Computer network systems are often subject to several types of attacks. For example the distributed Denial of Service (DDoS) attack introduces an excessive trafic load to a web server to make it unusable. A popular method for detecting attacks is to use the sequence of source IP addresses to detect possible anomalies. With the aim of predicting the next IP address, the Probability Density Function of the IP address sequence is estimated. Prediction of source IP address in the future access to the server is meant to detect anomalous requests. In this paper we consider the sequence of IP addresses as a numerical sequence and develop the nonlinear analysis of the numerical sequence. We used nonlinear analysis based on Volterra's Kernels and Hammerstein's models. The experiments carried out with datasets of source IP address sequences show that the prediction errors obtained with Hammerstein models are smaller than those obtained both with the Volterra Kernels and with the sequence clustering by means of the K-Means algorithm.
User modeling is an important task for web applications dealing with large trafic flows. They can be used for a variety of applications such as to predict future situations or classify current states. Furthermore, user modeling can improve detection or mitigation of Distributed Denial of Service (DDoS) attack [1, 2, 3], improve the quality of service (QoS) [4], individuate click fraud detection and optimize trafic management. In peer-to-peer (P2P) overlay networks, IP models can also be used for optimizing request routing [5]. Those techniques are used by severs for deciding how to manage the actual trafic. In this context, also outlier detection methods are often used if only one class is known. If, for example, an Intrusion Prevention System wants to mitigate DDoS attacks, it usually has only seen the normal trafic class before and it has to detect the outlier class by its diferent behavior. In this paper we deal with the management of DDos because nowadays it has become a major threat in the internet. Those attacks are done by using a large scaled networks of infected PCs (bots or zombies) that combine their bandwidth and computational power in order to overload a publicly available service and denial it for legal users. Due to the open structure of the internet, all public servers are vulnerable to DDoS attacks. The bots are usually acquired automatically by hackers who use software tools to scan through the network, detecting vulnerabilities and exploiting the target machine. Furthermore, there is also a strong need to mitigate DDoS attacks near the target, which seems to be the only solution to the problem in the current internet infrastructure. The aim of such a protection system is to limit their destabilizing efect on the server through identifying malicious requests. There are multiple strategies with dealing with DDoS attacks. The most efective ones are the near-target filtering solutions. They estimates normal user behavior based on IP packet header information. Then, during an attack the access of outliers is denied. One parameter that all methods have in common is the source IP address of the users. It is the main discriminant for DDoS trafic classification. However, the methods of storing IP addresses and estimating their density in the huge IP address space, are diferent. In this paper, we present a novel approach based on system identification techniques and, in particular, on the Hammerstein models.
Data driven identification of mathematical models of physical systems (i.e. nonlinear) starts with representing the systems as a black box. In other terms, while we may have access to the inputs and outputs, the internal mechanisms are totally unknown to us. Once a model type is chosen to represent the system, its parameters are estimated through an optimization algorithm so that eventually the model mimics at a certain level of fidelity the inner mechanism of the nonlinear system or process using its inputs and outputs. These approach is, for instance, widely used in the related big data analytics area (e.g., [6, 7, 8, 9, 10, 11, 12, 13, 14])
In this work, we consider a particular sub-class of nonlinear predictors: the
Linear-in-theparameters (LIP) predictors. LIP predictors are characterized by a linear dependence of the
predictor output on the predictor coeficients. Such predictors are inherently stable, and that
they can converge to a globally minimum solution (in contrast to other types of nonlinear filters
whose cost function may exhibit many local minima) avoiding the undesired possibility of getting
stuck in a local minimum. Let us consider a causal, time-invariant, finite-memory,continuous
nonlinear predictor as described in (
ˆ() = [( − 1), . . . , ( − )]
(
∞ ˆ() = ∑︁ ℎ()[( − )] =1 (2) where ℎ() a re proper coeficients. To make (2) realizable we truncate the series to the first In the general case, a linear-in-the-parameters nonlinear predictor is described by the inputoutput relationship reported in (4). where ⃗ is a row vector containing predictor coeficients and ⃗() is the corresponding column vector whose elements are nonlinear combinations and/or expansions of the input samples.
Linear prediction is a well known technique with a long history [15]. Given a time series ⃗, linear prediction is the optimum approximation of sample () with a linear combination of the most recent samples. That means that the linear predictor is described as eq. (5). terms, thus we obtain
ˆ() = ∑︁ ℎ()[( − )] =1 (3) (4) (5) (6) (7) (8) (9) where the coeficient and input vectors are reported in (7) and (8).
ˆ() = ∑︁ ℎ1()( − )
=1
ˆ() = ⃗ ⃗()
⃗ = [︀ ℎ1(
As well as Linear Prediction, Non Linear Prediction is the optimum approximation of sample () with a non linear combination of the most recent samples. Popular nonlinear predictors are based on Volterra series [16]. A Volterra predictor based on a Volterra series truncated to the second term is reported in (9).
1 2 2 ˆ() = ∑︁ ℎ1()( − ) + ∑︁ ∑︁ ℎ2(, )( − )( − )
=1 =1 = where the symmetry of the Volterra kernel (the ℎ coeficients) is considered. In matrix terms, the Volterra predictor is represented in (10).
ˆ() = ⃗ ⃗() = = = where the coeficient and input vectors are reported in (12) and (12).
⃗ ⃗ (10).
⃗
⃗
where the coeficient and input vectors of FLANN predictors are reported in (22) and (23).
︂]
︂[ ( − 1)
2( − 1) ( − 1)( − 2) . . . 2( − 2)
( − 2) . . . ( − 1)
=1 =1
∑︁ ∑︁ ℎ2(, ) sin[ ( − )] (13)
⎡ ℎ1(
ℎ1(2) . . . ℎ1( )
⎣ℎ2(
FLANN is a single layer neural network without hidden layer. The nonlinear relationships between input and output are captured through function expansion of the input signal exploiting suitable orthogonal polynomials. Many authors used for examples trigonometric, Legendre and Chebyshev polynomials. However, the most frequently used basis function used in FLANN for function expansion are trigonometric polynomials [17]. The FLANN predictor can be represented by eq.(13).
=1 ˆ() = ∑︁ ℎ1()( − ) + ∑︁ ∑︁ ℎ2(, ) cos[ ( − )]+ Also in this case the Flann predictor can be represented using the matrix form reported in ˆ() = ⃗ ⃗() ⎡ ( − 1) ( − 2)
. . . ( − ) = ⎣cos[ ( − 1)] cos[ ( − 2)] . . .
sin[ ( − 1)] sin[ ( − 2)] . . .
Previous research [18] shown that many real nonlinear systems, spanning from electromechanical systems to audio systems, can be modeled using a static non-linearity. These terms capture the system nonlinearities, in series with a linear function, which capture the system dynamics as shown in Figure 1.
FIR y(n)
Indeed, the front-end of the so called Hammerstein Model is formed by a nonlinear function
whose input is the system input. Of course the type of non-linearity depends on the actual
physical system to be modeled. The output of the nonlinear function is hidden and is fed
as input of the linear function. In the following, we assume that the non-linearity is a finite
polynomial expansion, and the linear dynamic is realized with a Finite Impulse Response (FIR)
iflter. Furthermore, in contrast with [ 18], we assume a mean error analysis and we postpone
the analysis in the robust framework in future work. In other word,
()
=
(2)2() + (3)3() + . . . ()()
=
On the other hand, the output of the FIR filter is:
() = ℎ0(
() = ∑︁ ℎ0()( − ) = ∑︁ ℎ0() ∑︁ ()( − ) = =1 =1 =2 Setting (, ) = ℎ0()() we write This equation can be written in matrix form as () = ∑︁ ∑︁ (, )( − ) =2 =1 ˆ() = ⃗ ⃗() ∑︁ ∑︁ ℎ0()()( − ) (19) =2 =1 (20)
So far we saw that all the predictors can be expressed, at time instant , as
ˆ() = ⃗ ⃗ () with diferent definitions of the input, ⃗ (), end parameters vectors ⃗ . There are two well known possibilities for estimating the optimal parameter vector.
The Minimum Mean Square estimation is based on the minimization of the mathematical expectation of the squared prediction error () = () − ˆ()
[2] = [(() − ˆ())2] = [(() − ⃗ ⃗ ())2] The minimization of (25) is obtain by setting to zero the Laplacian of the mathematical expectation of the squared prediction error:
∇ [2] = [∇ 2] = [2()∇ ] = 0
where
⃗
⃗
=
⎡ (
. . . (, 1) (, 2) . . . (, ) = ⎡ 2( − 2) 2( − 3) . . . 2( − ) ⎣ 3( − 2) 3( − 3) . . . 3( − ) ( − 2) ( − 3) . . . ( − 1) ( − 3) . . . ( − ) ⎤ ⎦ ⃗ = ⃗−1⃗ ⃗() = [⃗ ()⃗ ()]
⃗() = [()⃗ ()] ⃗() = ∑︀=1 ⃗ ()⃗ ()
⃗() = ∑︀ =1 ()()⃗ () which leads to the well known unique solution where is the statistical auto-correlation matrix of the input vector ⃗ () and is the statistical cross-correlation vector between the signal () and the input vector ⃗ (). The mathematical expectations of the auto and cross correlation are estimated using is the statistical auto-correlation matrix of the input vector ⃗ () and (22) (23) (24) (25) (26) (27) (28) (29) (30)
Let us consider a general second order terms of a Volterra predictor It can be generalized for higher order term as
− 1 − 1 () = ∑︁ ∑︁ ℎ2(, )( − )( − )
=0 =0 ∑︁ 1=1 · · · ∑︁ 1 · · · {︀ 1 (), · · · ()}︀ =1 ∑︁ (). =1 where
For the sake of simplicity and without loss of generality, we consider a Volterra predictor based
on a Volterra series truncated to the second term
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
() = |ℎ1(
, ℎ2 (2, 2)| () = ⃒⃒ ( − 1), · · · , ( − 1) , 2( − 1) ( − 1)( − 2), · · · , 2 ( − 2) | ˆ() = () ().
( ) = ∑︁ − [︀ ˆ() − () ()]︀ 2
=0
In order to estimate the best parameters , we consider the following loss function where − weights the relative importance of each squared error. In order to find the that minimizes the convex function (39) it is enough to impose its gradient to zero, i.e., That is equivalent to
∇ ( ) = 0 () () = () 1 () = () = ( − 1) + () () () = ( − 1) + −1 () () () = ˆ() − ( − 1) () () = ( − 1) + ()()
() = ( − 1) () Thus, inserting Eq (47) and Eq (45)in Eq (43) and rearranging the terms, we obtain where By recalling Eq. (46), we can write Eq. (48) as
where () is equal to Instead, matrix () in (43) can be written as −1 ( − 1) () + ()−1 ( − 1) () The previous equations can be resumed by the following system ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧⎪() = ( − 1) () ⎪⎪ () = + () () ⎪⎪ () = 1 √ ()+
() ⎪() = √1 [︀ ( − 1) − ()() ()]︀
⎪ ⎪
() ⎪⎩ () = ( − 1) + () () ⎪⎪ () = ˆ( − 1) − ()() () ⎪ where
It follows that the best can be computed by () = ∑︀ () = ∑︀ =0 − () () =0 − () () () = −1 () () () = ( − 1) + () () −1 () =
︀[ −1 ( − 1) − () ()−1 ( − 1)]︀ It should be noted that by using Eq (52) the estimation adapts in each step in order to decrease the error. Thus, the system structure is somehow similar to the Kalman filter.
() = () − () () It is worth noting that the computation of the predicted value from Eq. (38) requires 6tot +2t2ot operations, where tot = 1 + 2 (2 + 1) /2. (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53)
In order to prove the efectiveness of the proposed approach, in this section we present our experimental results for a real dataset. Specifically, we consider the requests made to the 1998 World Cup Web site between April 30, 1998 and July 26, 1998 1. During this period of time the site received 1,352,804,107 requests. The fields of the request structure contain the following information: (i) timestamp the time of the request, stored as the number of seconds since the Epoch. The timestamp has been converted to GMT to allow for portability. During the World Cup the local time was 2 hours ahead of GMT (+0200). In order to determine the local time, each timestamp must be adjusted by this amount; (ii) clientID a unique integer identifier for the client that issued the request; due to privacy concerns these mappings cannot be released; note that each clientID maps to exactly one IP address, and the mappings are preserved across the entire data set - that is if IP address 0.0.0.0 mapped to clientID on day then any request in any of the data sets containing clientID also came from IP address 0.0.0.0; (iii) objectID a unique integer identifier for the requested URL; these mappings are also 1-to-1 and are preserved across the entire data set; (iv) size the number of bytes in the response; (v) method the method contained in the client’s request (e.g., GET); (vi) status this field contains two pieces of information; the 2 highest order bits contain the HTTP version indicated in the client’s request (e.g., HTTP/1.0); the remaining 6 bits indicate the response status code (e.g., 200 OK); (vii) type the type of file requested, generally based on the file extension (.html), or the presence of a parameter list; ( viii) server indicates which server handled the request. The upper 3 bits indicate which region the server was at; the remaining bits indicate which server at the site handled the request.
In the dataset, 87 days are reported. We use the first one in order to initialise the estimator in (35) and we use the others as test set by using a rolling horizon method (as in [18]). Particularly, for each day we compute the estimation by using all the IP observations in the previous days [0, − 1]. The results are reported in Figure 2. The increase of errors in June is due to the sudden increment of diferent IP accessing the website due to the start of the competition (see [ 19]). It should be noted that the estimation error decrease exponentially despite dealing with several millions of IPs.
1ftp://ita.ee.lbl.gov/html/contrib/WorldCup.html
Since the computation of the optimal coeficient () may require some time, we measure the percentage of available data that our approach needs in order to provide good results. Particularly, in this experiment we consider the average estimation error done by the model at time by considering a subset of the IPs observed in interval [0, − 1]. The experimental results on real data cubes are depicted in Figure 3.
Hammersteinmodels VolterraKerners sequenceclusterings 0.1 0.2 0.3 0.4
It should be noted that the Hammerstain model outperform the results by the Volterra’s kernel as well as the clustering techniques. In more detail, the clustering techniques are the one less performing. This is due to the nature of the clustering techniques that exploit the geometric information of the data more than their time dependency. We highlight that despite the calculation of () is computational intensive, this does not efect the real time applicability of the method. In fact, the access decision is taken by considering the estimator ˆ() that is computed once per day. Thus the computation of () does not need to be fast.
In this paper, we presented a new way to deal with cyber attack by using Hammerstein models. Experimental results clearly confirm the efectiveness of the proposed techniques for a real data set, outperforming other well-known techniques. Future work will have two objectives. First, we want to consider the problem in a stochastic optimization settings, as for example in [20]. Second, we want to test the approach on other case studies, by also exploiting knowledge management methodologies (e.g., [21, 22]).
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