Secure localization in WSNs has attracted interest in the scientific literature (Li et al., 2005; Liu et al., 2007; Garg et al., 2012; Liu et al., 2019; Li et al., 2021; Beko and Tomic, 2021; Tomic and Beko, 2022; Mukhopadhyay et al., 2021; Tomic and Beko, 2024a; Tomic and Beko, 2024b). Still, there is no uniquely accepted solution and there is room for improvement in all aspects (localization accuracy, detection rates and complexity).

In Li et al. (2005), two secure localization solutions were explored: least median of squares (LMS) and radio frequency (RF) fingerprinting. LMS selects the subset of anchors with the least median residues, while RF fingerprinting employs a median-based distance metric. The work in Liu et al. (2007) introduced attack-resistant minimum mean square estimation (ARMMSE) and a voting scheme (VS). ARMMSE detects and removes malicious anchors based on inconsistencies, whereas VS assigns votes to grid cells based on distance measurements, identifying the target’s likely location. An iterative algorithm using gradient descent was proposed in Garg et al. (2012) for both uncoordinated and coordinated attacks. Malicious anchors were identified by their higher residues and excluded from the localization process. In Liu et al. (2019), a density-based spatial clustering method classified location points as normal or abnormal, followed by a sequential probability ratio test to authenticate anchors using received signal strength (RSS) and time of arrival (TOA) data. The study in Li et al. (2021) addressed secure localization and velocity estimation in mobile WSNs with malicious anchors, using a maximum a posteriori (MAP) estimator solved via variational message passing. In Beko and Tomic (2021), an initial location estimate was obtained via weighted central mass (WCM), followed by distance-based filtering and a generalized trust region sub-problem (GTRS) solved using a bisection method. This was extended in Tomic and Beko (2022) to a general range-based scenario using a generalized likelihood ratio test (GLRT) and law of cosines (LC). The work in Mukhopadhyay et al. (2021) proposed secure weighted least squares (SWLS) for uncoordinated spoofing attacks and l1-norm (LN-1E) for coordinated attacks. SWLS filters malicious nodes based on estimated noise standard deviation, while LN-1E uses 3D plane fitting and K-means clustering to separate malicious anchors. In Tomic and Beko (2024a), a robust min-max approach was formulated as a second-order cone programming (R-SOCP) problem and a robust GTRS (R-GTRS). Lastly, Tomic and Beko (2024b) introduced an alternating direction method of multipliers (ADMM) approach, applying a weighted least squares criterion within a decomposition-coordination iterative scheme.

Even though the methods in Li et al. (2005), Tomic and Beko (2024b) work well in the settings under scrutiny in the respective works, they either require additional knowledge on certain parameters, such as noise power or the maximum magnitude of an attack (e.g., Liu et al., 2007, Liu et al., 2019, Li et al., 2021, Mukhopadhyay et al., 2021, Tomic and Beko, 2024a) and/or convex relaxations/approximations that expand the set of possible solution leading to higher error (e.g., Li et al., 2005, Li et al., 2021, Tomic and Beko, 2024a) or are executed iteratively (e.g., Garg et al., 2012, Tomic and Beko, 2024b), leaving room for improvement in all aspects (localization accuracy, detection rates and complexity). These limitations could severely deteriorate their performance in scenarios where the required parameters are unknown or imperfectly known, i.e., where problem assumptions do not hold, resulting in the applied relaxations/approximations not being sufficiently tight or lead to burdensome computations that might even raise convergence issues. Furthermore, the majority of the existing works assume knowledge about the type of spoofing attacks (uncoordinated or coordinated) under which the network is beforehand and develop a solution for that specific type of attack. However, it is not possible to acquire such knowledge in practice. Therefore, there are still some challenges to address and room for improvement in all main aspects of the considered problem.

This work presents a geometric approach to compute intersection points between pairs of anchors and obtain an estimate of the target’s location through a VS and WCM, which is then exploited to detect attackers based on confidence intervals (CIs). Unlike (Liu et al. (2007), Garg et al. (2012), Beko and Tomic (2021), Tomic and Beko (2022), Mukhopadhyay et al. (2021)), the proposed algorithm does not make hard (binary) detection decisions, but assigns beliefs (votes) to each intersection point, leveraging information from malicious anchors when attack intensity is low. Besides, mistakenly removing a genuine anchor can severely degrade the localization accuracy, leading to severe and potentially lethal consequences in real life, such as collisions of autonomous vehicles with obstacles, another vehicles or infrastructure, tardy arrival at the desired location in the case of an emergency event, like wildfire and organ transplantation and similar. The highest-voted points are converted into probabilities and used as WCM weights for location estimation. Attack intensities are assessed per link using a maximum likelihood (ML) criterion, with attacker detection based on CIs at a predefined level. Due to its geometric nature, the method is adaptable to any range-based measurement. The primary goal is to advance secure localization beyond traditional systems, ensuring reliable malicious node detection and accurate target positioning. The main contributions of this work are threefold and are summarized in the following.

Design of a novel solution for target localization in randomly deployed sensor network in the presence of (uncoordinated and coordinated) spoofing attacks based on a new VS. The proposed localization estimator is a single-iteration scheme based on simple geometry, where points of interest extracted from the system model are clustered together and appraised by assigning votes to reach a set of the most trustworthy points. This set of points is then used to get an estimate on the target’s location through WCM.

Throughout the work, uppercase bold type, lowercase bold type, and regular type of fonts are used to denote matrices, vectors, and scalars, respectively. R n denotes the n -dimensional real Euclidean space. Symbol ( • ) T represent the transpose operator, while n k = n ! k ! ( n − k ) ! and p r o j S ( p ) denote the binomial coefficients and the projection of the point p onto the set S , respectively. The normal (Gaussian) distribution with mean μ and variance σ 2 is denoted by N ( μ , σ 2 ) . The N -dimensional identity matrix is denoted by I N and the M × N matrix of all zeros by 0 M × N (if no ambiguity can occur, subscripts are omitted). ‖ x ‖ denotes the vector norm defined by ‖ x ‖ = x T x , where x ∈ R n is a column vector. The i -th column of the matrix M is denoted by M i and an estimate of a parameter x with x ̂ .

The rest of this paper is structured as follows: Section 2 defines the problem, while Section 3 details the proposed solution. Section 4 presents performance comparisons, and Section 5 summarizes the key findings.

In this setting, the genuine anchors have a predefined transmitted power, while the malicious anchors change the transmitted power arbitrarily without notifying the network. The k -th RSS measurement sample ( k = 1 , … , K ) between the target and the i -th anchor can be modeled as P i , k = P 0 − 10 γ ⁡ log 10 ‖ x − a i ‖ d 0 + δ i + n i , k (1) where δ i = 0 , i ∈ H δ , i ∈ M ,

and M and H are, respectively, the set of malicious and honest nodes, P 0 is the RSS at a short reference distance d 0 (for simplicity referred to as the transmitted power), γ is the path loss exponent (PLE) representing the decay of the signal strength with distance, n i , k is the noise term modeled as n i , k ∼ N ( 0 , σ i , k 2 ) , and δ i ∈ R is the intensity of the spoofing attack of the i -th anchor. Note that in this work the measurement acquisition is assumed practically instantaneous; thus, ambient conditions do not change significantly during the measurement process, having very little to no impact on the deviation of the RSS measurements.

Note that, in contrast to Beko and Tomic (2021), where malicious anchors could only enlarge their distance measurements, in this work it is assumed that the attackers can either reduce or enlarge distance measurements. For simplicity, it is assumed that the variance of all measurements is equal (for every link and sample), i.e., for σ i , k 2 = σ 2 , i = 1 , … , N  and  k = 1 , … , K . Moreover, to easier combat outliers and also for the sake of notation simplicity and without loss of generality, the median of all K RSS measurements in (Equation 1) from the i -th anchor ( P i ) is used in the following derivations.

Figure 1a shows an uncoordinated attack, where two (represented by red squares) of the five anchors are malicious and report, independently, false distance measurements, aggravating the localization process. In the concrete case, both malicious anchors reduced their measurements independently from each other, resulting in two intersection points of the red dashed circles to be relatively remote from the main cluster of the intersection points and thus, from the true target location.

In coordinated attacks the malicious anchors communicate with each other to agree on a (false) location for the target. The idea is to make the network believe that the target is at a different location than it actually is. Similar to Mukhopadhyay et al. (2021), the coordinated attack can be modeled as P i , k = P 0 − 10 γ ⁡ log 10 ‖ x − a i ‖ d 0 + n i , k i ∈ H P 0 − 10 γ ⁡ log 10 ‖ x att − a i ‖ d 0 + n i , k i ∈ M , (2) where x att is the false location that the attackers agree upon.

Figure 1b shows an example of a coordinated attack, where two malicious anchors (represented by a red square) attempt to make the network think that the location of the target is the one represented by a blue triangle ( x att ) instead of the real target position represented by a blue cross ( x ) .

Note that (Equation 2) is a special case of (Equation 1), having δ i = 10 γ ⁡ log 10 ‖ x − a i ‖ ‖ x att − a i ‖ . Hence, this work adopts (Equation 1) as the general model for both types of spoofing attacks. From it, the probability density function of x parametrized by the median RSS observation at the i -th anchor P i is given by f x ; P i = 1 2 π σ 2 exp − P i − P 0 + 10 γ ⁡ log 10 ‖ x − a i ‖ d 0 − δ i 2 2 σ 2 , (3) where exp { • } denotes the exponential function. Minimizing (Equation 3) leads to the ML estimator Kay (1993). However, the ML estimator is non-convex and therefore difficult to tackle directly. In this work, a VS algorithm is introduced to estimate the target’s location instead.

At the beginning, all anchors are treated as honest. Hence, the set of malicious nodes is initialized as M = ∅ , whereas the set of honest nodes is H = { i : 1 ≤ i ≤ N } . Afterwards, one can construct circles, c i , centered at the known locations of the anchors and radii equivalent to the distance estimate, d ̂ i = d 0 1 0 P 0 − P i 10 γ , obtained from (Equation 1), of the respective anchor (the reader is referred to Figure 2a). The intersection points between all pairs of circles are used as points of interest for the voting scheme. The intersection points between a pair of circles (given that they exist) can be calculated Tomic and Beko (2022) as follows q i j ′ = q 0 + t  and  q i j ′ ′ = q 0 − t , for i = 1 , … , N − 1 , j = i + 1 , … , N , (4) where q 0 = a j − a i d i ̂ 2 − d j ̂ 2 2 ‖ a j − a i ‖ 2 + a i + a j 2 , t = u 2 ‖ a j − a i ‖ 2 T a j − a i  ,  T = 0 − 1 1 0 , with u = d i ̂ + d j ̂ 2 − ‖ a j − a i ‖ 2 ‖ a j − a i ‖ 2 − d j ̂ − d i ̂ 2 , according to Figure 2a. However, due to the presence of noise and possibly malicious/faulty nodes, a pair of circles might not intersect (the reader is referred to Figure 2b). In this case, the tuple set of anchors without intersection is defined as C = { ( i , j ) : c i ∩ c j = ∅ } , where the notation c i ∩ c j = ∅ is used to denote that the circles corresponding to the i -th and j -th anchors do not intersect. In that case, one can draw a line that passes through the respective pair of anchors, and compute the intersections between the circles and the drawn line as follows. q i , 1 = a 0 + a i − a 0 T b ̂ + a i − a 0 T b ̂ 2 − a 0 T a 0 + a i T a i − d i − 2 a 0 T a i , q i , 2 = a 0 + a i − a 0 T b ̂ − a i − a 0 T b ̂ 2 − a 0 T a 0 + a i T a i − d i − 2 a 0 T a i , q j , 1 = a 0 + a j − a 0 T b ̂ + a j − a 0 T b ̂ 2 − a 0 T a 0 + a j T a j − d j − 2 a 0 T a j , q j , 2 = a 0 + a j − a 0 T b ̂ − a j − a 0 T b ̂ 2 − a 0 T a 0 + a j T a j − d j − 2 a 0 T a j , where a 0 = a i + a j 2 is the position vector of the line, and b ̂ = ( a i − a j ) ‖ a i − a j ‖ is the unit vector that describes the line’s direction. Afterwards, the intersection points to be used in the voting-scheme, q i j ′ , q i j ′ ′ , are obtained as the middle of the intersection points between the line and the circles at both ends, according to q i j ′ = q i , 1 + q j , 1 2 ,     q i j ′ ′ = q i , 2 + q j , 2 2 (5)

The forged intersection points are required for the proposed algorithm in scenarios where the number of anchors is scarce and/or the genuine intersection points are in insufficient number. This idea has already been implemented in Tomic and Beko (2022), and the reasoning behind it is that when a pair of anchors is genuine, the intersection points would lay in the vicinity of the drawn line, making these forged intersection points a reasonable approximation of the real ones.

The voting scheme is a process to cluster and assign votes to the intersection points based on some criterion (for instance, their physical proximity). The main idea is to assign a value (vote) to each intersection point in order to find the most trustworthy ones. It is important to highlight that the localization process is done for a specific instant in time, therefore, past information of the network is not used to aid the voting-scheme. For the sake of simplicity, let us define the matrix Q = [ q i j ′    q i j ′ ′ ] ∈ R 2 × N 2 containing all intersection points, and the vector Q g ∈ R 2 as the g -th column of Q . This process iterates all pairs of anchors, and for each pair a hyperplane is computed as H i j = g : b ̂ T Q g = b ̂ T a 0 , (6)

which divides the problem space into two-half spaces. The intersection points are assigned to the upper half space, H i j ( u ) = { g : b ̂ T Q g > b ̂ T a 0 } , or to the lower half space, H i j ( l ) = { g : b ̂ T Q g < b ̂ T a 0 } , according to their physical location with respect to the hyperplane. Next, clusters composed of N − 1 elements that are physically the closest to each other in each half space are built: C i j ( u ) ⊆ H i j ( u ) as the upper cluster and C i j ( l ) ⊆ H i j ( l ) as the lower cluster. Lastly, votes, v g , are assigned to the points that belong to a cluster (if these exist), based on their distance to the hyperplane, see Figure 3. Vote for the g -th intersection point is calculated as v g = ∑ i = 1 N − 1 ∑ j = i + 1 N w g p r o j H i j Q g ∑ g : g ∈ C i j u p r o j H i j Q g + w g p r o j H i j Q g ∑ g : g ∈ C i j l p r o j H i j Q g ,   g ∈ C i j u ∪ C i j l (7) where w g = d j ̂ d i ̂ + d j ̂ ,  if  g ∈ H i j u d i ̂ d i ̂ + d j ̂ ,  if  g ∈ H i j l , (8) p r o j H i j Q g = ‖ Q g − e e T Q g + I 2 − e e T a 0 ‖ (9)

with e = T b ̂ , I 2 is the identity matrix of order two, p r o j H i j ( Q g ) denotes the distance of an intersection point, Q g , to it’s respective projection on the hyperplane, and w g is a weight based on the distances d i ̂ and d j ̂ which takes into account the anchor and the point Q g .

The procedure to compute the votes defined in (Equation 7) is explained in the following. For every point of interest Q g , with g = 1 , … , N 2 , one considers all combinations of pairs of anchors ( i , j ) , where i = 1 , … , N and j = i + 1 , … , N , and calculates the respective hyperplanes H i j according to (Equation 6). Then, each weight w g divides a single vote according to the proximity of H i j to a i and a j , such that more weight is assigned to the half space containing the anchor closer to the hyperplane (Equation 8). A cluster of N − 1 points physically closest to one another, C i j ( u ) and C i j ( l ) , are formed in each half space. If g ∈ C i j ( u ) or g ∈ C i j ( l ) , distances of the cluster points to H i j are calculated (Equation 9) and converted into weights by dividing the individual distances with the sum of all distances of the points in the cluster to the hyperplane; otherwise, sum zero and move on to the next anchor pair. Lastly, these weights are further weighted with w g and summed up to form a vote for the g -th point of interest. Hence, the intuition behind the votes defined in (Equation 7) is that it assigns greater weights (belief) to points closer to the hyperplane, since the correct cluster of (genuine) points should lie in its vicinity.

An estimate of the target’s location can be obtained by re-ordering the vote vector in a descending fashion, v ̃ = [ v ̃ h ] , such that v ̃ 1 ≥ v ̃ 2 ≥ … ≥ v ̃ N 2 . The first N − 1 votes correspond to the most trustworthy points; hence, the estimate is obtained by applying the WCM principle for the N − 1 (normalized) vote values as x ̂ = ∑ h = 1 N − 1 w ̃ h Q h ,    with    w ̃ h = v ̃ h ∑ h = 1 N − 1 v ̃ h . (10)

Considering (Equation 3), in the case of a malicious anchor, it is intuitive that the attack would shift the probability distribution according to the attack intensity. This shift of the distribution could cause the mean to fall outside of a CI. In other words, one could take advantage of it to detect attackers. However, as the attack intensity is unknown, it can be estimated by exploiting the estimated target location in (Equation 10) based on the ML principle. Similar can be done for the noise standard deviation and the expected (genuine) RSS value as δ i ̂ = ∑ k = 1 K P 0 − P i , k − 10 γ ⁡ log 10 ‖ x ̂ − a i ‖ K , (11) σ ̂ = 1 N ∑ i = 1 N 1 K − 1 ∑ k = 1 K α i − δ i ̂ 2 , (12) P ̂ i = P 0 − 10 γ ⁡ log 10 ‖ x ̂ − a i ‖ , (13) with α i , k = P i , k − P 0 + 10 γ ⁡ log 10 ‖ x ̂ − a i ‖ .

It is well known that 68% of the mass of a normal distribution falls within one standard deviation of the mean. Thus, if the measured RSS in (Equation 1) lays outside of the CI [ P ̂ i − σ ̂ , P ̂ i + σ ̂ ] , the respective anchor is classified as malicious, and the set of malicious nodes becomes M = { i : P i < P ̂ i − σ ̂ ∨ P i > P ̂ i + σ ̂ } . Notice that the symmetric band [ P ̂ i − σ ̂ , P ̂ i + σ ̂ ] used to flag an anchor as malicious might not be the optimal one, especially for anchors at very different ranges. Still, it effectively serves its intended purpose and achieves strong performance in practice; thus, its further enhancement is left for future work.

Note that, in contrast to Beko and Tomic (2021), Mukhopadhyay et al. (2021), the malicious node is possibly exploited in the localization process. This can be advantageous when the | δ i | is not high compared to noise power, and quantity of anchors prevails over quality.

The proposed algorithm is summarized in Algorithm 1. It should be noted that the proposed algorithm operates entirely locally at the target using a single network snapshot in time. While the inferred corruption status of the anchors could be reused by the same target over time (e.g., in tracking or navigation problems) or shared with other targets in a cooperative setting, such mechanisms fall outside the scope of this work; thus, no inter-target cooperation or information exchange is assumed or required by the algorithm.

Algorithm 1

Require:   N : Number of anchors in the network

The complexity analysis is highly relevant for the applicability of the algorithm, especially in real-time scenarios. Given that B max and B ADMM are respectively the maximum number of iterations for the GTRS-based and for the ADMM-based algorithms, Table 1 summarizes the worst-case computational complexity together with the average running time of the considered methods. The latter evaluation was performed with 100 Monte Carlo (MC) runs, for N = 6 , σ = 1 dB, δ = 10 dB, on a machine with the following characteristics: CPU: Intel(R) Core(TM) i5-1135G7 CPU @ 2.40 GHz, RAM: 16 GB, OS: Windows 11 Home, running MATLAB R2016b.

Essentially, all considered methods involve matrix addition, multiplication and transpose operations, and the LC-GTRS in Tomic and Beko (2022) and SWLS in Mukhopadhyay et al. (2021) require matrix inversion. These operations come with certain computational costs associated with them (e.g., O ( m 3 ) is the cost of matrix inversion, where m stands for the size of the square matrix). Nevertheless, one can note from the table that all considered algorithms have linear complexity in the dominant term, N . Regarding the complexity of the proposed VS method, its most expensive operation is the calculation of the votes in (Equation 7), where a series of matrix additions and multiplications is required in order to compute the projection of points of interest onto the hyperplane. However, dimensions of the matrices and vectors involved in these computations are fixed to y × y and y respectively, with y denoting the dimension of the space of interest (2-D or 3-D). Hence, although the operations in VS are computationally the least demanding, the proposed algorithm requires repeated actions (for each pair of anchors), which results in somewhat higher average running time than SWLS and R-GTRS. However, note that these actions could be done in parallel, so that the running time of VS is reduced significantly. Moreover, the GTRS-based and ADMM-based approaches require repeated actions, while LC-GTRS and LN-1E require two phases to obtain the final location estimation.

This section validates the performance of the considered algorithms in terms of localization accuracy and success of attacker detection through numerical simulations. All simulations disclose the results for N randomly deployed anchors and a single target (at a time), within a two-dimensional area of 25 × 25   m 2 with two malicious anchors. Moreover, anchors are randomly deployed N D = 1000 times and, for each localization setting, two randomly chosen anchors are considered malicious N A = 50 times.

The RSS measurements ¹ were obtained through (Equation 1), where the transmit power at the target node is set to P 0 = 15 dBm, the PLE is set at γ = 3 , which corresponds to propagation in an urban area, and K = 10 observation were considered. The main metric used to assess the localization performance is the root mean squared error (RMSE), RMSE = ∑ i = 1 M c ‖ x i − x ̂ i ‖ 2 M C (m), where x i and x ̂ i are, respectively, the true and the estimated target location in the i -th Monte Carlo run, i.e., M C = N D × N A .

It is worth mentioning here that SWLS in Mukhopadhyay et al. (2021) requires tuning the detection threshold by studying an empirical parameter, ζ . By fine-tuning the empirical threshold in the considered settings, it was concluded that the best localization results for SWLS were obtained for ζ = 1.5 ; thus, this value is adopted for SWLS in all presented simulations. Furthermore, R-GTRS in Tomic and Beko (2024a) requires knowledge on the magnitude of the attack intensity, i.e., | δ i | in (Equation 1). The true value δ i is given to R-GTRS in all presented simulations.

This section validates the performance of the considered algorithms in terms of localization accuracy and success of attacker detection through experimental measurements. The considered experimental configuration is illustrated in Figure 11. In all experiments, K = 50 was considered to localize a single target at a time, in the presence of two malicious anchors. Moreover, two randomly chosen anchors in each run out of N A = 100 runs are considered malicious, and their spoofing attacks were generated by simply injecting a value into the acquired experimental measurements.

While the proposed solution outperforms existing methods in localization and detection for medium-to-high attack intensity, some limitations and parameter choices are worth discussing. Certain parameters are fixed, such as the number of points of interest set to N − 1 for localization and the confidence interval for attacker detection set within one estimated noise standard deviation. The rationale for using N − 1 points is that not all N intersection points are necessary, as some may be affected by attacks or excessive noise. For smaller N , trimming unnecessary points improves accuracy. The confidence interval choice balances false detection (if too narrow) and missed detection (if too wide), with one standard deviation offering a reasonable trade-off. Although these parameters could be fine-tuned for different scenarios, numerical results indicate that the chosen values work well within the considered settings.

The proposed approach assigns beliefs (votes) to each intersection point, but under high noise or unfavorable attack coordination, it may mistakenly remove a genuine point, potentially impacting localization accuracy. However, numerical results confirm that such occurrences are infrequent and do not significantly degrade performance. While the final detection output is a binary decision (genuine or malicious anchor), this classification is solely for detection purposes and does not influence localization refinement. Like existing methods, the approach requires at least 50   % of anchors to be non-compromised, particularly in coordinated attacks, as a malicious majority would easily manipulate localization. A unique aspect of the method is its ability to generate points of interest even when anchor circles do not intersect. In extreme cases, all points might need to be fabricated, leading to inaccurate clusters. However, this scenario poses a challenge for all existing methods, as they inherently rely on intersection points for localization.

Although not all considered methods estimate the attack intensity directly in their detection procedure, one can obtain this estimate as explained here in (Equation 11). Hence, Figure 16 illustrates the average RMSE of δ = [ δ i ] T ( ARMSE δ ) in (dB) versus δ (dB) in the considered experimental uncoordinated attack scenario, Figure 16a, and the considered experimental coordinated attack scenario, Figure 16b, for N A = 100 . The figure shows that none of the considered approaches accomplishes an error below 7 (dB), which is due to the complexity of the scenario. Still, the proposed scheme exhibits a fairly good, but in general not the best, estimation performance, indicating that this estimation is not a prerequisite for high detection rate, which is somewhat intuitive. Finally, it is worth mentioning that the results of LN-1E are around 22 dBs in the considered setting, and are thus cut off for a better overview.