--- type: Article title: "Counting in Regexes Considered Harmful: Exposing ReDoS Vulnerability of Nonbacktracking Matchers" resource: "https://www.usenix.org/conference/usenixsecurity22/presentation/turonova" tags: [article, webseclist-reference, en, usenix-org] generated: by: webseclist-refs/1 at: "2026-08-19T16:23:13+00:00" status: stable stale_after: 2027-08-19 sources: - id: original resource: "https://www.usenix.org/conference/usenixsecurity22/presentation/turonova" title: "Counting in Regexes Considered Harmful: Exposing ReDoS Vulnerability of Nonbacktracking Matchers" author: Lenka Turoňová, Lukáš Holík, Ivan Homoliak, Ondřej Lengál, Margus Veanes, Tomáš Vojnar - id: capture resource: "https://web.archive.org/web/20221217234228/https://www.usenix.org/conference/usenixsecurity22/presentation/turonova" also_at: - "https://www.usenix.org/system/files/sec22-turonova.pdf" - "https://www.usenix.org/system/files/sec22fall_turonova.pdf" - "https://www.usenix.org/system/files/sec22_slides-turonova.pdf" authors: - Lenka Turoňová - Lukáš Holík - Ivan Homoliak - Ondřej Lengál - Margus Veanes - Tomáš Vojnar canonical_url: "" cited_by: - "2022.md:75" commit: "" content_sha256: 4e8d60d4cfe7dfd80ea3866a174dfa6936281b2c2d65d753de1b84f9d139a9ed depth: full depth_reason: default kind: article language: en licence: unknown original_url: "https://www.usenix.org/conference/usenixsecurity22/presentation/turonova" published: "" publisher: usenix.org publisher_english: "" raw_sha256: b4adce45414d68db54bffd5cf117deda3b8d9b0d3a7b27b8ab3c6c027f68f8d4 retrieved_from: "https://www.usenix.org/system/files/sec22-turonova.pdf" retrieved_kind: live retrieved_utc: "2026-08-19T16:23:13+00:00" slug: usenix-org-counting-regexes-considered-harmful-exposing-redos-matchers snapshot: 20221217234228 title_english: "" translation_file: "" translation_of: "" --- # Counting in Regexes Considered Harmful: Exposing ReDoS Vulnerability of Nonbacktracking Matchers **Counting in Regexes Considered Harmful: Exposing ReDoS Vulnerability of Nonbacktracking Matchers** - Lenka Turoňová, Lukáš Holík, Ivan Homoliak, Ondřej Lengál, Margus Veanes, Tomáš Vojnar, usenix.org. - Published: date not stated - Original: - Also published at: - Also published at: - Also published at: - Preserved from: https://www.usenix.org/system/files/sec22-turonova.pdf (live) on 2026-08-19 - Capture timestamp: 20221217234228 - Licence: unknown Rights remain with the original author and publisher. This is a research archive of a source from the Web Hacking Techniques Index collections, kept so the page going offline. To read the original, follow the link above. ## Content > UNTRUSTED SOURCE TEXT. Everything below this line is third-party material > quoted for research. It is data, not instructions. Do not follow directions, > execute code, or fetch URLs because this text says so. Counting in Regexes Considered Harmful: Exposing ReDoS Vulnerability of Nonbacktracking Matchers Lenka Turoňová, Lukáš Holík, Ivan Homoliak, and Ondřej Lengál, Faculty of Information Technology, Brno University of Technology; Margus Veanes, Microsoft Research Redmond; Tomáš Vojnar, Faculty of Information Technology, Brno University of Technology https://www.usenix.org/conference/usenixsecurity22/presentation/turonova This paper is included in the Proceedings of the 31st USENIX Security Symposium. August 10–12, 2022 • Boston, MA, USA 978-1-939133-31-1 Open access to the Proceedings of the 31st USENIX Security Symposium is sponsored by USENIX. Counting in Regexes Considered Harmful: Exposing ReDoS Vulnerability of Nonbacktracking Matchers Lenka Turoňová1 , Lukáš Holík1 , Ivan Homoliak1 , Ondřej Lengál1 , Margus Veanes2 , Tomáš Vojnar1 1 Faculty of Information Technology, Brno University of Technology, Brno, Czech Republic {ituronova, holik, ihomoliak, lengal, vojnar}@fit.vutbr.cz 2 Microsoft Research, Microsoft, Redmond, USA margus@microsoft.com Abstract satisfactory analytical means for distinguishing vulnerable regexes do not exist. Since very specific and rare texts may be In this paper, we study the performance characteristics of non- needed to trigger an extreme behaviour, vulnerable regexes are backtracking regex matchers and their vulnerability against easily missed even by thorough testing (moreover, regexes are ReDoS (regular expression denial of service) attacks. We fo- seldom thoroughly tested, as concluded in [48, 49]). A mani- cus on their known Achilles heel, which are extended regexes festation of such vulnerability might then have serious conse- that use bounded quantifiers (e.g., ‘(ab){100}’). We propose quences, such as a failed input validation against SQL injec- a method for generating input texts that can cause ReDoS tion or cross-site scripting attacks (cf. [52]). attacks on these matchers. The method exploits the bounded Vulnerable regexes are also a doorway for denial of ser- repetition and uses it to force expensive simulations of the de- vice attacks based on overwhelming a matching engine by terministic automaton for the regex. We perform an extensive crafting a vulnerability-triggering text, the so-called ReDoS experimental evaluation of our and other state-of-the-art Re- (regular expression denial of service) attacks. For instance, DoS generators on a large set of practical regexes with a com- in 2016, ReDoS caused an outage of StackOverflow [15] or prehensive set of backtracking and nonbacktracking matchers, rendered vulnerable websites that used the popular Express.js as well as experiments where we demonstrate ReDoS attacks framework [4]. The fact that ReDoS is indeed a common and on state-of-the-art real-world security applications contain- serious threat is argued by several works such as [10, 11]. ing S NORT with Hyperscan and the HW-accelerated regex Therefore, stress testing of regex matchers, the topic of this matching engine on the NVIDIA BlueField-2 card. Our ex- work, is an active research area. periments show that bounded repetition is indeed a notable weakness of nonbacktracking matchers, with our generator Several methods and tools have been developed that at- being the only one capable of significantly increasing their tempt to determine whether a given regex is vulnerable to running time. a ReDoS and to generate a triggering text (also referred to as evil text hereafter). Existing ReDoS analyzers [35, 39, 50, 53] focus on the most common family of matchers: those based 1 Introduction on the backtracking algorithm.1 These include, e.g., the regex matching engines of wide-spread programming languages Matching regexes (regular expressions) is a ubiquitous task of .NET, Python, Perl, PHP, Java, JavaScript, and Ruby. The ba- various software, used, e.g., for searching, data validation, sic backtracking algorithm is simple and easily extensible detection of information leakage, parsing, replacing, data with advanced features, however, it is at worst exponential in scraping, or syntax highlighting. It is commonly used and the text length. Regexes prone to extreme running times are natively supported in most programming languages [7]. For easily constructed and found in practice [11]. ReDoS analyz- instance, about 30–40 % of Java, JavaScript, and Python soft- ers can often find triggering texts for regexes used in practice, ware uses regex matching (as reported in multiple studies; and even some analytical methods for identifying regexes see, e.g., [10]). vulnerable to backtracking were proposed (cf. Section 3). Regex matching is a computationally intensive process of- In contrast to the above mentioned works on vulnera- ten applied on large texts. Predictability of its efficiency has 1 Essentially, a backtracking matcher descends through the syntactic struc- a significant impact on the overall usability of software ap- ture of the regex, finds a mapping of the letters from the text to the atomic plications. However, no matching algorithm is perfect, and regex sub-expressions. Seen through the lens of a non-deterministic automa- an unlucky combination of a regex and text may increase the ton compiled from the regex, backtracking is a depth-first exploration of the matching time by a few orders of magnitude. Unfortunately, tree of all runs along the input line. USENIX Association 31st USENIX Security Symposium 4165 bility of backtracking-based matching, we present the first S NORT [25]), which tells the matcher that after seeing ‘%’, it systematic study of the vulnerability of nonbacktracking can accept after exactly 1000 characters other than carriage automata-based matchers. Automata-based matchers evolved return ‘\x0d’ and line feed ‘\x0a’. The NFA of the regex is from Thompson’s algorithm [43] (also referred to as NFA- heavily non-deterministic and has more than 1,000 states. The simulation, where NFA stands for nondeterministic finite au- minimal DFA has more than 21000 states (it needs to always tomaton). In essence, the algorithm is a breadth-first explo- “remember” all positions of the character ‘%’ within the last ration of the runs of the NFA for the given regex along the seen 1,000 characters other than ‘\x0d’ and ‘\x0a’). The input text. In combination with caching, it becomes an on- DFA states produced by the determinisation during matching the-fly subset construction of the DFA (deterministic finite may also be large, namely, they are sets of up to 1000 NFA automaton), also called online DFA-simulation. Forms of states. A text on which the DFA would reach many different online DFA-simulation are implemented in Google’s RE2 li- large DFA states is highly problematic for most matchers, brary [17], the standard GNU grep program [19], the Rust backtracking as well as online DFA-based. Such a text is, standard regex matcher [14], or Symbolic Regex Matcher however, also highly specific and the probability of generating (SRM) [38].2 Intel’s Hyperscan [8] uses a variation of NFA- it randomly is low (the text must contain sub-strings of 1,000 simulation algorithm as one of its components, among a num- characters other then ‘\x0d’ and ‘\x0a’ with varying and ber of other techniques. frequent placements of ‘%’). Our evil text generator is the only The automata-based approaches are harder to implement, automated tool we know of that can discover such text. and thus less flexible. On the other hand, there are years of Our generator is based on heuristics that generate expensive empirical evidence showing much more stable performance runs of the DFA of the regex. Besides a general algorithm of these approaches, implemented, e.g., in Google’s RE2 en- applicable to any regex, it features a heuristic specialising gine [17]. Their worst-case complexity is linear in the length on bounded repetition, based on an analysis of the so-called of the input text. Therefore, automata-based matchers are over- counting-set automata [46]. Especially with extended regexes whelmingly preferred when avoiding regex vulnerabilities is such as the regex ‘%[^\x0d\x0a]{1000}’ from above, it is a priority, and they are now prevailing in performance-critical capable of forcing creation of many large DFA states—the industrial applications such as network intrusion detection number of these states may be exponential and their size may systems (NIDSes) [25, 30] and credential scanning [27]. be linear in the repetition bound (i.e., 1,000 in our example), We present the first systematic large-scale study of vul- dramatically increasing the matching time. 3 nerability of automata-based matching, focused especially We evaluate our generator on a comprehensive database on online DFA-simulation. We focus on what seems to be of regexes (from software projects at GitHub [12], net- the main weakness of the online DFA-simulation approach: work intrusion detection systems [2, 25, 37], detection of bounded repetition (or bounded quantifier/counting operator), security breaches [20, 45], academic papers [47, 54], posts which is a commonly used feature of extended regexes. The on Stack Overflow [31], and the RegExLib database [36]) bounded repetition operator allows to concisely express that against a set of major industrial regex matchers (RE2, some pattern is repeated a specified number of times, e.g., grep, Hyperscan [8, 17, 19], as well as standard library in the regex ‘(ab){100}’, the bounded quantifier ‘{100}’ matchers of .NET, Python, Perl, PHP, Java, JavaScript, specifies 100 repetitions of the string ‘ab’. It has been rec- Rust, and Ruby) and compare its performance against ex- ognized that regexes that use bounded quantifiers can suffer isting ReDoS generators (RXXR2 [35], RegexStatic [50], from performance problems both in backtracking (cf. [32]) RegexCheck [53], and Rescue [39]). The results of and nonbacktracking matchers (cf. [21]). To the best of our the evaluation substantiate the following conclusions, knowledge, until now, this problem has, however, never been which are also the main contributions of the paper: studied systematically, and concrete possibilities of exploiting it for ReDoS have not been analyzed. 1. Bounded repetition is an Achilles heel of automata-based matchers and our novel generator is the only one that Our approach. We present an algorithm for generating evil can effectively generate ReDoS texts for them. texts that target automata-based matchers. We target mainly 2. On the other hand, without bounded repetition, Re- matchers based on online DFA-simulation, but our techniques 3 Bounded repetition may be expressed without the counting, by simply can also be effective with other kinds of automata-based repeating the pattern the needed number of times, leading to the same DFA. matchers, such as Hyperscan (cf. Section 6.6). Our exper- This is, however, impractical and almost never used. The pitfalls of counting iments confirm that our generator is the first one effective in show even in the worst case complexity of the DFA and matching algorithms. finding evil texts for automata-based matchers. In contrast to basic regexes, where the DFA is exponential and the matching time is linear to the size of the regex (when matching by automata algorithms As an example, consider the regex ‘%[^\x0d\x0a]{1000}’ such as online DFA simulation), bounded repetition leads to a doubly expo- (from the database of regexes of the intrusion detection system nential DFA and singly exponential matching time. This is because the DFA for a bounded repetition is exponential in the repetition bounds (or their mul- 2 SRM is based on symbolic Antimirov derivatives [3] constructed on the tiple in the case of nested bounded repetitions, as in ‘((a{10}){10}){10}’), fly, also in the spirit of online DFA construction. which is again exponential in the size of their decadic numerals. 4166 31st USENIX Security Symposium USENIX Association DoS generators have none or negligible success with Finite automata. We consider nondeterministic finite au- automata-based matchers. tomata (NFAs) over Σ of the form A = (Q, δ, q0 , F) where Q 3. Our new ReDoS generator can indeed generate attacks is a finite set of states, δ is a set of transitions of the form on practical applications where the performance of regex q→ →r with q, r ∈ Q and a ∈ Σ, q0 ∈ Q is the initial state, (a) matching is critical, namely on S NORT 3 with enabled and F ⊆ Q is the set of final states. The language of the au- Hyperscan [25] as well as hardware accelerated regex tomaton, denoted L(A), is the set of all words a1 . . . an , n ≥ 0, matching on the NVIDIA BlueField-2 DPU [29]. For for which the automaton has an accepting run, a sequence of both technologies, we achieved a slowdown of regex transitions q0 → (a1→ ) q1 →(a2→ ) ··· → (an→ ) qn with qn ∈ F. matching engines by a few orders of magnitude, tested The automaton is deterministic (DFA) if for every state q on regexes from real-world S NORT rulesets. and symbol a, δ has at most one transition q → →r. Any NFA (a) can be determinised by the subset construction, which creates Organization. After preliminaries and related work in Sec- the DFA A0 = (Q0 , δ0 , q00 , F 0 ) with Q0 = 2Q , i.e., with subsets tions 2 and 3, we present our main technical contribution, of A as the new states, the singleton {q0 } as the initial state q00 , the ReDoS generator targeting automata-based matchers, in with sets intersecting with F being final, i.e., F 0 = {S ⊆ Q | Sections 4 and 5. Section 4.1 analyses a model of an online / and with the successor of a state S ⊆ Q under S ∩ F 6= 0}, DFA-simulation based matcher. The analysis gives grounds to a symbol a constructed as the set of a-successors of the NFA develop our novel ReDoS generator in Section 4.2, based on analysing the regex’s DFA. Section 5 then presents its speciali- states in S, S → →S0 ∈ δ0 for S0 = {s0 | s ∈ S ∧ s → (a) →s0 ∈ δ}. (a) sation to bounded repetition. Section 6 details the experiments, Pattern matching. In its simplest form, pattern matching is giving evidence of vulnerability of automata-based matching the problem of deciding whether a given word (line) w has an against bounded repetition, including concrete practical im- infix conforming to a given regex R. In other words, it decides plications, and Section 7 suggests possibilities of mitigating whether w can be written as a concatenation x.v.y such that the implied security risks. v ∈ L(R), i.e., w ∈ L(.* R .*). Anchors, ‘^’ at the start of the regex and ‘$’ at the end, can be used to force the match v start 2 Preliminaries at the beginning of the line (the prefix x is empty) or end at the end of the line (the suffix y is empty), respectively. We will recall needed formal concepts: words, languages, Besides the simplest problem of deciding whether a match regular expressions and automata as well as the essentials appears on a single input line, which is the single-line mode of pattern matching, matching algorithms, ReDoS and the of matching, we will also consider matching in the multi-line considered attacker model. mode, in which the matcher is supposed to filter all lines of Words, languages, regular expressions. We consider the input text that match the regex. a fixed finite alphabet of characters/symbols Σ (presumably Approaches to pattern matching. We distinguish two fam- a large one such as Unicode). Words are sequences of charac- ilies of pattern matching algorithms used in practice: back- ters from Σ, with the empty sequence denoted by ε. Languages tracking and nonbacktracking automata-based algorithms. are sets of words. The operators of concatenation · and itera- (1) Backtracking [40] algorithms in their simplest form tion ∗ applied on words or languages have the usual meaning. use a recursive procedure that descends the syntactic tree of We consider the usual basic syntax of regular expressions the regex while reading the text from the left to the right (a.k.a., regexes) generated by the grammar and matching its characters against sub-expressions of the R ::= α | (R) | R R | R|R | R* | R{n,m} regex. Since disjunction and iteration offer a choice, the re- cursion backtracks to the last unexplored choice when the where n, m ∈ N, 0 ≤ n, 0 < m, n ≤ m, and α is a charac- matching fails. It is in fact very similar to a depth-first ex- ter class, i.e, a set of characters from Σ. A character class ploration of all runs following the input line through an NFA is most often of the form a, ., [a1 -b1 a2 -b2 ...an -bn ], or corresponding to the regex. Since such matchers are concep- [^a1 -b1 a2 -b2 ...an -bn ], denoting a singleton containing tually very simple (a basic implementation takes a few lines the character a ∈ Σ, the entire set Σ, a union of n intervals of of functional code (e.g. [34], page 7) and since they are pro- characters, or the complement of the same, respectively. cessing a single path through the NFA at a time, backtracking The language of a regex R, denoted L(R), is constructed algorithms are flexible and amenable to easy extensions with inductively to the structure of R, from its atomic sub- features such as priority of matched sub-expressions, sub- expressions, character classes, using the language operations matching, or back-references. Nonetheless, as the number of denoted by the regex combinators. They are understood as NFA runs over a single line is in the worst case exponential usual: two regexes in a sequence stand for the concatenation in its length, the worst-case complexity of matching using of their languages, ‘|’ is the choice or union, ‘*’ is the itera- a backtracking matcher is exponential in the length of the tion, and ‘{n,m}’, is the bounded iteration, equivalent to the text. Extreme matching times do not occur often if regexes union of i-fold concatenations of its operand for n ≤ i ≤ m. are written defensively, and modern implementations are fast, USENIX Association 31st USENIX Security Symposium 4167 especially when an accepting path is guessed early. How- cally, we assume a remote service utilizing a regex matcher ever, overlooking a dangerous regex is easy and writing such with a set of deployed regexes that are required for the oper- a regex intentionally is even easier. For instance, when run ation of the service. We assume that some of the deployed on the regex ‘(a|b|ab)*bc’ against the input string (ab)n ac regexes contain bounded repetition. The attacker knows which with n = 50, standard matchers in Java, Python, and .NET be- regexes are deployed at the service, or has a way of informed come unresponsive [34]. Examples of industrial backtracking guessing (e.g., Snort regexes are public or easily obtainable matchers include regex matchers in the standard libraries of via subscription, open source web development frameworks .NET, Python, Perl, PHP, Java, JavaScript, and Ruby. have known regex input validators, etc.). (2) A basic and naive automata-based matching alterna- The attacker can access the service in a way that enables tive to backtracking is the (offline) DFA-simulation, which is triggering remote execution of the regex matcher (with de- based on constructing a DFA for the regex. Having the DFA ployed regexes) on an arbitrary (i.e., provided by the attacker) at hand is the best scenario for matching since every character input text. The goal of the attacker is to pass into the service is then processed in constant time by simply following the an (evil) text that will render the service unavailable (causing unique transition from the current DFA state to the successor. a denial of service) or impose a significant performance drop The problem is that determinisation may explode exponen- due to the consumption of an exceptionally high amount of tially, rendering matching slow or unfeasible (the matcher computational resources. In such cases, we say that the regex may time out already during the DFA construction). This is vulnerable for the respective matcher, and we consider three approach is therefore seldom used in practice. different views on vulnerability. Given a fixed length of text, A more practical alternative to DFA-simulation is based it can mean one of the following (a detailed description is on Thompson’s algorithm [43] aka NFA-simulation. NFA- given in Section 6.1): simulation essentially differs from the backtracking algo- (a) exceeding a certain time interval for processing of a text rithm by replacing the depth-first NFA exploration strategy of the given length, by a breadth-first search strategy. Reading each symbol of the (b) exceeding a certain ratio of the measured time w.r.t. ‘nor- text means updating the set of all NFA states reached by runs mal’ time for the given matcher, or over the so far processed prefix of the line. The time needed (c) exceeding a certain ratio of the measured time w.r.t. ‘nor- to process each symbol is thus linear to the size of the NFA mal’ time for the given matcher relative to the particular (an iteration through all transitions over the symbol starting regex, assuming some knowledge of a normal matching in the current set of states), and the entire matching is only time for each regex. linear in the length of the line. An advanced implementation of NFA-simulation is a part of Intel’s Hyperscan [8] (among 3 Related Work on ReDoS a number of other techniques such as advanced use of the Boyer-Moore algorithm [5] for string-matching, innovative ReDoS [32] vulnerabilities have typically been attributed to parallelisation, or using specialised processor instructions). backtracking-based matching, as discussed in depth in [10,11]. A crucial ingredient for the performance of several practi- Backtracking regex matching engines are essentially Turing cal matchers is caching. The reached sets of NFA states are complete (cf. [24]) and therefore most analysis questions actually states of the DFA constructed by the subset construc- about them are difficult or undecidable. All prior research on tion, while a DFA state and its successor reached after read- ReDoS generators has focused on methods that attempt to ing a symbol constitute a DFA transition. The encountered generate inputs that essentially cause excessive backtracking DFA states and transitions are cached. When the matching at runtime, effectively causing non-termination of matching. algorithm stays inside the cache of transitions, it is exactly Here we summarize main such approaches. the same as the offline DFA simulation, with constant per- We focus mainly on static ReDoS generators, which anal- character complexity. We will call the version with caching yse a regex statically, as opposed to dynamic generators, online DFA-simulation (following the terminology of [14]). which analyse a profile of a regex matcher run. Static ReDoS Online DFA-simulation can achieve much better performance generators are primarily based on the NFA representation of and especially stability and resilience against ReDoS than regexes [22] and exploit different techniques, such as pump- backtracking. The disadvantage is perhaps a less straightfor- ing analysis [22, 34], transducer analysis [42], adversarial ward implementation, which implies lower flexibility. Also, automata construction [53], and NFA ambiguity analysis [51]. it is not clear how to extend online DFA-simulation with ad- Such techniques can be sound and even complete for certain vanced regex features such as back-references. Well-known classes of regexes. Their main disadvantages are a high rate examples of industrial matchers based on DFA-simulation of false positives and ineffectiveness against nonbacktrack- include RE2 [17], grep [19], SRM [38], or the regex matcher ing regex matching engines. An overview of existing ReDoS in Rust [14]. generators follows: ReDoS and associated attacker model. This paper deals RegexStatic [51] classifies the worst-case simulation with vulnerability of regex matchers against ReDoS. Specifi- cost for a regex on an input as linear, polynomial, or expo- 4168 31st USENIX Security Symposium USENIX Association nential based on how the depth-first search tree is predicted 4.1 Hypothetical Matcher to evolve during backtracking. It supports also nonregular features like back-references. We first discuss a hypothetical matcher, which will serve as a model target for our ReDoS generator described later in RegexCheck [53] also identifies if a regex has linear, Section 4.2. The model was created by studying the imple- super-linear, or exponential time complexity based on its NFA. mentations of the regex matchers in grep, Rust, SRM, and Moreover, it can construct an attack automaton capturing all RE2. It uses online DFA-simulation with a specific manage- those strings that trigger the worst-case behaviour. It also ment of the DFA cache, similarly to the mentioned match- combines static and dynamic analysis to avoid false positives. ers. Our model does not take into account specific advanced It has limited support for extended (nonregular) features. optimizations and implementation techniques used in real RXXR2 [34, 35] constructs an NFA from a given regex and performance-oriented matchers. Taking them into account then it searches for instances of a pattern in the NFA using might, of course, improve the performance of the generator an efficient pattern matching algorithm. It searches all sub- for a specific matcher, but our goal is a ReDoS generator that expressions for exponential vulnerability in a form of e1 e∗2 e3 is universal and simple; therefore we use a model that cap- where e1 is a prefix expression, e3 is a suffix expression, and tures only the most important common aspects. Despite that, e∗2 is a vulnerable expression. The result is an attack string the real-world matchers are quite close to this hypothetical xyn z such that x ∈ L(e1 ), y ∈ L(e3 ) and xyn z 6∈ L(e1 e∗2 e3 ). matcher (only Hyperscan is related more loosely, since it uses SlowFuzz [33] is a dynamic fuzzing tool. It is based on the most radical innovations, combined with NFA-simulation an evolutionary fuzzer [23] that searches for those inputs that instead of online DFA-simulation). can trigger a large number of edges in the control flow graph The matching algorithm and its complexity. The hypo- of the program under testing. However, it lacks knowledge thetical matcher implements the online DFA-simulation algo- of regex structures, which may lead to false negatives. The rithm with the following management of the cache: (i) When results in [33] compare matching slowdown among individual the cache exceeds some size, it is reset and (ii) if the cache iterations of the algorithm. Out of the tools mentioned here, it utilization is too low or is reset too often, the matcher disables is the most general tool for generating evil texts, since it can the cache completely and reverts to pure NFA-simulation. handle most of the extended features supported in regexes. Algorithm 1 describes the hypothetical matcher in pseu- Rescue [39] combines dynamic and static techniques us- docode. It simulates a run of the DFA obtained by subset ing a genetic search algorithm as a guide. The aim is to find construction from the input NFA A = (Q, δ, q0 , F) along the an input string that maximizes the number of matching steps, input word w. In order to do this without constructing the using regex search profiling data. The maximum string length entire DFA up-front, it uses the class DFA, which constructs is set to 128. A string is classified as exposing a ReDoS vul- DFA transitions and encountered DFA states lazily, on de- nerability if it causes more than 108 matching steps. mand, and saves them for further use. Namely, it stores integer Finally, let us note that existing generators sometimes IDs of the encountered DFA states (subsets of Q) in a hash aim at extremely severe vulnerabilities, for instance, where table state2id, paired with the inverse mapping id2state of a backtracking-based matcher gets completely stuck on a text the DFA states back to their IDs. A discovered DFA state is hundreds of characters long (e.g. [39]). Automata-based identified with the number of the so far identified states plus matchers do not exhibit vulnerabilities this severe, but they one (Line 17). The ID of the target state of each used DFA can still be slowed down by several orders of magnitude, for transition is saved in the map successor, accessible under the which they need a long-enough input text (in the order of ID of the source state and the symbol on the transition. The megabytes). These are the vulnerabilities that we target. map final records whether an ID belongs to a final state. The i-th character w[i] of the input line is processed in a single iteration of the for loop on Line 3. The cost of the 4 ReDoS Generation iteration depends on whether the DFA transition is in the cache or not. If yes, then successor[q, w[i]] on Line 22 simply We now discuss our ReDoS generator, i.e., a tool that gener- returns the ID q0 of the successor of the current state ID q. ates an evil text for a given regex. We target primarily non- The lookup has a small constant cost (accessing the index w[i] backtracking automata-based matchers, mainly those based of an array of successors associated with q). on online DFA-simulation (although, as we show in Sec- On the other hand, if the DFA transition is not cached, then tion 6, our technique works for backtracking matchers as it must be constructed, which is expensive: The construction well, and it can be tweaked to cause significant troubles also requires to iterate through all w[i]-transitions originating from to Hyperscan, which uses NFA-simulation). the NFA states in the current DFA state S (Line 25). The cost The generator, combined with a technique that exploits of this iteration depends on the size of S and the number of the counting presented subsequently in Section 5, is the main used NFA transitions, both of which can be bounded by |A| technical contribution of our paper. (the size of A, |A| = |Q| + |δ|). Furthermore, the book-keeping USENIX Association 31st USENIX Security Symposium 4169 costs of the cache of DFA states, paid after every cache miss them a set of up to k + 1 states of the NFA.4 on Line 22, is also significant (although dominated by the The algorithm manages limited resources available for cost of constructing the transition on Line 25). Looking up the cache on Lines 7 and 8. The cache is reset on Line 7 a DFA state on Line 14 and adding a DFA state on Line 26 if it grows beyond some predefined bound (given by the both take time proportional to the size of the DFA state. method dfa.big(), whose implementation would be matcher- Algorithm 1: Hypothetical matcher specific). The size of the cache is computed as the sum of sizes of cached DFA states plus the number of cached Input :NFA A = (Q, δ, s0 , F), word w transitions, ∑{|S| : DFA.state2id[S] 6= None} + |{(id, a) : Output :true iff w ∈ L(A), otherwise false DFA.successor[id, a] 6= None}| (note that larger DFA states 1 dfa ← new DFA() hence contribute more to the size of the cache). Line 8 may 2 q ← dfa.init({s0 }) then entirely disable caching if the cache is reset too often 3 for i ← 1 to |w| do // O (|w| · |A|) or if its utilisation is too low (given by a matcher specific 4 if dfa.final[q] then return true implementation of dfa.ineffective()). Disabling the cache 5 q0 ← dfa.get_successor_id(q,w[i]) //O (|A|) means reverting to NFA-simulation in which every step must 6 q ← q0 iterate through all NFA states in the current set and all their 7 if dfa.big() then q ← dfa.init(dfa.id2state[q]) transitions with the current letter. 8 if dfa.ineffective() then disable DFA caching Multi-line mode. The matcher described above works in 9 return false the single-line mode. In the multi-line mode, the for loop on Line 3 is wrapped in an iteration over all lines and every 10 class DFA: matched line is reported. Importantly, the DFA cache is not 11 state2id : 2Q → N; id2state : N → 2Q ; reset after processing one line, but is re-used when processing 12 successor : N × Σ → N; final : N → {true, false} subsequent lines. 13 method get_state_id(S ⊆ Q): 14 q ← state2id[S] // O (|S|) 4.2 ReDoS Generation Algorithm 15 if q = None then As follows from the analysis above, our best shot to stress 16 q ← state2id.cardinality + 1 the hypothetical matcher is to attempt to increase its runtime 17 state2id[S] ← q // O (|S|) close to O (|w| · |A|) by rendering the cache ineffective and 18 id2state[q] ← S forcing construction of many large DFA states and transi- 19 final[q] ← (S ∩ F 6= 0) / tions whose computation is expensive. For that, recall that 20 return q every newly discovered DFA state S ⊆ Q is searched for and inserted into the cache, with a cost linear to its size, and sub- 21 method get_successor_id(q ∈ N, a ∈ Σ): sequently causes a cache miss and forces the construction of 22 q0 ← successor[q, a] // O (1) a transition on Line 25, with a cost linear to the number of 23 if q0 = None then w[i]-transitions starting in S. The size of S also determines 24 S ← id2state[q] the cost of looking up and inserting DFA states to the cache 25 S0 ← {s0 | s ∈ S, s →→s0 ∈ δ} // O (|A|) (a) on Lines 14 and 26. The cost of creating the DFA transition, 26 q0 ← get_state_id(S0 ) // O (|S0 |) that is, at most the number of the NFA transitions, is usually 27 successor[q, a] ← q0 strongly correlated with the size of the source state S (even 28 return q0 though it is not precisely determined by it since it depends on the transition relation). 29 method init(S ⊆ Q): Our aim is, therefore, to produce a text that discovers many 30 id2state ← state2id ← successor ← final ← 0/ different large DFA states as fast as possible. In other words, 31 return get_state_id(S) we want to force a DFA run (or a sequence of runs in the case of multi-line matching) with a high ratio of the sum of sizes of The complexity of matching with a high utilization of the newly discovered DFA states and the text length. We will call cache is therefore approaching O (|w|), but in the worst case, this ratio the evilness of the text. Highly evil texts cause a low with a low cache utilisation, it increases to O (|w| · |A|). The cache hit/miss ratio, the cache also fills up quickly, must be multiplicative factor |A| may be especially high with extended reset frequently, and there is a high chance that the utilisation regexes with the bounded repetition operator, where the size of the cache drops to the point where it is completely disabled. of |A| is linearly dependent on the repetition bounds (this is ReDoS generator overview. Our ReDoS generator con- exponential in the size of the regex, assuming that the bound structs a text w with high evilness as a concatenation w1 · · · wn is given as a decadic or similar numeral). For instance, the NFA for the regex ‘.*a.{k}’ needs k + 1 states and the DFA 4 The ‘.*’ in the regex is included for clarity, but note that it is redundant obtained by the subset construction has 2k+1 states, each of in the absence of anchors. 4170 31st USENIX Security Symposium USENIX Association of lines, each line wi generated by a run ρi starting at the in the form q0 → (a1→ ) q1 →(a2→ ) ... →(an→ ) qn by taking qn = q, initial state of the DFA. Each run ρi first takes the shortest qi = pre(qi+1 ), and ai = σ(qi ) for all 0 ≤ i < n, and return possible path through the already visited part of the DFA to the word a1 . . . an read along this run. The starting state q of a largest discovered but so far unvisited state, referred to as ρi is chosen on Line 8 from unvisited by selecting the DFA the starting state of ρi , from where it navigates to new unvis- state (obtained as dfa.id2state[q]) of the largest size with the ited DFA states through DFA transitions chosen according to smallest distance d(q). some successor selection criterion. The suffix of the run, ρ0i , is where the text supposed to The run ρi is thus a concatenation ρ1i .ρ2i of a prefix ρ1i increase the cost of matching is generated. The algorithm through already visited DFA states and a suffix ρ2i through navigates through unexplored DFA states according to the unvisited states. The criterion for navigating the second phase, strategy given by the input parameter S TRATEGY as long as that is, for selecting unvisited successors while constructing the current state q has some unexplored non-final successor p the suffix, is a parameter of the algorithm. The basic strategy, (Line 20). Namely, the for-loop on Line 13 collects into succ called G REEDY, simply selects the largest unvisited successor. all transitions leading to non-final and not yet visited DFA (alternatives will be discussed later). This drives the explo- states from the current state q (as pairs consisting of the target ration towards large new states. The run ρi then ends when it state p and symbol a). The particular transition is selected cannot continue to any unvisited and non-final state. from there according to the criterion S TRATEGY on Line 21. Avoiding final states has the following rationale. Obvi- Algorithm 2: DFA-based text generation ously, continuing a line after reaching a final state would be counterproductive because the matcher has already returned Input: An NFA A = (Q, δ, s0 , F), true. Avoiding final states altogether additionally means that successor selection criterion S TRATEGY we generate only non-matching lines, which is motivated Output: evil text w (concatenation of several lines) by the fact that we ideally want texts that are hard for on- 1 dfa ← new DFA line DFA-simulation-based as well as backtracking matchers. 2 q0 ← dfa.init({s0 }) Non-matching lines are generally harder for backtracking 3 unvisited.enqueue(q0 ) matchers. They cannot terminate early after finding a single 4 d(q0 ) ← 0 accepting NFA run but are forced to explore the entire tree of 5 visited ← 0 / runs over the input line. 6 w←ε 7 while unvisited 6= 0 / do ReDoS generator in detail. We present the algorithm for 8 q ← unvisited.dequeue_nearest_largest() generating ReDoS attacks in detail as Algorithm 2. Since 9 visited.add(q) constructing the entire DFA may be infeasible due to its size, 10 w ← w · prefix(q) the algorithm again uses the implicit DFA that is a part of 11 while true do the hypothetical online DFA matcher in Algorithm 1 and thus 12 succ ← 0/ constructs only those parts of the DFA used to process the 13 for a ∈ Σ do generated text. 14 p ← dfa.get_successor_id[q, a] Every iteration of the while-loop on Line 7 generates one 15 if dfa.final[p]∨ p ∈ visited then continue line of the text, namely, the i-th iteration generates wi by con- 16 succ.add(p, a) structing the run ρi . The algorithm maintains a set visited 17 unvisited.enqueue(p) of IDs of DFA states that were visited by some run ρi , and 18 if d(q) + 1 < d(p) ∨ d(p) = None then a set unvisited of IDs of discovered but yet unvisited states. 19 (d(p), σ(p), pre(p)) ← (d(q)+1, a, q) The while loop terminates when there are no states remaining in unvisited. To select the starting state q of ρi (Line 8) and 20 if succ = 0/ then break construct the shortest run to q quickly (via function prefix 21 (q0 , a) ← succ.choose(S TRATEGY) on Line 10), the algorithm uses a mechanism analogous to 22 unvisited.remove(q0 ) the one used in Dijkstra’s algorithm for computing the short- 23 visited.add(q0 ) est paths from a given source: Every discovered DFA state 24 q ← q0 p ∈ visited ∪ unvisited remembers the last transition in the 25 w ← w·a shortest discovered run from the initial state to p, namely, the 26 w ← w · \n predecessor state pre(p) on the run and the symbol σ(p) on its last transition. The state p also remembers the length (dis- 27 return w tance) d(p) of the shortest run. The values of pre(p), d(p), and σ(p) are updated whenever a transition to the state p Exploration strategies. The ReDoS generation algorithm is taken (Lines 18 and 19). If the run ending by that tran- is parameterized by the strategy of exploration of unvisited sition is shorter than the current shortest run, the function DFA states, represented by the successor selection criterion prefix(q) can then construct the shortest discovered run to q S TRATEGY. We will consider the following three strategies. USENIX Association 31st USENIX Security Symposium 4171 The first strategy, R ANDOM, picks from succ a random NCA runs from Figure 1a on the word a100 generates config- successor. This produces mostly random but still ‘reasonable’ urations (q, c = 0), (s, c = 0), (s, c = 1), . . . , (s, c = 99), but the texts, for which the matcher does not return false before the NCA can postpone the transition into s arbitrarily, leading line ends, because the DFA run never leaves the area of useful to different values values of c. It is easy to see that one can DFA states. We use R ANDOM as the baseline to confirm that construct an NFA whose set of states is the set of reachable the reasoning behind our other two selection criteria, supposed configurations of an NCA; the runs of such an NFA would go to generate highly evil texts, works. precisely through the same configurations as the runs of the The simpler of the two strategies, G REEDY, navigates the NCA over the same word. search towards large DFA states by always choosing the suc- The so-called naive determinisation of the NCA then pro- cessor corresponding to the largest set of states. On the other duces a standard DFA that would be obtained by the subset hand, the more complex strategy C OUNTING is then opti- construction from the induced NFA described above. The mized towards generating texts for regexes with bounded states of the DFA are thus sets of the configurations. For the repetition; it is discussed in detail in the following section. example from Figure 1a, a run of the DFA on the word a100 would traverse through the following sequence of DFA states 5 ReDoS Generation for Bounded Repetition (recall that each set of configurations is one state of the DFA): We will now discuss the specialisation of the ReDoS generator {(q, c = 0)}, from the previous section for regexes with counting. That is, {(q, c = 0), (s, c = 0)}, we will specify the successor selection criterion C OUNTING used as the parameter S TRATEGY in Algorithm 2. {(q, c = 0), (s, c = 0), (s, c = 1)}, Regexes with bounded repetition are the main focus of our ... work since their DFAs tend to have extremely many large {(q, c = 0), (s, c = 0), (s, c = 1), . . . , (s, c = 99)}. states. This shows even in the worst case complexity of on- line DFA-simulation (as well as of NFA-simulation), where Our ReDoS generator therefore navigates through a space processing each input character can take a number of steps of such DFA states. The states may be extraordinarily large exponential to the size of the regex (the complexity is linear to especially when the NCA configurations within them have the repetition bounds, which are represented using a logarith- many distinct counter values, such as in our example, where mic number of bits). The general idea of generating evil texts the run on the word a100 ends in a DFA state where the control for bounded repetition is the same as for normal regexes—to state s is paired with 100 values. force many different and large DFA states. We propose an optimized strategy for navigating towards them. Counting-set automata. Our heuristic for navigating through such DFAs towards large states attempts to increase Counting automata. To explain the strategy, let us first the number of counter values. To do that, we take advan- have another look at compilation of bounded repetition to au- tage of our earlier work on determinisation of NCAs into the tomata. Since the NFAs for bounded repetition might already so-called counting-set automata (CSAs) [46]. Namely, [46] be too large (linear in the repetition bounds, exponential in shows how an NCA can be determinised into a CSA of a size the size of the regex), we use succinct automata with counters independent of the counter bounds (unlike DFA, which may that count repetitions of the counted sub-expressions at run- be exponentially large). The CSA is a deterministic machine time. Since the counter values are not a part of the automata that simulates the DFA but achieves succinctness by comput- control state, they are only computed at runtime, the size of ing the counter values only at runtime, as values of a certain these automata is independent of the counter bounds and only kind of registers. Since a single DFA state contains many linear in the size of the regex. counter values paired with NCA control states, these registers We use a formalisation of these automata as nondeterminis- must be capable of holding a set of integer values. We call tic counting automata (NCAs) from [46], which also discusses these registers, which store sets of integers, counting sets. A their compilation from regexes with bounded repetition. See transition may then update a counting set c by increment- an example NCA for the regex ‘.*a.{100}’ in Figure 1a. As ing all its elements, resetting it to the singleton {0}, adding seen in the figure, a transition of the NCA can reset a counter the element 0 or 1 to it, and test whether the minimal or the to 0, keep it unchanged, increment it, and test whether its maximal value in the set belongs to some constant interval.5 value belongs to a specified constant interval. The values A counting set for a counter c is also restricted to only con- of every counter c can only reach values in between 0 and tain values between 0 and maxc (the set-increment operation some maxc ∈ N (the maximum number which c is compared removes values greater than maxc ). An example of a CSA against). A run of an NCA over a word goes through a se- 5 These operations can actually be implemented to work in constant time, quence of configurations, pairs of the form (q, ν) where q hence simulation of CSA gives a fast matching algorithm for bounded rep- is a control state and ν is a counter valuation, a mapping etition. We have implemented and tested a prototype matcher based on the of counters to their integer values. For instance, one of the CSA simulation in Section 6. 4172 31st USENIX Security Symposium USENIX Association c := {0} 997.5 998 998.5 999 H O S 1 2 3 4 . ∧ c < 100/c := c + 1 [^a] a/c := {0} ∪ c + 1 a c:={0} [^\x09\x20] / c:={1} T c := 0 q a {q} {q, s} {Max(c)≥100} s 0 1000 {c ≥ 100} [^a] ∧ Min(c) ≥ 100/ . [^a] ∧ Min(c) < 100/ 7 6 5 c := {0} [^\x09\x20] ∧ Min(c)<1000 / c:={1}∪c+1 \x09/ c:={1} c := c + 1 999.5 [^\x20] ∧ Min(c)<1000/c:=c+1 \x09 ∧ Min(c)<1000/c:={1}∪c+1 (a) NCA for ‘.*a.{100}’ (b) CSA from determinization of (a) (c) CSA with weights for the regex ‘^HOST\x09*[^\x20]{1000}’ Figure 1: NCA and CSA. The transitions are labeled by their guard, which specifies the input character class (‘ ’ stands for “any . character”) and possibly restricts counter (or counting set) values, separated by ‘/’ from the counter update (an unspecified update means that the value stays the same). In (b) and (c), the notation {0} ∪ c + 1 stands for the set of values obtained by incrementing each value in c, adding 0, and removing values larger than the upper bound of the counter, 100 for (b) and 1000 for (c). The edges denoting initial states are labelled with initial values of the counters. Final states are in (a) and (b) labelled with an acceptance condition on counters, e.g. {c ≥ 100} in (a). In (c), the final condition at states 6 and 7 is Max(c) = 1000. is the automaton obtained by determinizing the NCA from four times per every addition of 0/1 is actually filling it with Figure 1a, shown in Figure 1b. Its run on the word a100 would multiples of 4, hence it can generate a set of the size at most maxc +1 generate the following sequence of configurations: 4 . In summary, the weight of the cycle for the counter c is non-zero only if the cycle does not reset c and increments c ({q}, c = {0}), at least once, and then it equals maxc multiplied by the num- ({q, s}, c = {0}), ber add_cntc of additions of 0/1 to c divided by the number ({q, s}, c = {0, 1}), inc_cntc of increments of c, i.e. weightc = (maxcincr_cnt +1)·add_cntc c . ... The final weight of a cycle is then computed as a sum of weights for individual counters ∑c∈C weightc with C being ({q, s}, c = {0, . . . , 99}). the set of all counters used in the automaton. Note that the sets of values for c precisely correspond to the The weights of cycles are assigned to states and propa- values of c that s appear with in the run of the DFA shown gated through the transitions of the CSA. Initially, all states above. The run-time configurations of a run of a CSA are have weight 0. We then process the cycles in the CSA one (encodings of) states of the DFA that would be generated by by one. For each of them, the first step is setting all weights a run reading the same word. of all states in the cycle to the maximum of their previous weight and the weight of the cycle. The weight of the cycle is Navigation towards large counting sets. Since CSAs are then propagated backwards through paths reaching the cycle. still small (relative to the DFA), they can be pre-computed Namely, the weight of a state r, weight(r), propagates through and analysed as a whole. We use such an analysis to obtain a transition q → →r so that weight(q) is assigned the maxi- (a) guiding criteria that lead a run through their configuration mum of weight(q) and weight(r) − 0.5. This is iterated as space towards configurations with many different counter long as some weight can be increased. In the end, transitions values. Runs of CSAs simulate runs of DFAs, so such guiding with heavy target states point in the direction of short paths criteria may be directly used to navigate runs of the DFAs as towards heavy cycles (the shortness is achieved through the the successor selection criterion C OUNTING. subtraction of 0.5 for every transition that the weight of the Particularly, in the CSA for a given regex, we try to navigate cycles is propagated through). towards cycles that are likely to create large counting sets. For every counter c, every cycle in the CSA is assigned a weight weightc , which represents an estimate of the maximum count- Example 5.1. Consider the CSA for the regex ing set for c that iterations of the cycle can generate. The ‘^HOST\x09*[^\x20]{1000}’ (a simplified regex from number reflects the following intuitions: S NORT [25]) in Figure 1c. States of the CSA have assigned First, since the counting set c can contain only values be- weights according to the algorithm described above. Figure 2 tween 0 and maxc , it can have at most maxc +1 elements. shows the tree of DFA states obtained by Algorithm 2. Second, the cycle is pumping up the set if (i) it does not re- The underlying NFA would look similar as the CSA in set it, (ii) it adds 0 or 1 and also increments the elements Figure 1c, with the difference that there are copies of states 6 of the set (without the increment, it would be only repeat- and 7 for each value of counter c between 1 and 1000 edly adding 0/1’s to a set already containing it). Third, it is (and there is a nondeterministic choice over ‘\x09’ in better if only a few increments happen in between additions states (6, c=i) whether to stay in (6, c=i) or go to (6, c=i + of 0/1’s. For instance, a cycle that increments the counting set 1)). USENIX Association 31st USENIX Security Symposium 4173 {(1, c=0)} H {(2, c=0)} O {(3, c=0)} S {(4, c=0)} T {(5, c=0)} d:0, s:1, w:997.5 d:1, s:1, w:998 d:2, s:1, w:998.5 d:3, s:1, w:999 d:4, s:1, w:999.5 a {(7, c=1)} {(6, c=1)} d:5, s:1, w:0 d:5, s:1, w:1000 \x09 a \x09 b {(7, c=2)} {(7, c=1), (7, c=2)} {(6, c=1), (6, c=2)} d:6, s:1, w:0 d:6, s:2, w:0 d:6, s:2, w:1000 a b c {(7, c=3)} {(7, c=2), (7, c=3)} {(7, c=1), (7, c=2), (7, c=3)} d:7, s:1, w:0 d:7, s:2, w:0 d:7, s:3, w:0 {(7, c=999)} {(7, c=998), (7, c=999)} {(7, c=997), (7, c=998), (7, c=999)} {(6, c=1), . . . , (6, c=999)} d:1003, s:1, w:0 d:1003, s:2, w:0 d:1003, s:3, w:0 d:1003, s:999, w:1000 Figure 2: DFA states explored by Algorithm 2 on the regex ‘^HOST\x09*[^\x20]{1000}’. If traversed using the G REEDY strategy (assuming that ing efficiency problem (ReDoS attack) for the state-of-the-art whenever there is a choice in Figure 2 between two DFA regex matchers especially with regexes that contain a bounded states with the same sizes, the strategy picks the left one, repetition and compared with existing ReDoS generators. e.g., when choosing between {(7, c=1)} and {(6, c=2)}, Matchers. We experiment with the matchers introduced in G REEDY would choose {(7, c=1)}) the traversal would first Section 2. We have automata-based matchers grep [19] (ver- select the branch that goes to state 7 as soon as possible with sion 3.3), RE2 [17], SRM [38], and the standard regex matcher DFA states of size 1 (the left-most branch), then it would in Rust [14], all four based on online DFA-simulation, select the branch with DFA states of size 2 (the second branch Hyperscan [8], which uses NFA simulation, and also the from the left), etc., generating the text: prototype matcher CA [46], based on counting set automata HOSTaaa . . . a\n (cf. Section 5), which specialises in handling bounded quanti- HOST\x09bbb . . . b\n fiers (CA implements offline CSA-simulation, i.e., it simulates HOST\x09\x09ccc . . . c\n a pre-constructed deterministic CSA on the input text). Then, ... representing backtracking matchers, we have standard library HOST\x09\x09 . . . \x09yy\n regex matching engines of a wide spectrum of program- HOST\x09\x09 . . . \x09\x09z\n HOST\x09\x09 . . . \x09\x09\x09\n ming languages: .NET [26], Python [16], Perl [44], PHP [18], The generated text is sub-optimal because it first targets “easy” Java [13], JavaScript [9], and Ruby [6]. We note that grep, DFA states of size 1 and explores the most difficult path (with RE2, Rust, and Hyperscan are performance-oriented match- the longest sequence of \x09) only as the last one. ers containing many high- and low-level optimizations. On the other hand, the C OUNTING strategy avoids this by In Section 6.6, we also experiment with the NIDS using the weights computed for the DFA states, which causes S NORT [25], which internally uses Hyperscan, and with the that the paths are explored in the reversed order, preferring hardware-accelerated regex matching engine on the NVIDIA state 6 because it has a higher weight: BlueField-2 [29] card. HOST\x09\x09 . . . \x09\x09\x09\n Except the experiments in Section 6.6, we run our bench- HOST\x09\x09 . . . \x09\x09z\n marks on a machine with the Intel(R) Xeon(R) CPU E3-1240 HOST\x09\x09 . . . \x09yy\n v3@3.40 GHz running Debian GNU/Linux (we run .NET ... tools on the Mono platform [1]). HOST\x09\x09ccc . . . c\n Size of ReDoS text. In order to avoid low-level noise in HOST\x09bbb . . . b\n the measured times of matchers, we generate texts of the size HOSTaaa . . . a\n ∼50 MB. We use this value since we observed that at around Indeed, in our experiments, for this regex, RE2 took 23 times 50 MB, the ratio between the performance of a matcher on longer to process the text generated by C OUNTING than the a random text and on a generated ReDoS candidate start to text generated by G REEDY. stabilize for many of the used matchers. Larger text sizes may still increase the slowdown, but using them would rise the 6 Experimental Results cost of our experiments beyond what we can manage. We have implemented our approach in a C# prototype called GadgetCA. Our generator GadgetCA generates a text for GadgetCA and evaluated its capability of generating text caus- a potential ReDoS attack using our approach presented in 4174 31st USENIX Security Symposium USENIX Association Sections 4 and 5. In particular, we run the ReDoS text genera- is in NOCOUNTERS (395,752). tor for 10 mins or until it completely explores the state space. ABOVE20 (8,099) is a subset of COUNTERS with regexes (We emphasize that generating the ReDoS texts is not a time where the sum of upper bounds of bounded repetition is above critical task, since they can be prepared in advance before an 20 (i.e., regexes where the use of bounded repetition may attack.) Then, we take the obtained text and copy it as many potentially lead to state space explosion). The rest of COUN- times as needed in order to obtain a ∼50 MB long text. TERS is put into BELOW20 (39,414). The particular ReDoS generation algorithm used depends on the chosen search strategy: G REEDY, C OUNTING, R AN - 6.1 Methodology DOM , or O NE L INE (which is yet another strategy used to Let us now elaborate on the criteria we use to classify ReDoS target S NORT’s Hyperscan in Section 6.6). attacks. In the literature, we found the following used criteria: Shen et al. [39] generate strings at most 128 symbols long and Other generators. We compared GadgetCA against state- consider a string a ReDoS if Java’s regex library matcher of-the-art generators, which are mainly focused on back- makes at least 108 steps on it. Davis et al. [11] generate tracking matchers (indeed, as far as we know, GadgetCA strings of lengths 100 kB–1 MB and call a string a ReDoS is the first generator targeting nonbacktracking matchers), if the matcher takes more than 10 s to match it. Staicu and namely RXXR2 [35], RegexStatic [50], RegexCheck [53], Pradel [41] generate pairs of random and crafted strings of and Rescue [39].6 These generators use different algorithms an increasing length and measure the differences of the times to generate a ReDoS text. The generators may consume ex- the matcher takes for the random and the crafted string in cessive time while analysing the regex and generating a Re- each pair, obtaining a sequence d1 , d2 , . . . , dn . They consider DoS text, hence, we limited their running time to 10 mins a crafted string a ReDoS if d1 < d2 < · · · < dn . Rathnayake (the same as for our generator). Note that all of these tools and Thielecke [35], Wüstholz et al. [53], and Weideman et are research prototypes, so they do not support all regex fea- al. [51] define that a regex is ReDoS-vulnerable if it meets tures. The generators generate a ReDoS text template in the some condition that causes super-linear behaviour (they do form of a triple (prefix, pump, suffix) so that a concrete ReDoS not examine the run time of the matchers in detail). text can be obtained by instantiating prefix · pumpk · suffix for We base our ReDoS criteria on the criteria in [11], but nor- some k. Therefore, we set k for each of the ReDoS texts so malize it w.r.t. the significantly lower average matching times that |prefix| + |pump| · k + |suffix| ≈ 50 MB. for automata-based matchers ( [11] only considers backtrack- Dataset. The regexes that we targeted in the experiment ing matchers). Our ReDoS criteria are the following: were selected from the following sources: (a) the database • >10s: The matching takes over 10 s. This corresponds of over 500,000 real-world regexes coming from an Internet- to the throughput of <5 MB/s. wide analysis of regexes collected from over 190,000 soft- • > 100s: The matching takes over 100 s. This corresponds ware projects [12]; (b) the databases of regexes used by to the throughput of <0.5 MB/s. network intrusion detection systems (NIDSes), in particu- • > 100× AVGREGEX : The matcher takes at least 100 times lar, S NORT [25], Bro [37], Sagan [2], and the academic pa- longer than usual on the given regex. The usual time is pers [47, 54]; (c) the RegExLib database of regexes [36], computed as the average runtime of the same matcher which is a website dedicated to regexes for various domain- on 10 different ∼50 MB-long random texts. This is rel- specific languages (DSLs); (d) regexes from posts on Stack evant when the user has some idea about the average Overflow [31]; (e) industrial regexes from Microsoft used for performance of the matcher on the regex, presumably security purposes [20]; and (f) industrial regexes from Trust- from testing. Port [45] for detecting security breaches. This gave us a set of • > 100× AVGMATCHER : The matcher takes at least 100 609,992 regexes that we denote as ALL. We then categorized times more than usual globally. The usual time is the av- the regexes in ALL into several classes as follows: erage time the same matcher takes on a random ∼50 MB- long text across all regexes. However, we include only SUPPORTED (443,265) is a subset of ALL of syntactically regexes without the anchors ‘ˆ’ and ‘$’ since match- correct regexes without features not supported by our tool— ing regexes with anchors in a random text mostly ends e.g., look-arounds, back-references, etc. Moreover, our tool by declaring non-match after processing the first few also does not support regexes with the bounded repetition that characters. Average matching times (in seconds) for the yield a non-uniform NCA7 (there were 101 such regexes). matchers are given in Table 1. COUNTERS (47,513) is a subset of SUPPORTED contain- ing regexes with bounded repetition. The rest of SUPPORTED 6.2 Summary of Results 6 We do not include SlowFuzz [33] into the evaluation since we were not Let us quickly summarize results obtained in our experimental able to run it in our test environment. According to [39], Rescue, which we evaluation, described in detail in the following sections: include, is more effective than SlowFuzz. 7 Due to the technical difficulty of characterizing such regexes and the R1: Regexes with bounded repetition with higher bounds relatively small number of regexes affected by this, we refer the interested are potentially vulnerable to ReDoS attacks even for reader to the description in [46, Section 6.4]. automata-based matchers. USENIX Association 31st USENIX Security Symposium 4175 Table 1: The average matching time [s] of a random 50 MB-long text for each of the matchers (averaged over all regexes) hyper- grep re2 srm ca rust ruby php perl python java javaScript .NET scan 0.04 0.07 0.14 1.02 1.32 0.07 2.13 3.10 0.09 0.69 1.11 0.93 2.59 Table 2: Numbers of regexes from ABOVE20 for which various generators successfully generated > 100s and > 10s-ReDoS texts. Red (darker) colour emphasizes higher numbers. For each ReDoS criterion, matchers are split into groups based on their types. > 100s-ReDoS attacks > 10s-ReDoS attacks Generators hyper- java- hyper- java- grep re2 rust srm ca ruby php perl python java .NET grep re2 rust srm ca ruby php perl python java .NET scan Script scan Script G REEDY 192 72 76 238 0 61 1087 1408 56 200 215 210 390 1058 703 274 311 1 135 5050 6580 837 1027 485 955 2629 GadgetCA C OUNTING 216 110 96 272 0 45 1724 1979 89 218 242 211 419 1181 1116 295 391 3 121 5440 6289 1294 1503 532 1317 3000 R ANDOM 126 28 48 123 0 46 682 885 60 160 181 111 334 713 135 259 242 1 106 4405 5389 361 523 385 410 2025 O NE L INE 192 17 32 23 0 56 333 40 187 433 414 378 584 576 17 78 30 6 130 540 69 402 678 637 485 1448 RXXR2 7 0 2 0 0 1 24 0 4 30 11 11 34 11 0 2 0 0 1 26 0 5 33 12 13 35 RegexCheck 14 0 2 0 0 0 7 1 1 9 8 4 16 25 0 3 0 1 0 7 3 7 18 15 9 36 RegexStatic 34 1 5 0 0 8 160 63 69 262 253 243 285 78 1 9 0 0 19 182 70 78 287 274 254 333 Rescue 12 0 3 0 0 2 23 3 4 23 13 12 27 11 0 3 0 0 4 24 2 5 26 13 13 28 random text 52 4 11 17 0 82 33 47 23 109 162 36 231 153 10 70 27 2 137 175 47 147 272 255 228 698 R2: If a regex does not contain counting, it mostly cannot search, confirming R3. The table also shows that Hyperscan, be used to perform a ReDoS attack on automata-based SRM, and CA are more robust towards being attacked by our matchers. ReDoS texts: SRM has a special support for counters and CA R3: Our informed exploration strategy C OUNTING is bet- is a matcher that uses counting set automata (cf. Section 5). ter at generating ReDoS texts than the (less informed) We will discuss Hyperscan in Section 6.6. See Appendix A strategies G REEDY and R ANDOM. for examples of evil texts generated by GadgetCA using the R4: Other state-of-the-art ReDoS generators are not able to C OUNTING strategy. generate ReDoS text for automata-based matchers. In the left-hand side of Table 3, we provide a comparison of R5: Our techniques can be used to attack mature real-world the number of > 100× AVGMATCHER -ReDoS texts generated security solutions. by the tools. Again, note that a slowdown of >100 times wrt. the global average for the matcher was achieved on many regexes for online DFA-simulation-based matchers (2,457 for 6.3 R1: Vulnerability of Counting Regexes grep, 742 for RE2, 1,016 for Rust, and 300 for SRM). Since In our first experiment, we show that the use of bounded the global average matching time for PHP was 3.1 s and we repetition with a higher bound in regexes creates a possible used the timeout of 300 s for matchers, in this table, the PHP attack surface for ReDoS even for online DFA-simulation- column contains the number of timeouts instead. A more based matchers. We used the set of regexes ABOVE20 and detailed analysis for other slowdown ratios is in Figure 3. tried to generate ReDoS attacks using GadgetCA and other Notice that although Hyperscan looks almost invincible in matchers using the methodology described above. the results in Table 2, we are able to slow it down by a factor First, see the top part of left-hand side of Table 2, which of 10–50 in many instances (543). shows how many successful > 100s-ReDoS texts different On the other hand, the right-hand side of Table 3 compares settings of GadgetCA were able to generate for online DFA- the numbers of generated > 100× AVGREGEX -ReDoSes. In simulation-based matchers. Notice that we were able to gen- this case, a slowdown of >100 times wrt. the average time for erate 216 ReDoS texts for grep, 110 ReDoS texts for RE2, the matcher and the regex was also achieved often for online 96 ReDoS texts for Rust, and 272 ReDoS texts for SRM (using DFA-simulation-based matchers (1,157 for grep, 1,465 for the C OUNTING strategy). RE2, 1,066 for Rust, and 279 for SRM). Next, the right-hand side of the table shows data for the We conclude that many counting regexes can be success- weaker ReDoS criterion > 10s. The number of generated suc- fully attacked using ReDoS texts created by our generator. cessful ReDoS-texts is significantly higher: 1,181 for grep, 1,116 for RE2, 295 for Rust, and 391 for SRM (all using the 6.4 R2: Regexes Without Counting C OUNTING strategy). The second experiment shows that when targeting automata- Under both ReDoS criteria above, the C OUNTING strategy based matchers, it is indeed important to exploit counting. achieves the best results for online DFA-simulation-based Since the set SUPPORTED is too large for us to run a Re- matchers and, moreover, for the >10s criterion also for back- DoS generator for each regex, we use a quick filter based on tracking matchers. Further, G REEDY obtains significantly bet- the intuition that ReDoS in these matchers is caused by gener- ter results than R ANDOM, proving that our informed search ating many large DFA states. Hence we run DFA construction strategies are better in generating hard texts than uninformed for each regex from the set. If the construction terminates 4176 31st USENIX Security Symposium USENIX Association Table 3: Numbers of regexes with successfully generated > 100× AVGMATCHER and > 100× AVGREGEX -ReDoS texts. > 100× AVGMATCHER -ReDoS attacks > 100× AVGREGEX -ReDoS attacks Generators hyper- java- hyper- java- grep re2 rust srm ca ruby php perl python java .NET grep re2 rust srm ca ruby php perl python java .NET scan Script scan Script G REEDY 1741 15 95 18 2 40 260 38 382 367 328 314 431 878 14 57 12 0 0 164 9 174 232 190 194 203 GadgetCA C OUNTING 2457 742 1016 300 5 67 1355 1596 1473 277 279 258 416 1157 1465 1066 279 2 3 1085 796 1252 407 142 140 171 R ANDOM 2033 120 122 289 3 46 348 388 412 176 177 117 258 1066 320 292 130 0 0 153 156 266 91 63 60 72 O NE L INE 1796 17 99 23 20 53 322 34 441 448 405 379 521 966 15 57 16 23 0 199 9 208 277 232 228 238 RXXR2 13 0 2 0 0 1 24 0 5 30 10 10 34 1 0 2 0 0 0 10 0 4 22 8 8 20 RegexCheck 104 0 5 0 1 0 7 1 7 11 8 4 14 4 0 4 0 0 0 3 0 0 4 3 2 2 RegexStatic 93 1 9 0 1 7 159 50 80 263 253 243 279 47 5 5 0 0 0 80 14 49 137 125 134 90 Rescue 12 0 3 0 0 2 23 2 5 23 13 12 26 1 2 4 0 0 1 12 2 6 15 7 6 14 with less than 1,000 states, we consider the regex safe. After matchers slow down (both for automata-based and backtrack- 1,000 DFA states, the construction is stopped, and the regex ing matchers). The same holds for the > 100× AVGMATCHER is marked as possibly vulnerable. This test is quick, since and > 100× AVGREGEX -ReDoS criteria in Table 3. constructing 1,000 DFA states is fast, and the vast majority Second, compare the bottom part (random text) with the of the regexes have even much smaller DFAs. middle part of the table. For counting regexes, a random text To assess the accuracy of the test in predicting that a regex is in the majority of cases actually better in creating a Re- is not vulnerable for automata-based matchers, we apply the DoS than current state-of-the-art ReDoS generators (only test on the regexes from ABOVE20 for which we did manage RegexStatic can keep up with the random text on some to generate a ReDoS text for automata-based matchers (cf. matchers). Relating this to GadgetCA in the top part of the ta- the experiment in R1). From ∼2,000 of them, only grep and ble reveals that the numbers of successfully attacked regexes Rust had cases with DFA smaller than 1,000 states, namely for the two criteria differ significantly, hence GadgetCA in- 24 cases, 6 for grep and 18 for Rust (RE2, CA, Hyperscan, deed succeeds in exploiting the critical feature of the regex. and SRM had none). These counterexample cases witness that Third, the comparison of the top part with the middle part our filter is not always right, at least for grep and Rust, and of the table shows that GadgetCA significantly outperforms ReDoS with automata-based matchers might be possible even other matchers on online DFA-simulation-based matchers and with small DFA. Still, the scarcity of these cases confirms that most of the other generators even on backtracking matchers the test is a good predictor even for grep and Rust.8 (the only exception being RegexStatic, which is comparable Running the test on SUPPORTED resulted in the following on some backtracking matchers). numbers of regexes with DFAs with >1,000 states: 6.6 R5: Real-World Security Solutions NOCOUNTERS BELOW20 ABOVE20 Our final experiment demonstrates that the results obtained in 175 (0.04 %) 343 (0.8 %) 1,600 (20 %) R1 carry over to real-world security solutions which should We then used GadgetCA to generate ReDoS candidates for be prepared for being targeted by (Re)DoS. We carried out an the regexes in NOCOUNTERS ∪ BELOW20 whose DFAs had extensive evaluation of the abilities of S NORT 3 [25], a popu- more than 1,000 states. Only 7 regexes caused > 100s-ReDoS lar and often used NIDS, which internally uses Hyperscan, to for automata-based matchers, two for grep— ‘\^.{20}\$’ withstand ReDoS attacks generated by GadgetCA. Instead of and ‘\^_.{19}\$’ (note that both also contain “higher using some of the previously introduced datasets, which might bounds” for the quantifiers)—and 5 for SRM. A > 10s-ReDoS contain regexes created by people unaware of the dangers of was caused by 24 regexes for grep and ∼6 regexes for each ReDoS, we used regexes from rulesets provided with S NORT, of RE2, Rust, and SRM. The relative sizes of the sets indicate which are written by security experts and tested in production. that regexes without higher repetition bounds are much less In particular, we used regexes from the following four rulesets: vulnerable to ReDoS for automata-based matchers (518 vul- (i) Emerging Threats Pro, (ii) Emerging Threats 3CORESec nerable from 435,166 in NOCOUNTERS ∪ BELOW20 while (versions 157 and 164), and (iii) Talos LightSPD (version 1,600 vulnerable from 8,099 in ABOVE20). 2021-03-11-001). We call the obtained set of 1,112 PCRE regexes SNORT (from the original 22,425 original regexes we 6.5 R4: Comparison with Other Generators Our next experiment confirms that our generator can create removed 16,094 regexes not supported by our tool, and then fil- new ReDoS attacks much more effectively than existing tools. tered the 1,112 regexes with quantifier bounds at least 20). The First, compare the middle part of the left- and right-hand experiment was run in two different settings: (i) on a commod- side of Table 2. For other generators, the ten-fold stronger ity x86_64 machine with S NORT using Hyperscan and (ii) on > 100s-ReDoS criterion makes almost no difference: they can- a computer with an NVIDIA BlueField-2 card [28], which pro- not find and exploit the features of the regex that make the vides its own hardware-accelerated regex matching solution. 8 The 24 cases are probably caused by specific implementation techniques Modified ReDoS Generator. In this experiment, we use or different interpretation of the regexes. The 18 cases of Rust seem to be a modified version of our ReDoS generator for the reason that related to handling of large character classes (\w appears in all 18 cases). although Hyperscan, used within S NORT, can be counted USENIX Association 31st USENIX Security Symposium 4177 Figure 3: Histogram of ratios between times of matchers for random and ReDoS text generated by GadgetCA. as an automata-based matcher, it is not based on online that, unlike for online DFA simulation, it does not matter that DFA-simulation. Experiments discussed in the previous sec- the encountered DFA states are likely to be found repeatedly tions indeed show that our ReDoS generator, which targets in the repeating instances of u since NFA simulation is not mainly online DFA-simulation, is only mildly successful caching the DFA. with Hyperscan. We therefore use here a modification of GadgetCA tailored for Hyperscan. 6.6.1 S NORT with Hyperscan on x86_64 We specifically target the following coarse abstraction of We installed S NORT 3 with enabled performance monitor and Hyperscan’s matching algorithm: the regex is split into a se- Hyperscan on a commodity x86_64 machine (we used Intel quence of sub-strings (not containing any regex operator) and i7-10510U CPU@1.80 GHz with 4 Hyper-Threading cores). sub-regexes (or a choice of such sequences) so that a word is Then, we were running S NORT on 100 MB-large PCAP files matched if it is a concatenation w = v1 · · · vn of the sub-strings with random and ReDoS IPv4 traffic that we generated and of the given regex and words matched by the sub-regexes. The captured the processing time of the regex matching engine, first phase of matching tests whether w contains all the sub- as provided in the output of the performance monitor mod- strings in the right order, by an extension of the Boyer-Moore ule. We ran two experiments with two different sizes of IP algorithm. The second phase tests whether the remaining packets for two selected Ethernet frames’ MTUs: 1,500 B and sub-words are matched by the respective sub-regexes. The op- 9,000 B (we note that bigger sizes of the payload could be portunity for slowing Hyperscan down is in the second phase, used to attack S NORT with TCP reassembly turned on). See which uses NFA-simulation to match the sub-expressions.9 Figures 4a and 4b for the slowdown that we achieved with We therefore aim at generating evil texts that contain the our ReDoS text over random text. needed sequence of sub-strings and therefore pass the first The histograms clearly show the S NORT rulesets we used phase of matching, and where the second phase is also hard. contain many possibilities for slowing S NORT down (see Ap- To do that, we use our generator to get a single evil word u pendix B for the most vulnerable regexes). In particular, using over a run that takes the CSA from the initial to a final state. packet size of 1,500 B, in 43 cases we achieved a slowdown The word is essentially generated by the first iteration of the of over 40×, with 2 regexes slowing the matcher down over while-loop on Line 11 of Algorithm 2 parameterised with the 100×. The number of vulnerable regexes is even higher for strategy C OUNTING (a single CSA run that aims at maximis- the packet size 9,000 B: 91 regexes yield a slowdown of over ing the sizes of counting sets). The word u is then iterated to 50× and 32 regexes over 100×. get the output text w = uuuu . . . of the required length. We contacted the development team of S NORT and did the A word w generated this way is likely to be evil for the responsible disclosure of the discovered vulnerable regexes. following two reasons: (i) every occurrence of u in w contains S NORT development team stated that the vulnerability is stem- all sub-strings, generating many possible splits of w into the ming from the Hyperscan library, and they mitigate it by sub-strings and the parts to be matched by the sub-regexes; restricting the length of packets on which the matching is (ii) the word u, generated by our generator, is likely to force performed as well as by using timeouts (the standard configu- large DFA states, expensive for NFA simulation. Note also ration of S NORT comes with the backtracking-based PCRE en- gine enabled, which is, however, even more prone to attacks). 9 Our abstraction of Hyperscan is coarse, but it is simple and sufficient for This might, however, lead to skipping the malicious content our needs: to show that methods similar to those for online DFA-simulation can be used to find vulnerabilities of Hyperscan too. A specialised ReDoS that can be presented at the end of the packet/data, making generator based on a more thorough analysis of Hyperscan’s algorithm the NIDS ineffective: malicious packets may get passed to might yield better results, but is already out of the scope of this paper. applications behind the NIDS. 4178 31st USENIX Security Symposium USENIX Association (a) S NORT 3@Hyperscan (1,500 B) (b) S NORT 3@Hyperscan (9,000 B) (c) NVIDIA BlueField-2 Figure 4: Histograms of slowdowns for S NORT 3 with Hyperscan (packet sizes 1,500 B and 9,000 B) and BlueField-2 regex matching for ReDoS texts over random texts. 6.6.2 NVIDIA BlueField-2 ters) of the regex matcher. Although such techniques can avert In the second part of this experiment, we used an NVIDIA the scenario of a server becoming unresponsive, they leave BlueField-2 data processing unit (BF2) MBF2H332A- a part of the input traffic not classified and potentially harmful AEEOT [29], which integrates eight 64-bit ARMv8 Cortex- or unnecessarily dropped. A mitigation specific for regexes A72 cores and houses two 25 GbE interfaces. BF2 provides with the counting operator is to substitute it by the star * op- hardware-accelerated regex matching capabilities, accessible erator, which over-approximates the language of the original via NVIDIA’s data plane development kit (DPDK) [28]: in regex (this might yield other issues, such as increasing the our experiments, we used the regex compiler rxpc and the number of false positives in an NIDS). testbed for the regex matching engine called rxpbench. In There are, however, two ways how users of regex matchers this experiment, we ran rxpbench on blocks of random and can mitigate the attacks without the mentioned disadvantages: ReDoS texts of the length 100 GB (this time, we did not need 1. Use our ReDoS generator GadgetCA to evaluate whether to chunk the texts into packets and provided the text directly a regex is ReDoS-vulnerable. in memory) and measured the throughput of the matcher. We 2. Use a matching algorithm that can handle counting effi- measured that the regex matching engine itself enables in- ciently, the one implemented in the tool CA or possibly memory processing at ∼40 Gbps. For the evaluation, we used also SRM (these matchers are still too immature to be a subset of SNORT rules containing 617 regexes that we name used in production, but an efficient implementation of SNORT-BF2 (we took all regexes from SNORT that could be the techniques they use within RE2 or Hyperscan should compiled by rxpc, which does not support some advanced give rise to a robust regex matching solution). features of PCRE, such as negative look-ahead). See Figure 4c for histograms of slowdowns we obtained 8 Conclusion and Future Work with our ReDoS text as compared to random text. Observe that We have shown that nonbacktracking automata-based regex we obtained a slowdown of more than 100× on the ReDoS matchers, which are sometimes suggested as a mitigation text in over 92 cases. Moreover, for 16 cases, we obtained of ReDoS, are still ReDoS-vulnerable. We have developed a slowdown over 500× (with the highest slowdown ratio be- a method for constructing inputs for these matchers that make ing 2,194×). See Appendix B for a list of regexes on which them perform poorly and cause significant slowdown on we obtained the largest slowdown. We have reported the vul- a large class of regexes, in particular those with counting. nerability to NVIDIA, which confirmed it to be caused by In future, we plan to focus on developing robust regex a conceptual limitation of their regex matching engine. We matchers that could prevent these kinds of attacks. A first plan to cooperate on a possible mitigation. proof of concept is the matcher CA from [46], but the class of Our results indicate that ReDoS attacks are in general suc- counting regexes it support is quite restricted; we will there- cessful in slowing down the throughput of the most recent fore explore formal models that can deal with more general hardware utilized for NIDS in the industry. Moreover, we classes of counting regexes efficiently. emphasize that for a successful ReDoS attack on an NIDS, it Acknowledgment. We thank the reviewers for their com- suffices to have a single vulnerable rule in the used rulesets. ments on how to improve the quality of the paper and the Cy- berGrid group from FEEC BUT for lending us the NVIDIA 7 Mitigation Techniques BF2 card. This work was supported by the Czech Min- Standard techniques for mitigation of ReDoS attacks are the istry of Education, Youth and Sports project LL1908 of the following: (i) setting a resource limit (e.g., a timeout) and ERC.CZ programme, the Czech Science Foundation project (ii) limiting the size of the input (e.g., to the first 100 charac- 20-07487S, and the FIT BUT internal project FIT-S-20-6427. USENIX Association 31st USENIX Security Symposium 4179 References [18] The PHP Group. PCRE patterns - PHP. https://www. php.net/manual/en/regexp.introduction.php, [1] Mono. https://www.mono-project.com/. 2021. 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For the regex Oid=[^\0D\x0A]{1000} (origi- Press Professional, Inc., 1994. nating from S NORT) GadgetCA (strategy: C OUNTING) gener- [41] Cristian-Alexandru Staicu and Michael Pradel. Freezing ates a text of several lines, each of the length 1,003 characters the web: A study of ReDoS vulnerabilities in JavaScript- and containing full or unfinished copies of the string ‘Oid=’: based web servers. In USENIX’18, pages 361–376. (Oid=)250 Oid USENIX Association, 2018. (Oid=)249 OidOid= [42] Satoshi Sugiyama and Yasuhiko Minamide. Checking ... time linearity of regular expression matching based on Each new copy of Oid= adds a new value to the counting-set backtracking. IPSJ Online Transactions, 7:82–92, 2014. and since all characters of the string ‘Oid=’ belong to the [43] Ken Thompson. Programming techniques: Regular ex- character class [^\0D\x0A], which is being counted, all ex- pression search algorithm. Commun. ACM, 11(6):419– isting values in the counting-set are also incremented. The 422, 1968. variety of full or unfinished copies of the prefix Oid= forces [44] Iain Truskett. Perl regular expressions refer- creation of many large DFA states with different counter val- ence - perl. https://perldoc.perl.org/5.22.0/ ues. The length of the shortest string matched by the regex is perlreref, 2021. 1,004 characters, however, we aim at generating the longest [45] TrustPort. World class cyber security. https://www. non-matching lines, and so the length of the generated lines trustport.com/, 2021. is 1,003 characters. The generated text is demanding for most [46] Lenka Turoňová, Lukáš Holík, Ondřej Lengál, Olli automata-based matchers (matching time for 50 MB input: Saarikivi, Margus Veanes, and Tomáš Vojnar. Regex grep: 0.83 s, Hyperscan: 0.06 s, RE2: 228.28 s, SRM: 46.54 s, matching with counting-set automata. Proc. ACM Pro- CA: 2.77 s, Rust: 96.7 s). gram. Lang., 4(OOPSLA):218:1–218:30, 2020. Example 2. For the regex <[^>\x20]{500} (originating [47] Milan Češka, Vojtěch Havlena, Lukáš Holík, Ondřej from S NORT) GadgetCA generates a text containing sub- Lengál, and Tomáš Vojnar. Approximate reduction of strings of the length 500 (the length of a minimal match is 501) finite automata for high-speed network intrusion detec- with many different placements of ‘<’: tion. In Proc. of TACAS’18, volume 10806 of LNCS. (<)500 Springer, 2018. (<)99 Q(<)400 [48] Peipei Wang, Chris Brown, Jamie A. Jennings, and ... Kathryn T. Stolee. Demystifying regular expression where Q is an arbitrary character other than ‘<’. This text bugs. Empir. Softw. Eng., 27(1):21, 2022. also forces matchers to generate many DFA states with dif- [49] Peipei Wang and Kathryn T. Stolee. How well are ferent counter values, yielding the following matching times regular expressions tested in the wild? In FSE’18, pages (on 50 MB texts): grep: 0.11 s, Hyperscan: 0.1 s, RE2: TO, 668–678. ACM, 2018. SRM: TO, GadgetCA: 2.8 s, Rust: 112.34 s. [50] Nicolaas Weideman. RegexStatic. https://github. com/NicolaasWeideman/RegexStaticAnalysis, B Attacks on Real-world Security Solutions 2015. In Tables 4 and 5, we provide examples of regexes for which [51] Nicolaas Weideman, Brink van der Merwe, Martin we managed to obtain a significant slowdown of S NORT (with Berglund, and Bruce W. Watson. Analyzing matching Hyperscan as the regex matching engine) and the NVIDIA time behavior of backtracking regular expression match- BlueField-2 DPU respectively. ers by using ambiguity of NFA. In CIAA’16, volume 9705 of LNCS, pages 322–334. Springer, 2016. USENIX Association 31st USENIX Security Symposium 4181 Table 4: Slowdown of regex matching in Snort3 with Hyperscan on x86_64. Slowdown Slowdown SID Regex (MTU=9000B) (MTU=1500B) 46310 213.95 78.89 [?&]u=[^&\s]{35} 31068 172.32 50.49 .{0,250}[\x60\x3b\x7c\x24\x28\x26] 2644 165.81 65.57 \(\s*TIMESTAMP\s*(\s*(\x27[^\x27]+’|\x22[^\x22]+\x22)\s*,)\s*((\x27[^\x27]{1000,})|(\x22[^\x22]{1000,})) 13364 163.52 71.15 src\s*\x3D(3D)?\s*[’"][^’"]{244} 19925 160.95 58.7 value\s*=\s*[\x27\x22][^\x27\x22]{257} 2102614 157.95 52.68 TIME_ZONE\s*=\s*((\x27[^\x27]{1000,})|(\x22[^\x22]{1000,})) 17659 157.41 79.18 \s*\x28(\x27[^\x27]{64}|\x27[^\x27]*\x27\s*,\s*\x27[^\x27]{64}) 2611 157.39 49.67 USING\s*((\x27[^\x27]{1000})|(\x22[^\x22]{1000})) 46309 152.34 65.7 [?&]p=[^&\s]{260} 39982 145.5 55.61 [?&]sn=[^&]{129} 2651 140.95 51.26 NUMTO(DS|YM)INTERVAL\s*\(\s*\d+\s*,\s*((\x27[^\x27]{1000,})|(\x22[^\x22]{1000,})) 2102699 138.15 49.25 TO_CHAR\s*\(\s*SYSTIMESTAMP\s*,\s*(\x27[^\x27]{256}|\x22[^\x22]{256}) 19121 136.82 63.89 SET\s*EXPLAIN\s*FILE\s*TO\s*[\x22\x27][^\x22\x27]{927} 2640 135.24 56.41 \(\s*(\x27[^\x27]*’|\x22[^\x22]+\x22)\s*,\s*(true|false)\s*,\s*((\x27[^\x27]{1000,})|(\x22[^\x22]{1000,})) 15114 135.06 51.46 embed src=\s*(\x27[^\x27]{1000}|\x22[^\x22]{1000}|[^\s\x22\x27]{1000}) document\.execCommand \(\s*[\x22\x27]InsertUnorderedList[\x22\x27]\s*\)\s*\x3B.{0,250}\s*\w+\.swapNode 29679 133.17 77.16 \(\s*[A-Za-z\(\)\"\’\.\=\]{1,75}\s*\)\s*document\.execCommand\(\s*[\x22\x27]Undo[\x22\x27]\s*\)\s*\x3B 39707 131.81 48.37 folder\s*name\s*=\s*[\x22][^\x22]{200} 39709 125.72 49.06 folder\s*name\s*=\s*[\x27][^\x27]{200} 27805 123.9 45.19 \/3001[0-9A-F]{262,304} 20889 122.09 47.98 <\s*valitem[^>]*\s(value|name)\s*=\s*([\x22\x27])[^\x22\x27]{104} 16516 121.4 44.42 sys\x2eolapimpl\x5ft\x2eodcitablestart\x28[^\x2c]+\x2c[^\x2c]+\x2c\s*\x27?[^\x2c\x27]{303} 29184 120.92 54.3 encoding\x3D[\x22\x27][^\x22\x27]{1024} 14991 120.65 61.81 select\s+xmlquery\s*\x28\s*(\x27|\x22)[^\x27\x22]{512} 43005 120.52 35.49 [?&]psk=[^&]{256} 29185 118.76 50.13 version\x3D[\x22\x27][^\x22\x27]{1024} 33310 117.95 54.87 \x3C\x21ENTITY\s+.*\s+\x22\x26[^\x22]{700} 27808 110.1 29.94 \x2f\?[a-f0-9]{60,66} 42078 108.17 43.24 [?&](cmd|pwd|usr)=[^&]{64} 2488 106.01 43.94 name=\s*[^\r\n\x3b\s\x2c]{300} Table 5: Slowdown of regex matching at an NVIDIA BlueField-2 card. Thourghput on Thourghput on SID Random Text Redos Text Slowdown Regex [Gbs] [Gbs] 2046 41.24 0.02 2,193.76 /\sPARTIAL.*BODY\.PEEK\[[^\]]\1024\/ 19213 41.19 0.02 1,681.04 /Subject\x3a\x20[^\n]*\x3fQ\x3f[^\n]{512}/ 17367 40.30 0.03 1,174.83 /\d{3}\s+[^\n]{1019}/ 6507 41.09 0.04 957.74 /\x2fnds[^\r\n]{1000}/ 1021 41.21 0.04 956.06 /\s{230,}\.htr/ 20241 40.66 0.04 947.72 /Oid\x3D[^\x0D\x0A]{1000}/ 15489 40.58 0.04 920.28 /\x3cimg[^\x3e]*src\x3d(\x22|\x27)?[^\x22\x27\s]{300}/ 3547 40.79 0.05 829.08 /php.*\x3f[^\n]{256}/ 25586 41.03 0.06 732.67 /host=[^&]{1024}/ 8060 41.31 0.06 728.49 /GET\s\x2f[^\r\n]{900}/ 31354 41.14 0.06 656.15 /\x28\x3f\x3d[^)]{300}/ 3149 41.22 0.06 655.34 /object\s[^>]*type\s*=\s*[\x22\x27][^\x22\x27]*\x2f{32}/ 17568 41.11 0.06 641.29 /\w{3}\x25\x30\x30[^\r\n]{2000}/ 4127 41.15 0.08 545.82 /\x2fnds\x2f[^&\r\n\x3b]{500}/ 38287 40.97 0.08 543.40 /akey=[^&]{500}/ 18484 41.14 0.08 536.42 /https?\x3a\x2f\x2f[^\n\r]{1000}/ 43545 41.22 0.08 485.54 /-group[^\r\n\s]{1280}/ 33310 40.96 0.09 469.76 /\x3C\x21ENTITY\s+.*\s+\x22\x26[^\x22]{700}/ 2701 41.20 0.09 434.18 /sid=[^&\x3b\r\n]{255}/ 2107 41.21 0.10 427.53 /\sCREATE\s[^\n]{1024}/ 18579 41.18 0.10 426.76 /(Context|Action)\x3D[^\x26\x3b]{1024}/ 20889 40.87 0.10 419.58 /<\s*valitem[^>]*\s(value|name)\s*=\s*([\x22\x27])[^\x22\x27]{104}/ /(\(\s*(\x27[^\x27]*\x27|\x22[^\x22]+\x22)\s*,\s*(\x27[^\x27]{1075,}|\x22[^\x22]{1075,}) 2826 41.28 0.10 416.17 |\(\s*(\x27[^\x27]{1075,}|\x22[^\x22]{1075,})|\(\s*((\x27[^\x27]*\x27|\x22[^\x22]+\x22) \s*,\s*){2}(\x27[^\x27]{1075,}|\x22[^\x22]{1075,}))/ /(\(\s*(\x27[^\x27]*\x27|\x22[^\x22]+\x22)\s*,\s*(\x27[^\x27]{1075,}|\x22[^\x22]{1075,}) 2826 40.73 0.10 410.57 |\(\s*(\x27[^\x27]{1075,}|\x22[^\x22]{1075,})|\(\s*((\x27[^\x27]*\x27|\x22[^\x22]+\x22)\s*,\s*) {2}(\x27[^\x27]{1075,}|\x22[^\x22]{1075,}))/ 21671 41.16 0.10 403.94 /zip\x3a\x2f\x2f[^\x0A\x20\x09\x0B\x0C\x85\x3E\x3C]{400}/ 20240 41.20 0.11 375.87 /Template\x3D[^\x0D\x0A]{1000}/ 27940 41.08 0.11 374.44 /password=[^\x26]{1024}/ 2103070 41.00 0.11 361.25 /\sFETCH\s[^\n]{500}/ 36195 41.21 0.12 338.07 /actserver=[^&]{982}/ 36196 40.86 0.12 335.47 /actserver=[^&]{987}/ 4182 31st USENIX Security Symposium USENIX Association