Semantics-Preserving Simplification
of Real-World Firewall Rule Sets
Cornelius Diekmann, Lars Hupel, and Georg Carle
Technische Universität München
Abstract. The security provided by a firewall for a computer network
almost completely depends on the rules it enforces. For over a decade, it
has been a well-known and unsolved problem that the quality of many
firewall rule sets is insufficient. Therefore, there are many tools to an-
alyze them. However, we found that none of the available tools could
handle typical, real-world iptables rulesets. This is due to the complex
chain model used by iptables, but also to the vast amount of possible
match conditions that occur in real-world firewalls, many of which are
not understood by academic and open source tools.
In this paper, we provide algorithms to transform firewall rulesets. We
reduce the execution model to a simple list model and use ternary logic
to abstract over all unknown match conditions. These transformations
enable existing tools to understand real-world firewall rules, which we
demonstrate on four decently-sized rulesets. Using the Isabelle theorem
prover, we formally show that all our algorithms preserve the firewall’s
filtering behavior.
Keywords: Computer Networks, Firewalls, Isabelle, Netfilter Iptables, Semantics
1 Introduction
Firewalls are a fundamental security mechanism for computer networks. Several
firewall solutions, ranging from open source [2,28,29] to commercial [3,13], exist.
Operating and managing firewalls is challenging as rulesets are usually written
manually. While vulnerabilities in the firewall software itself are comparatively
rare, it has been known for over a decade [32] that many firewalls enforce poorly
written rulesets. However, the prevalent methodology for configuring firewalls
has not changed. Consequently, studies regularly report insufficient quality of
firewall rulesets [7, 12, 18,21, 27, 31, 33, 34].
Therefore, several tools [18–22,25,30,33] have been developed to ease firewall
management and reveal configuration errors. However, when we tried to analyze
real-world firewalls with the publicly available tools, none of them could handle
our firewall rules. We found that the firewall model of the available tools is too
In this paper, we address the following fundamental problem: Many tools do
not understand real-world firewall rules. To solve the problem, we transform and
simplify the rules such that they are understood by the respective tools.
Chain INPUT (policy ACCEPT)
target prot source destination
DROP tcp tcp dpt:22
DROP tcp multiport dports
DROP udp multiport dports
DROP all
Chain DOS_PROTECT (1 references)
target prot source destination
RETURN icmp icmptype 8 limit:
avg 1/sec burst 5
DROP icmp icmptype 8
RETURN tcp tcp flags:0x17/0x04
limit: avg 1/sec burst 5
DROP tcp tcp flags:0x17/0x04
RETURN tcp tcp flags:0x17/0x02
limit: avg 10000/sec burst 100
DROP tcp tcp flags:0x17/0x02
Fig. 1. Linux iptables ruleset of a Synology NAS (network attached storage) device
To demonstrate the problem by example, we decided to use ITVal [19] be-
cause it natively supports iptables [28], is open source, and supports calls to user-
defined chains. However, ITVal’s firewall model is representative of the model
used by the majority of tools; therefore, the problems described here also apply
to a vast range of other tools. Firewall models used in related work are surveyed
in Sect. 2. For this example, we use the firewall rules in Fig. 1, taken from an NAS
device. The ruleset reads as follows: First, incoming packets are sent to the user-
defined DOS_PROTECT chain, where some rate limiting is applied. Afterwards, the
firewall allows all packets which belong to already established connections. This
is generally considered good practice. Then, some services, identified by their
ports, are blocked. Finally, the firewall allows all packets from the local network and discards all other packets. We used ITVal to partition the IP
space into equivalence classes (i.e. ranges with the same access rights) [20]. The
expected result is a set of two IP ranges: the local network and
the “rest”. However, ITVal erroneously only reports one IP range: the universe.
Removing the first two rules (in particular the call in the DOS_PROTECT chain)
lets ITVal compute the expected result.
We identified two main problems which prevent tools from “understanding”
real-world firewalls. First, calling and returning from custom chains, due to the
possibility of complex nested chain calls. Second, more seriously, most tools do
not understand the firewall’s match conditions. In the above example, the rate
limiting is not understood. The problem of unknown match conditions cannot
simply be solved by implementing the rate limiting feature for the respective
tool. The major reason is that the underlying algorithm might not be capable of
dealing with this special case. Additionally, firewalls, such as iptables, support
numerous match conditions and several new ones are added in every release.
expect even more match conditions for nftables [29] in the future since they can
be written as simple userspace programs [17]. Therefore, it is virtually impossible
to write a tool which understands all possible match conditions.
In this paper, we build a fundamental prerequisite to enable tool-supported
analysis of real-world firewalls: We present several steps of semantics-preserving
ruleset simplification, which lead to a ruleset that is “understandable” to subse-
quent analysis tools: First, we unfold all calls to and returns from user-defined
chains. This process is exact and valid for arbitrary match conditions. After-
wards, we process unknown match conditions. For that, we embed a ternary-logic
semantics into the firewall’s semantics. Due to ternary logic, all match conditions
not understood by subsequent analysis tools can be treated as always yielding
an unknown result. In a next step, all unknown conditions can be removed. This
introduces an over- and underapproximation ruleset, called upper/lower closure.
Guarantees about the original ruleset dropping/allowing a packet can be given
by using the respective closure ruleset.
To summarize, we provide the following novel contributions:
1. a formal semantics of iptables packet filtering (Sect. 4),
2. chain unfolding: transforming a ruleset in the complex chain model to a
ruleset in the simple list model (Sect. 5),
3. an embedded semantics with ternary logic, supporting arbitrary match con-
ditions, introducing a lower/upper closure of accepted packets (Sect. 6), and
4. normalization and translation of complex logical expressions to an iptables-
compatible format, discovering a meta-logical firewall algebra (Sect. 7).
We evaluate applicability on large real-world firewalls in Sect. 8. All proofs
are machine-verified with Isabelle [24] (Sect. 3). Therefore, the correctness of
all obtained results only depends on a small and well-established mathematical
kernel and the iptables semantics (Fig. 2).
2 Firewall Models in the Literature and Related Work
Packets are routed through the firewall and the firewall needs to decide whether
to allow or deny a packet. A firewall ruleset determines the firewall’s filtering
behavior. The firewall inspects its ruleset for each single, arbitrary packet to
determine the action to apply to the packet. The ruleset can be viewed as a
list of rules; usually it is processed sequentially and the first matching rule is
The literature agrees on the definition of a single firewall rule. It consists of a
predicate (the match expression) and an action. If the match expression applies
to a packet, the action is performed. Usually, a packet is scrutinized by several
rules. Zhang et al. [34] specify a common format for packet filtering rules. The
action is either “allow” or “deny”, which directly corresponds to the firewall’s
filtering decision. The ruleset is processed strictly sequentially. Yuan et al. [33]
call this the simple list model. ITVal also supports calls to user-defined chains
as an action. This allows “jumping” within the ruleset without having a final
filtering decision yet. This is called the complex chain model [33]. Zhang et al. [34]
support matching on the following packet header fields: IP source and destination
address, protocol, and port on layer 4. This model is commonly found in the
literature [4, 5, 25, 33, 34]. ITVal extends these match conditions with flags (e.g.
TCP SYN) and connection states (INVALID, NEW, ESTABLISHED, RELATED). The
state matching is treated as just another match condition.
This model is similar
to Margrave’s model for IOS [21]. When comparing these features to the simple
firewall in Fig. 1, it becomes obvious that none of these tools supports that
We are not aware of any tool which uses a model fundamentally different
than those described in the previous paragraph. Our model enhances existing
work in that we use ternary logic to support arbitrary match conditions. To an-
alyze a large iptables firewall, the authors of Margrave [21] translated it to basic
Cisco IOS access lists [3] by hand. With our simplification, we can automatically
remove all features not understood by basic Cisco IOS. This enables transla-
tion of any iptables firewall to a basic Cisco access lists which is guaranteed to
drop no more packets than the original iptables firewall. This opens up all tools
available only for Cisco IOS access lists, e.g. Margrave [21] and Header Space
Analysis [15].
3 Formal Verification with Isabelle
We verified all proofs with Isabelle, using its standard Higher-Order Logic (HOL).
The corresponding theory files are publicly available. An interested reader may
consult the detailed (100+ pages) proof document.
Notation. We use pseudo code close to SML and Isabelle. Function ap-
plication is written without parentheses, e.g. f a denotes function f applied
to parameter a. We write for prepending a single element to a list, e.g.
ab [c, d] = [a, b, c, d], and ∶∶ for appending lists, e.g. [a, b][c, d] = [a, b, c, d].
The empty list is written as []. [f a. a l] denotes a list comprehension, i.e.
applying f to every element a of list l. [f x y. x l
, y l
] denotes the list
comprehension where f is applied to each combination of elements of the lists l
and l
. For f x y = (x, y), this returns the cartesian product of l
and l
4 Semantics of iptables
We formalized the semantics of a subset of iptables. The semantics focuses on
access control, which is done in the INPUT, OUTUT, and FORWARD chain. Thus
packet modification (e.g. NAT) is not considered (and also not allowed in these
Match conditions, e.g. source and protocol TCP, are called
primitives. A primitive matcher γ decides whether a packet matches a primitive.
Formally, based on a set X of primitives and a set of packets P , a primitive
matcher γ is a binary relation over X and P . The semantics supports arbi-
trary packet models and match conditions, hence both remain abstract in our
In one firewall rule, several primitives can be specified. Their logical con-
nective is conjunction, for example src and tcp. Disjunction
is omitted because it is neither needed for the formalization nor supported by
iptables; this is consistent with the model by Jeffrey and Samak [14]. Primitives
can be combined in an algebra of match expressions M
mexpr = x for x X ¬ mexpr mexpr mexpr True
For a primitive matcher γ and a match expression m M
, we write m
if a packet p P matches m, essentially lifting γ to a relation over M
and P ,
with the connectives defined as usual. With completely generic P , X, and γ, the
semantics can be considered to have access to an oracle which understands all
possible match conditions.
Furthermore, we support the following actions, modeled closely after iptables:
Accept, Reject, Drop, Log, Empty, Call c for a chain c, and Return. A rule can
be defined as a tuple (m, a) for a match expression m and an action a. A
list (or sequence) of rules is called a chain. For example, the beginning of the
DOS_PROTECT chain in Fig. 1 is [(icmp icmptype 8 limit: . . . , Return), . . . ].
A set of chains associated with a name is called a ruleset. Let Γ denote
the mapping from chain names to chains. For example, Γ DOS_PROTECT returns
the contents of the DOS_PROTECT chain. We assume that Γ is well-formed that
means, if a Call c action occurs in a ruleset, then the chain named c is defined
in Γ . This assumption is justified as the Linux kernel only accepts well-formed
The semantics of a firewall w.r.t. to a given packet p, a background ruleset
Γ , and a primitive matcher γ can be defined as a relation over the currently
active chain and the state before and the state after processing this chain. The
semantics is specified in Fig. 2. The expression p
rs, t
states that
starting with state t, after processing the chain rs, the resulting state is t
. For
a packet p, our semantics focuses on firewall filtering decisions. Therefore, only
the following three states are necessary: The firewall may allow ( ) or deny ( )
the packet, or it may not have come to a decision yet (
We will now discuss the most important rules. The Accept rule describes the
following: if the packet p matches the match expression m, then the firewall with
no filtering decision (
) processes the singleton chain [(m, Accept)] by switching
to the allow state. Both the Drop and Reject rules deny a packet; the difference
is only in whether the firewall generates some informational message, which does
not influence filtering. The NoMatch rule specifies that if the firewall has not
come to a filtering decision yet, it can process any non-matching rule without
changing its state. The Decision rule specifies that as soon as the firewall made
[], t
[(m, Accept)],
[(m, Drop)],
[(m, Reject)],
¬ m
[(m, a)],
rs, t
t p
, t
p p
Γ c,
[(m, Call c)],
p Γ c = rs
, Return) ∶∶ rs
p p
[(m, Call c)],
[(m, Log)],
[(m, Empty)],
(for any primitive matcher γ and any well-formed ruleset Γ )
Fig. 2. Big Step semantics for iptables
a filtering decision, it does not change its decision. The Seq rule specifies that
if the firewall has not come to a filtering decision and it processes the chain rs
which results in state t and starting from t processes the chain rs
which results
in state t
, then both chains can be processed sequentially, ending in state t
The CallResult rule specifies that if a matching Call to a chain named c
occurs, the resulting state t is the result of processing the chain Γ c. Likewise,
the CallReturn rule specifies that if processing a prefix rs
of the called chain
does not lead to a filtering decision and directly afterwards, a matching Return
rule occurs, the called chain is processed without result.
The Log rule does not
influence the filtering behavior. Similarly, the Empty rule does not result in a
filtering decision. An Empty rule, i.e. a rule without an action, occurs if iptables
only updates its internal state, e.g. updating packet counters.
The subsequent parts of this paper are all based on these semantics. When-
ever we provide a procedure P to operate on chains, we proved that the firewall’s
filtering behavior is preserved, formally:
P rs, t
iff p
rs, t
All our proofs are machine-verified with Isabelle. Therefore, once the reader is
convinced of the semantics as specified in Fig. 2, the correctness of all subsequent
theorems follows automatically without any hidden assumptions or limitations.
The rules in Fig. 2 are designed such that every rule can be inspected indi-
vidually. However, considering all of them together, it is not immediately clear
whether the result depends on the order of their application to a concrete ruleset
and packet. Theorem 1 states that the semantics is deterministic, i.e. only one
uniquely defined outcome is possible.
Theorem 1 (Determinism).
If p
rs, s
t and p
rs, s
then t = t
5 Custom Chain Unfolding
In this section, we present algorithms to convert a ruleset from the complex
chain model to the simple list model.
The function pr (“process return”) iterates over a chain. If a Return rule
is encountered, all subsequent rules are amended by adding the Return rule’s
negated match expression as a conjunct. Intuitively, if a Return rule occurs in
a chain, all following rules of this chain can only be reached if the Return rule
does not match.
add-match m
rs = [(m m
, a). (m, a) rs]
pr [] = []
pr ((m, Return) rs) = add-match (¬m) (pr rs)
pr ((m, a) rs) = (m, a) ∶∶ pr rs
The function pc (“process call”) iterates over a chain, unfolding one level of
Call rules. If a Call to the chain c occurs, the chain itself (i.e. Γ c) is inserted
instead of the Call. However, Returns in the chain need to be processed and the
match expression for the original Call needs to be added to the inserted chain.
pc [] = []
pc ((m, Call c) rs) = add-match m (pr (Γ c)) pc rs
pc ((m, a) rs) = (m, a) pc rs
The procedure pc can be applied arbitrarily many times and preserves the
semantics. It is sound and complete.
Theorem 2 (Soundness and Completeness).
rs, t
iff p
rs, t
In each iteration, the algorithm unfolds one level of Calls. The algorithm
needs to be applied until the result no longer changes. Note that the semantics
[(¬ (icmp icmptype 8 limit: . . . ) icmp icmptype 8, Drop) ,
(¬ (icmp icmptype 8 limit: . . . ) ¬ (tcp tcp flags:0x17/0x04 limit: . . . )
tcp tcp flags:0x17/0x04, Drop), . . . , (src, Accept) , . . . ]
Fig. 3. Unfolded Synology Firewall
allows non-terminating rulesets; however, the only rulesets that are interesting
for analysis are the ones actually accepted by the Linux kernel.
Since it rejects
rulesets with loops, both our algorithm and the resulting ruleset are guaranteed
to terminate.
Corollary 1. Every ruleset (with only Accept, Drop, Reject, Log, Empty,
Call, Return actions) accepted by the Linux kernel can be unfolded completely
while preserving its filtering behavior.
In addition to unfolding calls, the following transformations applied to any
ruleset preserve the semantics:
Replacing Reject actions with Drop actions,
Removing Empty and Log rules,
Simplifying match expressions which contain True or ¬ True.
For some given primitive matcher, specific optimizations may also be per-
formed, e.g. rewriting src to True.
Therefore, after unfolding and optimizing, a chain which only contains Allow
or Drop actions is left. In the subsequent sections, we require this as a precondi-
tion. As an example, recall the firewall in Fig. 1. Its INPUT chain after unfolding
and optimizing is listed in Fig. 3. Observe that the computed match expressions
are beyond iptable’s expressiveness. An algorithm to normalize the rules to an
iptables-compatible format will be described in Sect. 7.
6 Unknown Primitives
As we argued earlier, it is infeasible to support all possible primitives of a firewall.
Suppose a new firewall module is created which provides the ssh_blacklisted
and ssh_innocent primitives. The former applies if an IP address has had too
many invalid SSH login attempts in the past; the latter is the opposite of the
former. Since we made up these primitives, no existing tool will support them.
However, a new version of iptables could implement them or they can be provided
as third-party kernel modules. Therefore, our ruleset transformations must take
unknown primitives into account. To achieve this, we lift the primitive matcher
γ to ternary logic, adding Unknown as matching outcome. We embed this new
“approximate” semantics into the semantics described in the previous sections.
Thus, it becomes easier to construct matchers tailored to the primitives sup-
ported by a particular tool.
6.1 Ternary Matching
Logical conjunction and negation on ternary values are as before, with these
additional rules for Unknown operands (commutative cases omitted):
True Unknown = Unknown False Unknown = False ¬ Unknown = Unknown
These rules correspond to Kleene’s 3-valued logic [16] and are well-suited for
firewall semantics: The first equation states that, if one condition matches, the
final result only depends on the other condition. The next equation states that
a rule cannot match if one of its conditions does not match. Finally, by negating
an unknown value, no additional information can be inferred.
We demonstrate this by example: the two rulesets [(ssh_blacklisted, Drop)]
and [(True, Call c)] where Γ c = [(ssh_innocent, Return), (True, Drop)] have
exactly the same filtering behavior. After unfolding, the second ruleset collapses
to [(¬ ssh_innocent, Drop)]. Both the ssh_blacklisted and the ssh_innocent
primitives are Unknown to our matcher. Thus, since both rulesets have the same
filtering behavior, a packet matching Unknown in the first ruleset should also
match ¬ Unknown in the second ruleset matches.
6.2 Closures
In the ternary semantics, it may be unknown whether a rule applies to a packet.
Therefore, the matching semantics are extended with an “in-doubt”-tactic. This
tactic is consulted if the result of a match expression is Unknown. It decides
whether a rule applies.
We introduce the in-doubt-allow and in-doubt-deny tactics. The first tactic
forces a match if the rule’s action is Accept and a mismatch if it is Drop. The
second tactic behaves in the opposite manner. Note that an unfolded ruleset is
necessary, since no behavior can be specified for Call and Return actions.
We denote the exact Boolean semantics with and embedded ternary
semantics with an arbitrary tactic α with
”. In particular, α = allow for
in-doubt-allow and α = deny analogously.
are related to the in-doubt-tactics as follows: considering the
set of all accepted packets, in-doubt-allow is an overapproximation, whereas in-
doubt-deny is an underapproximation. In other words, if accepts a packet,
also accepts the packet. Thus, from the opposite perspective, the
in-doubt-allow tactic can be used to guarantee that a packet is certainly dropped.
Likewise, if denies a packet, then
also denies this packet. Thus, the
in-doubt-deny tactic can be used to guarantee that a packet is certainly accepted.
For example, the unfolded firewall of Fig. 1 contains rules which drop a packet
if a limit is exceeded. If this rate limiting is not understood by γ, the in-doubt-
allow tactic will never apply this rule, while with the in-doubt-deny tactic, it is
applied universally.
We say that the Boolean and the ternary matchers agree iff they return the
same result or the ternary matcher returns Unknown. Interpreting this definition,
the ternary matcher may always return Unknown and the Boolean matcher serves
as an oracle which knows the correct result. Note that we never explicitly specify
anything about the Boolean matcher; therefore the model is universally valid,
i.e. the proof holds for an arbitrary oracle.
If the exact and ternary matcher agree, then the set of all packets allowed
by the in-doubt-deny tactic is a subset of the packets allowed by the exact se-
mantics, which in turn is a subset of the packets allowed by the in-doubt-allow
tactic. Therefore, we call all packets accepted by
the lower closure, i.e.
the semantics which accepts at most the packets that the exact semantics ac-
cepts. Likewise, we call all packets accepted by
the upper closure, i.e. the
semantics which accepts at least the packets that the exact semantics accepts.
Every packet which is not in the upper closure is guaranteed to be dropped by
the firewall.
Theorem 3 (Lower and Upper Closure of Allowed Packets).
p. p
p. p
p. p
The opposite holds for the set of denied packets.
For the example in Fig. 1, we computed the closures (without the RELATED,
ESTABLISHED rule, see Sect. 6.4) and a ternary matcher which only understands
IP addresses and layer 4 protocols. The lower closure is the empty set since
rate limiting could apply to any packet. The upper closure is the set of packets
originating from
6.3 Removing Unknown Matches
In this section, as a final optimization, we remove all unknown primitives. We
call this algorithm pu (“process unknowns”). For this step, the specific ternary
matcher and the choice for the in-doubt-tactic must be known.
In every rule, top-level unknown primitives can be rewritten to True or
¬True. For example, let m
be a primitive which is unknown to γ. Then, for in-
doubt-allow, (m
, Accept) is equal to (True, Accept) and (m
, Drop) is equal
to (¬True, Drop). Similarly, negated unknown primitives and conjunctions of
(negated) unknown primitives can be rewritten.
Hence, the base cases of pu are straightforward. However, the case of a
negated conjunction of match expressions requires some care.The following equa-
tion represents the De Morgan rule, specialized to the in-doubt-allow tactic.
pu (¬(m
), a) =
True if pu (¬m
, a) = True
True if pu (¬m
, a) = True
pu (¬m
, a) if pu (¬m
, a) = ¬ True
pu (¬m
, a) if pu (¬m
, a) = ¬ True
¬(¬pu (¬m
, a) ¬pu (¬m
, a)) otherwise
The ¬ Unknown = Unknown equation is responsible for the complicated nature
of the De Morgan rule. Fortunately, we machine-verified all our algorithms. For
example, during our research, we wrote a seemingly simple (but incorrect) ver-
sion of pu and everybody agreed that the algorithm looks correct. In the early
empirical evaluation, with yet unfinished proofs, we did not observe our bug.
Only because of the failed correctness proof did we realize that we introduced
an equation that only holds in Boolean logic.
Theorem 4 (Soundness and Completeness).
[pu r. r rs], t
iff p
rs, t
Theorem 5. Algorithm pu removes all unknown primitive match expressions.
An algorithm for the in-doubt-deny tactic (with the same equation for the
De Morgan case) can be specified in a similar way. Thus,
can be treated as
if it were defined only on Boolean logic with only known match expressions.
As an example, we examine the ruleset of the upper closure of Fig. 1 (with-
out the RELATED,ESTABLISHED rule, see Sect. 6.4) for a ternary matcher which
only understands IP addresses and layer 4 protocols. The ruleset is simplified to
[(src, Accept), (True, Drop)]. ITVal can now directly com-
pute the correct results on this ruleset.
Since firewalls process rules sequentially, the first rule has no dependency on any
previous rules. Similarly, rules at the beginning have very low dependencies on
other rules. Therefore, firewall rules in the beginning can be inspected manually,
whereas the complexity of manual inspection increases with every additional
preceding rule.
It is good practice [9] to start a firewall with an ESTABLISHED (and some-
times RELATED) rule. This also happens in Fig. 1 after the rate limiting. The
ESTABLISHED rule usually matches most of the packets [9],
which is important
for performance; however, when analyzing the filtering behavior of a firewall, it is
important to consider how a connection can be brought to this state. Therefore,
we remove this rule and only focus on the connection setup.
The ESTABLISHED rule essentially allows packet flows in the opposite direc-
tion of all subsequent rules [6]. Unless there are special security requirements
(which is not the case in any of our analyzed scenarios), the ESTABLISHED rule
can be excluded when analyzing the connection setup [6, Corollary 1].
If the
ESTABLISHED rule is removed and in the subsequent rules, for example, a prim-
itive state NEW occurs, our ternary matcher returns Unknown. The closure pro-
cedures handle these cases automatically, without the need for any additional
7 Normalization
Ruleset unfolding may result in non-atomic match expressions, e.g. ¬ (a b).
iptables only supports match expressions in Negation Normal Form (NNF).
There, a negation may only occur before a primitive, not before compound
expressions. For example, ¬ (src ip) tcp is a valid NNF formula, whereas
¬ ((src ip) tcp) is not. We normalize match expressions to NNF, using the
following observations:
The De Morgan rule can be applied to match expressions, splitting one rule
into two. For example, [(¬ (src ip tcp), Allow)] and [(¬ src ip, Allow),
(¬ tcp, Allow)] are equivalent. This introduces a “meta-logical” disjunction con-
sisting of a sequence of consecutive rules with a shared action. For example,
, a), (m
, a)] is equivalent to [(m
, a)].
For sequences of rules with the same action, a distributive law akin to com-
mon Boolean logic holds. For example, the conjunction of the two rulesets
, a), (m
, a)] and [(m
, a), (m
, a)] is equivalent to the ruleset [(m
a), (m
, a), (m
, a), (m
, a)]. This can be illustrated with a
situation where a = Accept and a packet needs to pass two firewalls in a row.
We can now construct a procedure which converts a rule with a complex
match expression to a sequence of rules with match expressions in NNF. It is
independent of the particular primitive matcher and the in-doubt tactic used.
The algorithm n (“normalize”) of type M
) is defined as follows:
n True = [True]
n (m
) = [x y. x n m
, y n m
n (¬ (m
)) = n (¬m
) n (¬m
n (¬¬m) = n m
n (¬True) = []
n x = [x]
for x X
n (¬x) = [¬x]
The second equation corresponds to the distributive law, the third to the De
Morgan rule. For example, n (¬(src ip tcp)) = [¬src ip, ¬tcp]. The fifth
rule states that non-matching rules can be removed completely.
The unfolded ruleset of Fig. 3, which consists of 9 rules, can be normalized to
a ruleset of 20 rules (due to distributivity). In the worst case, normalization can
cause an exponential blowup. Our evaluation shows that this is not a problem
in practice, even for large rulesets. This is because rulesets are usually managed
manually, which naturally limits their complexity to a level processible by state-
of-the-art hardware.
Theorem 6. n always terminates, all match expressions in the returned list
are in NNF, and their conjunction is equivalent to the original expression.
We show soundness and completeness w.r.t. arbitrary γ, α, and primitives.
Hence, it also holds for the Boolean semantics. In general, proofs about the
ternary semantics are stronger (the ternary primitive matcher can simulate the
Boolean matcher).
Theorem 7 (Soundness and Completeness).
, a). m
n m], t
iff p
[(m, a)], t
After having been normalized by n, the rules can mostly be fed back to
iptables. For some specific primitives, iptables imposes additional restrictions,
e.g. that at most one primitive of a type may be present in a single rule. For
our evaluation, we only need to solve this issue for IP address ranges in CIDR
notation [10]. We introduced and verified another transformation which com-
putes intersection of IP address ranges, which returns at most one range. This
is sufficient to process all rulesets we encountered during evaluation.
8 Evaluation
In this section, we demonstrate the applicability of our ruleset preprocessing.
Usually, network administrators are not inclined towards publishing their firewall
ruleset because of potential negative security implications. For this evaluation, we
have obtained approximately 20k real-world rules and the permission to publish
them. In addition to the running example in Fig. 1 (a small real-world firewall),
we tested our algorithms on four other real-world firewalls. We put focus on the
third ruleset, because it is one of the largest and the most interesting one.
For our analysis, we wanted to know how the firewall partitions the IPv4
space. Therefore, we used a matcher γ which only understands source/destina-
tion IP addresses and the layer 4 protocols TCP and UDP. Our algorithms do
not require special processing capabilities, they can be executed within seconds
on a common off-the-shelf 4 GB RAM laptop.
Ruleset 1 is taken from a Shorewall [8] firewall, running on a home router, with
around 500 rules. We verified that our algorithms correctly unfold, preprocess,
and simplify this ruleset. We expected to see, in both the upper and lower closure,
that the firewall drops packets from private IP ranges. However, we could not
see this in the upper closure and verified that the firewall does indeed not block
such packets if their connection is in a certain state. The administrator of the
firewall confirmed this issue and is currently investigating it.
Ruleset 2 is taken from a small firewall script found online [1]. Although it only
contains about 50 rules, we found that it contains a serious mistake. We assume
the author accidentally confused iptables -I (insert at top) and -A (append at
tail) options. We saw this after unfolding, as the firewall allows nearly all pack-
ets at the beginning. Subsequent rules are shadowed and cannot apply. However,
these rules come with a documentation of their intended purpose, such as “drop
reserved addresses”, which highlights the error. We verified the erroneous behav-
ior by installing the firewall on our systems. The author is currently investigating
this issue. Thus, our unfolding algorithm alone can provide valuable insights.
Ruleset 3 & 4 are taken from the main firewall of our lab (Chair for Network
Architectures and Services). One snapshot was taken 2013 with 2800 rules and
one snapshot was taken 2014, containing around 4000 rules. It is obvious that
these rulesets have historically grown. About ten years ago, these two rulesets
would have been the largest real-world rulesets ever analyzed in academia [32].
We present the analysis results of the 2013 version of the firewall. Details can
be found in the additional material. We removed the first three rules. The first
rule was the ESTABLISHED rule, as discussed in Sect. 6.4. Our focus was put on
the second rule when we calculated the lower closure: this rule was responsible
for the lower closure being the empty set. Upon closer inspection of this rule, we
realized that it was ‘dead’, i.e. it can never apply. We confirmed this observation
by changing the target to a Log action on the real firewall and could never see
a hit of this rule for months. Due to our analysis, this rule could be removed.
The third rule performed SSH rate limiting (a Drop rule). We removed this
rule because we had a very good understanding of it. Keeping it would not
influence correctness of the upper closure, but lead to a smaller lower closure
than necessary.
First, we tested the ruleset with the well-maintained Firewall Builder [22].
The original ruleset could not be imported by Firewall Builder due to 22 er-
rors, caused by unknown match expressions. Using the calculated upper closure,
Firewall Builder could import this ruleset without any problems.
Next, we tested ITVal’s IP space partitioning query [20]. On our original
ruleset with 2800 rules, ITVal completed the query with around 3 GB of RAM
in around 1 min. Analyzing ITVal’s debug output, we found that most of the rules
were not understood correctly due to unknown primitives. Thus, the results were
spurious. We could verify this as, obviously dropped by our firewall,
was grouped into the same class as the rest of the Internet. In contrast, using
the upper and lower closure ruleset, ITVal correctly identifies as its
own class.
We found another interesting result about the ITVal tool: The (optimized)
upper closure ruleset only contains around 1000 rules and the lower closure only
around 500 rules. Thus, we expected that ITVal could process these rulesets
significantly faster. However, the opposite is the case: ITVal requires more than
10 times the resources (both CPU and RAM, we had to move the analysis to
a > 40 GB RAM cluster) to finish the analysis of the closures. We assume that
this is due to the fact that ITVal now understands all rules.
9 Conclusion
This work was motivated by the fact that we could not find any tool which
helped analyzing our lab’s and other firewall rulesets. Though much related work
about firewall analysis exists, all academic firewall models are too simplistic to
be applicable to those real-world rulesets. With the transformations presented
in this paper, they became processable by existing tools. With only a small
amount of manual inspection, we found previously unknown issues in four real-
world firewalls.
We introduced an approximation to reduce even further the complexity of
real-world firewalls for subsequent analysis. In our evaluation, we found that the
approximation is good enough to provide meaningful results. In particular, using
further tools, we were finally able to provide our administrator with a meaningful
answer to the question of how our firewall partitions the IP space.
Our transformations can be extended for different firewall configurations. A
user must only provide a primitive matcher for the firewall match conditions she
wishes to support. Since we use ternary logic, a user can specify “unknown” as
matching outcome, which makes definition of new primitive matchers very easy.
The resulting firewall ruleset conforms to the simple list model in Boolean logic
(i.e. the common model found in the literature).
Future work includes increasing the accuracy of the approximation by pro-
viding more feature-rich primitive matchers and directly implementing firewall
analysis algorithms in Isabelle to formally verify them. Another planned appli-
cation is to assist firewall migration between different vendors and migrating
legacy firewall systems to new technologies. In particular, such a migration can
be easily prototyped by installing a new firewall in chain with the legacy firewall
such that packets need to pass both systems: with the assumption that users
only complain if services no longer work, the formal argument in this paper
proves that the new firewall with an upper closure ruleset operates without user
complaints. A new fast firewall with a lower closure ruleset allows bypassing a
slow legacy firewall, probably removing a network bottleneck, without security
The analyzed firewall rulesets can be found at
Our Isabelle formalization can be obtained from
A special thanks goes to Andreas Korsten for valuable discussions. We thank
Julius Michaelis for contributing his Shorewall firewall. We express our grat-
itude to both for agreeing to publish their firewalls. In addition, Julius and
Lars Noschinski contributed proofs to the formalization of the IP address space.
Manuel Eberl, Lukas Schwaighofer, and Fabian Immler commented on early
drafts of this paper. This work was greatly inspired by Tobias Nipkow’s and
Gerwin Klein’s book on semantics in Isabelle [23].
This work has been supported by the German Federal Ministry of Education
and Research (BMBF), EUREKA project SASER, grant 16BP12304, and by the
European Commission, FP7 project EINS, grant 288021.
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As of version 1.4.21 (Linux kernel 3.13), iptables supports more than 50 match
Firewalls can be stateful or stateless. Most firewalls nowadays are stateful, which
means the firewall remembers and tracks information of previously seen packets, e.g.
the TCP connection a packet belongs to and the state of this connection. ITVal does
not track the state of connections. Match conditions on connection states are treated
exactly the same as matches on a packet header. In general, focusing on rulesets and not
firewall implementation, matching on iptables conntrack states is exactly as matching
on any other (stateless) condition. However, internally, not only the packet header is
consulted but also the current connection tables. Note that existing firewall analysis
tools also largely ignore state [21]. In our semantics, we also model stateless matching.
Note that the other direction is considered easy [26], because basic Cisco IOS access
lists have “no nice features” [11]. Note that there also are Advanced Access Lists.
The semantics gets stuck if a Return occurs on top-level. However, this is not a
problem since we make sure that this cannot happen. iptables specifies that a Return
on top-level in a built-in chain is allowed and in this corner case, the chain’s default
policy is executed. To comply with this behavior, we always start analysis of a ruleset
as follows: [(True, Call start-chain), (True, default-policy)], where the start chain is
one of iptables built-in INPUT, FORWARD, or OUTPUT chains with a default policy of either
Accept or Drop.
A rule without an action can also be used to mark a packet for later handling.
This marking may influence the filtering decision. Since our primitive matchers and
packets are completely generic, this case can be represented within our model: Instead
of updating the firewall’s internal state, an additional “ghost field” must be introduced
in the packet model. Since packets are immutable, this field cannot be set by a rule but
the packet must be given to the firewall with the final value of the ghost field already
set. Hence, an analysis must be carried out with an arbitrary value of the ghost fields.
We admit that this model is very unwieldy. However, when later embedding the more
practical ternary semantics, we want to mention that all primitives which mark a packet
for later processing can be considered “unknown” and are correctly abstracted by these
The relevant check is in mark_source_chains, file source/net/ipv4/netfilter/
ip_tables.c of the Linux kernel version 3.2.
The final decision ( or ) for Call and Return rules depends on the called/calling
We revalidated this observation in September 2014 and found that in our firewall,
which has seen more than 15 billion packets (19+ Terabyte data) since the last reboot,
more than 95% of all packets matched the first RELATED,ESTABLISHED rule.
The same can be concluded for reflexive ACLs in Cisco’s IOS Firewall [3].
Since match expressions do not contain disjunctions, any match expression in NNF
is trivially also in Disjunctive Normal Form (DNF).