holds if for every NFA , PA , and a set , we have that (a) , (b) , and (c) .
C1(a) ensures that the error computed by the reduction algorithm indeed overapproximates the exact probabilistic distance, C1(b) ensures that the algorithm can (in the worst case, by applying the reduction at every state of ) for any output a result of the size , and C1(c) ensures that when no error is to be introduced at any state, we obtain the original automaton.
Lemma 3
Algorithm 1 is correct if C1 holds.
Proof
Follows straightforwardly from Condition C1. ∎
4.2 A General Algorithm for ErrorDriven Reduction
In Algorithm 2, we provide a highlevel method of computing the errordriven reduction. The algorithm is in many ways similar to Algorithm 1; it also computes a set ofstates where an error is to be introduced, but an important difference is that we compute an approximation of the error in each step and only add to if it does not raise the error over the threshold . Note that the does not need to be monotone, so it may be advantageous to traverse all states from and not terminate as soon as the threshold is reached. The correctness of Algorithm 2 also depends on C1.
Lemma 4
Algorithm 2 is correct if C1 holds.
Proof
Follows straightforwardly from Condition C1. ∎
4.3 Pruning Reduction
The pruning reduction is based on identifying a set of states to be removed from an NFA , underapproximating the language of . In particular, for , the pruning reduction finds a set and restricts to , followed by removing useless states, to construct a reduced automaton . Note that the natural decision problem corresponding to this reduction is also PSPACEcomplete.
Lemma 5 ()
Consider an NFA , a PA , a bound on the number of states , and an error bound . It is PSPACEcomplete to determine whether there exists a subset of states of the size such that .
Although Lemma 5 shows that the pruning reduction is as hard as a general reduction (cf. Lemma 2), the pruning reduction is more amenable to the use of heuristics like the greedy algorithms from §4.1 and §4.2. We instantiate , , and in these highlevel algorithms in the following way (the subscript means pruning):
where is defined as follows. Because of the use of in , for a pair of sets s.t. , it holds that may, in general, yield the same automaton as . Hence, we define a partial order on as iff and , and use to denote the set of minimal elements wrt and . The value of the approximation is therefore the minimum of the sum of errors over all sets from .
Note that the size of can again be exponential, and thus we employ a greedy approach for guessing an optimal . Clearly, this cannot affect the soundness of the algorithm, but only decreases the precision of the bound on the distance. Our experiments indicate that for automata appearing in NIDSes, this simplification has typically only a negligible impact on the precision of the bounds.
For computing the state labelling, we provide the following three functions, which differ in the precision they provide and the difficulty of their computation (naturally, more precise labellings are harder to compute): , and . Given an NFA and a PA , they generate the labellings , and , respectively, defined as
A state label approximates the error of the words removed from when is removed. More concretely, is a rough estimate saying that the error can be bounded by the sum of probabilities of the banguages of all final states reachable from (in the worst case, all those final states might become unreachable). Note that (1) counts the error of a word accepted in two different final states of twice, and (2) also considers words that are accepted in some final state in without going through . The labelling deals with (1) by computing the total probability of the banguage of the set of all final states reachable from , and the labelling in addition also deals with (2) by only considering words that traverse through (they can still be accepted in some final state not in though, so even is still imprecise). Note that if is unambiguous then .
When computing the label of , we first modify to obtain accepting the language related to the particular labelling. Then, we compute the value of using the algorithm from §3.1. Recall that this step is in general costly, due to the determinisation/disambiguation of . The key property of the labelling computation resides in the fact that if is composed of several disjoint subautomata, the automaton is typically much smaller than and thus the computation of the label is considerable less demanding. Since the automata appearing in regex matching for NIDS are composed of the union of “tentacles”, the particular s are very small, which enables efficient componentwise computation of the labels.
The following lemma states the correctness of using the pruning reduction as an instantiation of Algorithms 1 and 2 and also the relation among , , and .
Lemma 6 ()
For every , the functions , , and satisfy C1. Moreover, consider an NFA , a PA , and let for . Then, for each , we have .
4.4 Selfloop Reduction
The main idea of the selfloop reduction is to overapproximate the language of by adding selfloops over every symbol at selected states. This makes some states of redundant, allowing them to be removed without introducing any more error. Given an NFA , the selfloop reduction searches for a set of states , which will have selfloops added, and removes other transitions leading out of these states, making some states unreachable. The unreachable states are then removed.
Formally, let be the NFA whose transition function is defined, for all and , as if and otherwise. As with the pruning reduction, the natural decision problem corresponding to the selfloop reduction is also PSPACEcomplete.
Lemma 7 ()
Consider an NFA , a PA , a bound on the number of states , and an error bound . It is PSPACEcomplete to determine whether there exists a subset of states of the size such that .
The required functions in the error and sizedriven reduction algorithms are instantiated in the following way (the subcript means selfloop):
where is defined in a similar manner as in the previous section (using a partial order defined similarly to ; in this case, the order has a single minimal element, though).
The functions , and compute the state labellings , and for an NFA and a PA defined as follows:
Above, for a PA and a word is defined as (i.e., similarly as but with the final weights discarded), and for is defined as .
Intuitively, the state labelling computes the probability that is reached from an initial state, so if is pumped up with all possible word endings, this is the maximum possible error introduced by the added word endings. This has the following sources of imprecision: (1) the probability of some words may be included twice, e.g., when , the probabilities of all words from are included twice in because , and (2) can also contain probabilities of words that are already accepted on a run traversing . The state labelling deals with (1) by considering the probability of the language , and deals also with (2) by subtracting from the result of the probabilities of the words that pass through and are accepted.
The computation of the state labellings for the selfloop reduction is done in a similar way as the computation of the state labellings for the pruning reduction (cf. §4.3). For a computation of one can use the same algorithm as for , only the final vector for PA is set to . The correctness of Algorithms 1 and 2 when instantiated using the selfloop reduction is stated in the following lemma.
Lemma 8 ()
For every , the functions , , and satisfy C1. Moreover, consider an NFA , a PA , and let for . Then, for each , we have .
5 Reduction of NFAs in Network Intrusion Detection Systems
We have implemented our approach in a Python prototype named Appreal (APProximate REduction of Automata and Languages)^{4}^{4}4https://github.com/vhavlena/appreal/tree/tacas18 and evaluated it on the use case of network intrusion detection using Snort [1], a popular open source NIDS. The version of Appreal used for the evaluation in the current paper is available as an artifact [39] for the TACAS’18 artifact virtual machine [40].
5.1 Network Traffic Model
The reduction we describe in this paper is driven by a probabilistic model representing a distribution over , and the formal guarantees are also wrt this model. We use learning to obtain a model of network traffic over the 8bit ASCII alphabet at a given network point. Our model is created from several gigabytes of network traffic from a measuring point of the CESNET Internet provider connected to a 100 Gbps backbone link (unfortunately, we cannot provide the traffic dump since it may contain sensitive data).
Learning a PA representing the network traffic faithfully is hard. The PA cannot be too specific—although the number of different packets that can occur is finite, it is still extremely large (a conservative estimate assuming the most common scenario Ethernet/IPv4/TCP would still yield a number over ). If we assigned nonzero probabilities only to the packets from the dump (which are less than ), the obtained model would completely ignore virtually all packets that might appear on the network, and, moreover, the model would also be very large (millions of states), making it difficult to use in our algorithms. A generalization of the obtained traffic is therefore needed.
A natural solution is to exploit results from the area of PA learning, such as [41, 42]. Indeed, we experimented with the use of Alergia [41], a learning algorithm that constructs a PA from a prefix tree (where edges are labelled with multiplicities) by merging nodes that are “similar.” The automata that we obtained were, however, too general. In particular, the constructed automata destroyed the structure of network protocols—the merging was too permissive and the generalization merged distant states, which introduced loops over a very large substructure in the automaton (such a case usually does not correspond to the design of network protocols). As a result, the obtained PA more or less represented the Poisson distribution, having essentially no value for us.
In §5.2, we focus on the detection of malicious traffic transmitted over HTTP. We take advantage of this fact and create a PA representing the traffic while taking into account the structure of HTTP. We start by manually creating a DFA that represents the highlevel structure of HTTP. Then, we proceed by feeding 34,191 HTTP packets from our sample into the DFA, at the same time taking notes about how many times every state is reached and how many times every transition is taken. The resulting PA (of 52 states) is then obtained from the DFA and the labels in the obvious way.
The described method yields automata that are much better than those obtained using Alergia in our experiments. A disadvantage of the method is that it is only semiautomatic—the basic DFA needed to be provided by an expert. We have yet to find an algorithm that would suit our needs for learning more general network traffic.
5.2 Evaluation
We start this section by introducing the experimental setting, namely, the integration of our reduction techniques into the tool chain implementing efficient regex matching, the concrete settings of Appreal, and the evaluation environment. Afterwards, we discuss the results evaluating the quality of the obtained approximate reductions as well as of the provided error bounds. Finally, we present the performance of our approach and discuss its key aspects. Due to the lack of space, we selected the most interesting results demonstrating the potential as well as the limitations of our approach.
General setting.
Snort detects malicious network traffic based on rules that contain conditions. The conditions may take into consideration, among others, network addresses, ports, or Perl compatible regular expressions (PCREs) that the packet payload should match. In our evaluation, we always select a subset of Snort rules, extract the PCREs from them, and use Netbench [20] to transform them into a single NFA . Before applying Appreal, we use the stateoftheart NFA reduction tool Reduce [43] to decrease the size of . Reduce performs a languagepreserving reduction of using advanced variants of simulation [32] (in the experiment reported in Table 3, we skip the use of Reduce at this step as discussed in the performance evaluation). The automaton obtained as the result of Reduce is the input of Appreal, which performs one of the approximate reductions from §4 wrt the traffic model , yielding . After the approximate reduction, we, one more time, use Reduce and obtain the result .
Settings of Appreal.
In the use case of NIDS prefiltering, it may be important to never introduce a false negative, i.e., to never drop a malicious packet. Therefore, we focus our evaluation on the selfloop reduction (§4.4). In particular, we use the state labelling function , since it provides a good tradeoff between the precision and the computational demands (recall that the computation of can exploit the “tentacle” structure of the NFAs we work with). We give more attention to the sizedriven reduction (§4.1) since, in our setting, a bound on the available FPGA resources is typically given and the task is to create an NFA with the smallest error that fits inside. The order over states used in §4.1 and §4.2 is defined as .
Evaluation environment.
All experiments run on a 64bit Linux Debian workstation with the Intel Core(TM) i5661 CPU running at 3.33 GHz with 16 GiB of RAM.
Description of tables.
In the caption of every table, we provide the name of the input file (in the directory regexps/tacas18/ of the repository of Appreal) with the selection of Snort regexes used in the particular experiment, together with the type of the reduction (size or errordriven). All reductions are overapproximating (selfloop reduction). We further provide the size of the input automaton , the size after the initial processing by Reduce (), and the time of this reduction (). Finally, we list the times of computing the state labelling on (), the exact probabilistic distance (), and also the number of lookup tables () consumed on the targeted FPGA (Xilinx Virtex 7 H580T) when was synthesized (more on this in §5.3). The meaning of the columns in the tables is the following:

is the parameter of the reduction. In particular, is used for the sizedriven reduction and denotes the desired reduction ration for an input NFA and the desired size of the output . On the other hand, is the desired maximum error on the output for the errordriven reduction.

shows the number of states of the automaton after the reduction by Appreal and the time the reduction took (we omit it when it is not interesting).

contains the number of states of the NFA obtained after applying Reduce on and the time used by Reduce at this step (omitted when not interesting).

shows the estimation of the error of as determined by the reduction itself, i.e., it is the probabilistic distance computed by the function in §4.

contains the values of that we computed after the reduction in order to evaluate the precision of the result given in Error bound. The computation of this value is very expensive () since it inherently requires determinisation of the whole automaton . We do not provide it in Table 3 (presenting the results for the automaton with 1,352 states) because the determinisation ran out of memory (the step is not required in the reduction process).

shows the error that we obtained when compared with on an HTTP traffic sample, in particular the ratio of packets misclassified by to the total number of packets in the sample (242,468). Comparing Exact error with Traffic error gives us a feedback about the fidelity of the traffic model . We note that there are no guarantees on the relationship between Exact error and Traffic error.

is the number of LUTs consumed by when synthesized into the FPGA. Hardware synthesis is a costly step so we provide this value only for selected NFAs.
center Error Exact Traffic bound error error LUTs 0.1 9 (0.65 s) 9 (0.4 s) 0.0704 0.0704 0.0685 — 0.2 19 (0.66 s) 19 (0.5 s) 0.0677 0.0677 0.0648 — 0.3 29 (0.69 s) 26 (0.9 s) 0.0279 0.0278 0.0598 154 0.4 39 (0.68 s) 36 (1.1 s) 0.0032 0.0032 0.0008 — 0.5 49 (0.68 s) 44 (1.4 s) 2.8e05 2.8e05 4.1e06 — 0.6 58 (0.69 s) 49 (1.7 s) 8.7e08 8.7e08 0.0 224 0.8 78 (0.69 s) 75 (2.7 s) 2.4e17 2.4e17 0.0 297 
center Error Exact Traffic bound error error 0.08 3 3 0.0724 0.0724 0.0720 0.07 4 4 0.0700 0.0700 0.0683 0.04 35 32 0.0267 0.0212 0.0036 0.02 36 33 0.0105 0.0096 0.0032 0.001 41 38 0.0005 0.0005 0.0003 1e04 47 41 7.7e05 7.7e05 1.2e05 1e05 51 47 6.6e06 6.6e06 0.0 
s, s, 3.8–6.5 s, and
.
Approximation errors
center Error Exact Traffic bound error error 0.1 11 (1.1s) 5 (0.4s) 1.0 0.9972 0.9957 0.2 22 (1.1s) 14 (0.6s) 1.0 0.8341 0.2313 0.3 33 (1.1s) 24 (0.7s) 0.081 0.0770 0.0067 0.4 44 (1.1s) 37 (1.6s) 0.0005 0.0005 0.0010 0.5 56 (1.1s) 49 (1.2s) 3.3e06 3.306 0.0010 0.6 67 (1.1s) 61 (1.9s) 1.2e09 1.2e09 8.7e05 0.7 78 (1.1s) 72 (2.4s) 4.8e12 4.8e12 1.2e05 0.9 100 (1.1s) 93 (4.7s) 3.7e16 1.1e15 0.0
Table 1 presents the results of the selfloop reduction for the NFA describing regexes from httpmalicious. We can observe that the differences between the upper bounds on the probabilistic distance and its real value are negligible (typically in the order of or less). We can also see that the probabilistic distance agrees with the traffic error. This indicates a good quality of the traffic model employed in the reduction process. Further, we can see that our approach can provide useful tradeoffs between the reduction error and the reduction factor. Finally, Table 1(b) shows that a significant reduction is obtained when the error threshold is increased from 0.04 to 0.07.
Table 2 presents the results of the sizedriven selfloop reduction for NFA describing httpattacks regexes. We can observe that the error bounds provide again a very good approximation of the real probabilistic distance. On the other hand, the difference between the probabilistic distance and the traffic error is larger than for . Since all experiments use the same probabilistic automaton and the same traffic, this discrepancy is accounted to the different set of packets that are incorrectly accepted by . If the probability of these packets is adequately captured in the traffic model, the difference between the distance and the traffic error is small and vice versa. This also explains an even larger difference in Table 3 (presenting the results for constructed from httpbackdoor regexes) for . Here, the traffic error is very small and caused by a small set of packets (approx. 70), whose probability is not correctly captured in the traffic model. Despite this problem, the results clearly show that our approach still provides significant reductions while keeping the traffic error small: about a 5fold reduction is obtained for the traffic error 0.03 % and a 10fold reduction is obtained for the traffic error 6.3 %. We discuss the practical impact of such a reduction in §5.3.
Performance of the approximate reduction
center Error Traffic bound error LUTs 0.1 135 (1.2 m) 8 (2.6 s) 1.0 0.997 202 0.2 270 (1.2 m) 111 (5.2 s) 0.0012 0.0631 579 0.3 405 (1.2 m) 233 (9.8 s) 3.4e08 0.0003 894 0.4 540 (1.3 m) 351 (21.7 s) 1.0e12 0.0003 1063 0.5 676 (1.3 m) 473 (41.8 s) 1.2e17 0.0 1249 0.7 946 (1.4 m) 739 (2.1 m) 8.3e30 0.0 1735 0.9 1216 (1.5 m) 983 (5.6 m) 1.3e52 0.0 2033
In all our experiments (Tables 1–3), we can observe that the most timeconsuming step of the reduction process is the computation of state labellings (it takes at least 90 % of the total time). The crucial observation is that the structure of the NFAs fundamentally affects the performance of this step. Although after Reduce, the size of is very similar to the size of , computing takes more time (28.3 min vs. 38.7 s). The key reason behind this slowdown is the determinisation (or alternatively disambiguation) process required by the product construction underlying the state labelling computation (cf. §4.4). For , the process results in a significantly larger product when compared to the product for . The size of the product directly determines the time and space complexity of solving the linear equation system required for computing the state labelling.
As explained in §4, the computation of the state labelling can exploit the “tentacle” structure of the NFAs appearing in NIDSes and thus can be done componentwise. On the other hand, our experiments reveal that the use of Reduce typically breaks this structure and thus the componentwise computation cannot be effectively used. For the NFA , this behaviour does not have any major performance impact as the determinisation leads to a moderatesized automaton and the state labelling computation takes less than 40 s. On the other hand, this behaviour has a dramatic effect for the NFA . By disabling the initial application of Reduce and thus preserving the original structure of , we were able to speed up the state label computation from 28.3 min to 1.5 min. Note that other steps of the approximate reduction took a similar time as before disabling Reduce and also that the tradeoffs between the error and the reduction factor were similar. Surprisingly, disabling Reduce caused that the computation of the exact probabilistic distance became computationally infeasible because the determinisation ran out of memory.
Due to the size of the NFA , the impact of disabling the initial application of Reduce is even more fundamental. In particular, computing the state labelling took only 19.9 min, in contrast to running out of memory when the Reduce is applied in the first step (therefore, the input automaton is not processed by Reduce in Table 3; we still give the number of LUTs of its reduced version for comparison, though). Note that the size of also slows down other reduction steps (the greedy algorithm and the final Reduce reduction). We can, however, clearly see that computing the state labelling is still the most timeconsuming step.
5.3 The Real Impact in an FPGAAccelerated NIDS
Further, we also evaluated some of the obtained automata in the setting of [5] implementing a highspeed NIDS prefilter. In that setting, the amount of resources available for the regex matching engine is 15,000 LUTs^{5}^{5}5We omit the analysis of flipflop consumption because in our setting it is dominated by the LUT consumption. and the frequency of the engine is 200 MHz. We synthesized NFAs that use a 32bitwide data path, corresponding to processing 4 ASCII characters at once, which is—according to the analysis in [5]—the best tradeoff between the utilization of the chip resources and the maximum achievable frequency. A simple analysis shows that the throughput of one automaton is 6.4 Gbps, so in order to reach the desired link speed of 100 Gbps, 16 units are required, and 63 units are needed to handle 400 Gbps. With the given amount of LUTs, we are therefore bounded by 937 LUTs for 100 Gbps and 238 LUTs for 400 Gbps.
We focused on the consumption of LUTs by an implementation of the regex matching engines for httpbackdoor () and httpmalicious ().

100 Gbps: For this speed, can be used without any approximate reduction as it is small enough to fit in the available space. On the other hand, without the approximate reduction is way too large to fit (at most 6 units fit inside the available space, yielding the throughput of only 38.4 Gbps, which is unacceptable). The column LUTs in Table 3 shows that using our framework, we are able to reduce such that it uses 894 LUTs (for = 0.3), and so all the needed 16 units fit into the FPGA, yielding the throughput over 100 Gbps and the theoretical error bound of a false positive wrt the model .

400 Gbps: Regex matching at this speed is extremely challenging. The only reduced version of that fits in the available space is the one for the value with the error bound almost 1. The situation is better for . In the exact version, at most 39 units can fit inside the FPGA with the maximum throughput of 249.6 Gbps. On the other hand, when using our approximate reduction framework, we are able to place 63 units into the FPGA, each of the size 224 LUTs ( = 0.6) with the throughput over 400 Gbps and the theoretical error bound of a false positive wrt the model .
6 Conclusion
We have proposed a novel approach for approximate reduction of NFAs used in network traffic filtering. Our approach is based on a proposal of a probabilistic distance of the original and reduced automaton using a probabilistic model of the input network traffic, which characterizes the significance of particular packets. We characterized the computational complexity of approximate reductions based on the described distance and proposed a sequence of heuristics allowing one to perform the approximate reduction in an efficient way. Our experimental results are quite encouraging and show that we can often achieve a very significant reduction for a negligible loss of precision. We showed that using our approach, FPGAaccelerated network filtering on large traffic speeds can be applied on regexes of malicious traffic where it could not be applied before.
In the future, we plan to investigate other approximate reductions of the NFAs, maybe using some variant of abstraction from abstract regular model checking [44], adapted for the given probabilistic setting. Another important issue for the future is to develop better ways of learning a suitable probabilistic model of the input traffic.
Data Availability Statement and Acknowledgements.
The tool used for the experimental evaluation in the current study is available in the following figshare repository: https://doi.org/10.6084/m9.figshare.5907055. We thank Jan Kořenek, Vlastimil Košař, and Denis Matoušek for their help with translating regexes into automata and synthesis of FPGA designs, and Martin Žádník for providing us with the backbone network traffic. We thank Stefan Kiefer for helping us proving the PSPACE part of Lemma 1 and Petr Peringer for testing our artifact. The work on this paper was supported by the Czech Science Foundation project 1617538S, the IT4IXS: IT4Innovations Excellence in Science project (LQ1602), and the FIT BUT internal project FITS174014.
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Appendix 0.A Proofs of Lemmas
Some of the proofs use the PA defined as where .
models an exponential distribution over the words from
(wrt their length). In particular, assigns every word the probability . We use to assign every word over a nonzero probability; any other PA with the same property would work as well.See 1
Proof
We prove the first and the second part of the lemma independently.

Computing is PSPACEcomplete for an NFA . The membership in PSPACE can be shown as follows. The computation described at the end of §3.1 corresponds to solving a linear equation system. The system has an exponential size because of the blowup caused by the determinisation/disambiguation of required by the product construction. The equation system can, however, be constructed by a PSPACE transducer . Moreover, as solving linear equation systems can be done using a polylogarithmicspace transducer , one can combine these two transducers to obtain a PSPACE algorithm. Details of the construction follow:

First, we construct a transducer that, given an NFA and a PA on its input, constructs a system of linear equations of unknowns for and representing the product of and , where is a deterministic automaton obtained from using the standard subset construction. The system is defined as follows (cf. [37]):
