Spoofing Attack Results Determination in Code Domain Using a Spoofing Process Equation

When a user receiver is tracking an authentic signal, a spoofing signal can be transmitted to the user antenna. The question is under what conditions does the tracking point of the receiver move from the authentic signal to the spoofing signal? In this study, we develop a spoofing process equation (SPE) that can be used to calculate the tracking point of the delay lock loop (DLL) at regular chip intervals for the entire spoofing process. The condition for a successful spoofing signal is analyzed using the SPE. To derive the SPE, parameters, such as the signal strength, sweep velocity, loop filter order, and DLL bandwidth are considered. The success or failure of a spoofing attack is determined for a specific spoofing signal using the SPE. In addition, a correlation between each parameter for a successful spoofing attack could be obtained through the SPE. The simulation results show that the SPE performance is largely consistent with that of general DLL methods, even though the computational load of SPE is very low.


Introduction
The security and safety aspects of global navigation satellite systems (GNSSs) have been receiving significant attention from researchers and the general public, because the use of GNSSs has been increasing in modern society [1]. Because the power of a GNSS signal coming from the ground is very low, the signal is exposed to different types of radio interferences [2]. Moreover, in contrast to military signals, safety and security issues are not considered for civilian GNSS signals. A civilian signal is not encrypted, and the details of such a signal are open [3]. In other words, anyone can intentionally transmit a fake signal to deceive the user.
Some of the types of intentional interferences include jamming, meaconing, and spoofing [4][5][6]. The aim of jamming is to prevent a user from receiving the authentic signal by transmitting another signal with a significantly greater power than that of the authentic signal. A meaconing attack involves transmitting another signal collected at a different location or time. If a meaconing attack is successful, the receiver would end up providing navigation information, such as the location and time, at which the meaconing signal was collected. The most dangerous type of interference is a spoofing attack. If the receiver captures a spoofing signal, the navigation solution can be controlled by the spoofer [7].
There are two main technological approaches in spoofing studies: spoofing attacks and anti-spoofing techniques. Many spoofing attack tests have been conducted over the past few years.
A portable GPS spoofer was developed, and a spoofing attack test was demonstrated for a target receiver [8]. Although this experiment was conducted with a very short distance between the spoofer and the target receiver, it was possible to develop a practical spoofer with low cost. Moreover, successful spoofing tests were carried out against an unmanned aerial vehicle [9], a ship [10], and a mobile device [11]. These studies have shown that spoofing attacks could be executed in real situations. Moreover, many anti-spoofing techniques have been studied for receiver security and safety. A maximum likelihood estimation-based positioning technique was applied to the detection of spoofing signals and correction of navigation solution [12]. In another study, a cross-correlation approach between two GNSS receivers was used to detect the spoofing signal [13]. In Ref [14], an extended coupled amplitude delay lock loop (DLL) architecture was applied to spoofing detection. A pseudorange difference-based anti-spoofing algorithm was introduced [15]. In Ref [16], spoofing detection was performed using a machine learning algorithm such as a neural network. In other studies, antenna-aided techniques [17] and inertial measurements unit-aided techniques [18] have been developed.
Although the aforementioned studies report on spoofing attacks and anti-spoofing techniques, few have analyzed the conditions and circumstances required for a successful spoofing attack. In [19], the spoofing attack results were presented considering the time, position, and power offset. However, only the effects of the spoofing parameters on the spoofing attack results were studied. In [9], a spoofing signal with a 10 dB greater power than that of the authentic signal was transmitted to successfully deceive a drone. However, to avoid as much as possible the detection of a spoofing signal at the victim receiver, it is better to transmit the signal with the minimum power possible for a successful spoofing attack.
In this paper, we analyzed the conditions for a successful spoofing attack in the code domain. The spoofing parameters considered in this study are the spoofing signal strength, spoofing sweep velocity (Doppler offset), DLL order, and bandwidth. With the increase in the spoofing signal strength or DLL bandwidth, the probability of a successful spoofing attack increases. If the sweep velocity increases, the probability of a successful spoofing attack is reduced because of the increase in the Doppler offset between the authentic signal and the spoofing signal. However, for a specific spoofing signal, it is difficult to determine whether a spoofing attack would be successful when the bandwidth is more than a certain level. It is also difficult to determine the correlation between each parameter for a successful spoofing attack. In this research, we develop a spoofing process equation (SPE) for the entire spoofing process. Generally, to determine whether a specific spoofing signal would be successful, it is necessary to perform an iterative DLL calculation during the entire spoofing process. The concept of the SPE is to reduce the number of iterative DLL tracking calculations during the spoofing process by increasing the integration time. Moreover, we express the entire DLL calculation process in the form of an nth order polynomial. The spoofing attack results could be obtained in one single calculation through the SPE. The following are the contributions of this study:

•
We develop an SPE that can be used to express the entire spoofing process in the form of an nth order polynomial.

•
We obtain the spoofing results in one single calculation using the SPE and determine the correlation between each parameter based on the boundary line which distinguishes between successful and unsuccessful spoofing attacks. • For a particular receiver, the minimum power of a spoofing signal for a successful spoofing attack could be estimated via the SPE.
The remainder of this paper is organized as follows: Section 2 introduces the auto correlation function (ACF) models of the authentic and spoofing signals. Section 3 presents simulations of a spoofing scenario using the ACF models. Section 4 presents the derivation process of the SPE. Section 5 shows the simulation results obtained using the SPE. Finally, the discussions and conclusions are provided in Sections 6 and 7, respectively.

Authentic and Spoofing Signal ACF Model
In this section, the authentic and spoofing signal models are presented. To generate a spoofing signal aligned with the authentic signal, the spoofer should estimate the position and velocity of the target receiver. Figure 1 shows a brief illustration of a spoofing scenario. First, the spoofer estimates the position and velocity of the victim receiver using radar [6]. The spoofer can then calculate the aligned spoofing signal by compensating for the spoofer processing delay and transmission delay. Moreover, the power of the spoofing signal should be greater than that of the authentic signal. Therefore, it is necessary to compensate for the propagation loss depending on the distance between the spoofer and the victim receiver. The signal received at the victim receiver antenna can be represented using a complex baseband model as follows: where, • s(t) denotes the total received signal; • C denotes the pseudorandom code; • τ a (t) is the code phase of the authentic signal; • τ s (t) is the code phase of the spoofing signal; • P s is the spoofing power advantage; • φ a (t) is the carrier phase of the authentic signal; • φ s (t) is the carrier phase of the spoofing signal; • n(t) is the complex zero-mean white Gaussian noise (AWGN).

Authentic and Spoofing Signal ACF Model
In this section, the authentic and spoofing signal models are presented. To generate a spoofing signal aligned with the authentic signal, the spoofer should estimate the position and velocity of the target receiver. Figure 1 shows a brief illustration of a spoofing scenario. First, the spoofer estimates the position and velocity of the victim receiver using radar [6]. The spoofer can then calculate the aligned spoofing signal by compensating for the spoofer processing delay and transmission delay. Moreover, the power of the spoofing signal should be greater than that of the authentic signal. Therefore, it is necessary to compensate for the propagation loss depending on the distance between the spoofer and the victim receiver. The signal received at the victim receiver antenna can be represented using a complex baseband model as follows: (1) In the receiver, a correlation process is implemented to track the input signal s(t). Figure 2 shows the ACF model of s(t). The blue triangle indicates the ACF of the authentic signal, whereas the red triangle indicates the ACF of the spoofing signal. The horizontal axis represents the chip offset, and the vertical axis represents the normalized correlator output (the amplitude of the authentic signal is 1). The parameters, shown in Figure 2, can be expressed as follows: where: • y 1 indicates the left line of the ACF of the authentic signal; • y 2 indicates the right line of the ACF of the authentic signal; • R a (τ) is the ACF of the authentic signal; • D is the difference in the code phases between spoofing and authentic signals; • y s1 indicates the left line of the ACF of the spoofing signal; • y s2 indicates the right line of the ACF of the spoofing signal; • a s is the slope of the ACF of the spoofing signal; • R s (τ) is the ACF of the authentic signal; • R(τ) is the ACF of the total signal; • XE is the accumulation result with the replica code separated 0.5 chip early; • XP is the accumulation result with the replica code; • XL is the accumulation result with the replica code separated 0.5 chip late.

Authentic and Spoofing Signal ACF Model
In this section, the authentic and spoofing signal models are presented. To generate a spoofing signal aligned with the authentic signal, the spoofer should estimate the position and velocity of the target receiver. Figure 1 shows a brief illustration of a spoofing scenario. First, the spoofer estimates the position and velocity of the victim receiver using radar [6]. The spoofer can then calculate the aligned spoofing signal by compensating for the spoofer processing delay and transmission delay. Moreover, the power of the spoofing signal should be greater than that of the authentic signal. Therefore, it is necessary to compensate for the propagation loss depending on the distance between the spoofer and the victim receiver. The signal received at the victim receiver antenna can be represented using a complex baseband model as follows: where, • ( ) s t denotes the total received signal; • C denotes the pseudorandom code;    The XP could be written as [20]: where: In the next section, we explain the change in the replica code phase, τ, depending on the success or failure of a spoofing attack. In our simulation, we assume that the code phase and Doppler frequency of the replica are perfectly aligned with the authentic signal before the spoofing signal approaches. This implies that ∆τ a [1] and ∆ f a [1] are zero.

Spoofing Scenario Simulation Using ACF Model
Using the ACF models explained in Section 2, we conduct the spoofing simulation. We assume that the authentic signal is stationary and that the spoofing signal is moving from right to left with a static velocity. This simulation is done without noise. In general, the DLL discriminator is used to calculate the feedback output using XE and XL and thereby track the incoming signal. The replica code phase gradually aligns with the code start point of the incoming signal during DLL code tracking. In our spoofing simulation, the DLL initially tracks the authentic signal. When the spoofing signal approaches and overlays with the authentic signal, the ACF changes. Figure 3 shows the sequential ACF variation during the spoofing simulation. The green circle indicates the XP, which is the prompt of the DLL. The position of XP in each figure is different. The position of XP is determined by the shape of the ACF and the positions of previous XP and DLL settings. Figure 4a,b show the τ histories of the two spoofing simulations. The only difference between the two simulations is the DLL bandwidth. In Figure 4, the black lines indicate the code phase distance between the authentic and spoofing signals. Because the authentic signal is fixed at zero, this can be considered the position of the spoofing signal relative to the authentic signal in the code domain. We now focus on Figure 4a. The green arrow indicates the start point of τ. At the start of the simulation, τ is zero. The spoofing signal approaches the authentic signal from a distance of two chips. When the spoofing signal reaches a distance of 1.5 chips, τ starts to gradually increase, as the spoofing signal starts to affect XL. A blue arrow indicates the point where τ is increasing. The peak point of the total ACF R(τ) is always the same as that of the spoofing ACF R a (τ). Therefore, τ moves to the peak point of R(τ) until the discriminator output becomes zero. After the spoofing signal passes the authentic signal, the peak point of R(τ) is located on the negative side, as shown in Figure 3c. Finally, τ follows the spoofing signal. The orange arrow represents the final value of the τ. In the case of Figure 4a, τ follows the spoofing signal, and therefore, the spoofing attack is successful. Figure 4b shows the other spoofing simulation case. In Figure 4b, the final value of τ returns to zero. τ seems to chase the spoofing signal, as indicated using the dotted black line, but eventually returns to its location. This implies that the spoofing attack is a failure. The difference between the two simulations is that the DLL bandwidth. The DLL bandwidth is 5 Hz in the first simulation, whereas it is 3 Hz in the second simulation. As shown in the simulation results, the greater the bandwidth of the receiver, the more vulnerable it is to a spoofing attack. Moreover, the higher the strength of the spoofing signal, the higher is the probability of a successful spoofing attack. The faster the spoofing signal sweeps, the more likely it is that the spoofing attack will fail. Table 1 lists the changes in the spoofing attack results with respect to increases in the bandwidth, signal strength and sweep velocity. However, it is difficult to determine how strong a signal should be for a successful spoofing attack. It is also difficult to obtain a correlation between the different parameters for a successful spoofing attack.
is the carrier phase difference between the local replica and the spoofing signal.
In the next section, we explain the change in the replica code phase, τ , depending on the success or failure of a spoofing attack. In our simulation, we assume that the code phase and Doppler frequency of the replica are perfectly aligned with the authentic signal before the spoofing signal approaches. This implies that   • is the carrier phase difference between the local replica and the spoofing signal.
In the next section, we explain the change in the replica code phase, τ , depending on the success or failure of a spoofing attack. In our simulation, we assume that the code phase and Doppler frequency of the replica are perfectly aligned with the authentic signal before the spoofing signal approaches. This implies that

Conventional Approach for τ Calculation
XP is calculated through DLL using the ACF and previous XP. The first-order DLL can be expressed as follows: where ∆τ, T, and B indicate the discriminator output, integration time, and DLL bandwidth, respectively. In general, the spoofing attack results can be obtained by determining which signal the DLL is tracking after the spoofing signal completely sweeps the authentic signal. In other words, if the integration time of the receiver is 1 ms, it is necessary to repeatedly calculate the equation thousands of times to obtain the spoofing attack results. This calculation can be expressed as follows: For a specific spoofing attack scenario, a lot of computations are required to calculate the final replica code phase τ [n]. Moreover, it is necessary to know τ and ACF at all previous epochs. Thus, the final τ value can be written as follows:

Proposed Approach for τ Calculation
In this subsection, we propose a method to compute the spoofing attack results by calculating each epoch at a certain chip interval (CI). The entire spoofing process is summarized in a mathematical equation, i.e., the SPE, and the spoofing results are obtained by one calculation using the SPE. Figure 5 shows the results of the τ estimation with respect to the CI. The blue lines indicate the calculation results of τ per 1 ms. Red circles indicate the calculation results of τ per specific CI. The equation for calculating the time interval (TI) in terms of the chip interval can be expressed as follows: where V s denotes the spoofing sweep velocity (m/s), and the number 293 indicates the wavelength of the C/A code in meter-scale.    Figure 6b shows the region enclosed in the black box shown in Figure  6a. If the spoofing attack is successful, the absolute value of τ exceeds 0.5 at the point where D is −1, and if it fails, the absolute value of the prompt is lower than 0.5.      Figure 7a shows the τ estimates for various spoofing parameters listed in Table 3. It is noteworthy that the absolute value of τ at D is −1. As shown in Figure 7b, if the absolute value of τ exceeds 0.5 chip when D is −1, the DLL tracks the spoofing signal.  The success or failure of a spoofing attack can be determined by looking at the absolute value of τ at the point where D is −1. Figure 6b shows the region enclosed in the black box shown in Figure 6a. If the spoofing attack is successful, the absolute value of τ exceeds 0.5 at the point where D is −1, and if it fails, the absolute value of the prompt is lower than 0.5. Figure 7a shows the τ estimates for various spoofing parameters listed in Table 3. It is noteworthy that the absolute value of τ at D is −1. As shown in Figure 7b, if the absolute value of τ exceeds 0.5 chip when D is −1, the DLL tracks the spoofing signal.   Figure 7a shows the τ estimates for various spoofing parameters listed in Table 3. It is noteworthy that the absolute value of τ at D is −1. As shown in Figure 7b, if the absolute value of τ exceeds 0.5 chip when D is −1, the DLL tracks the spoofing signal.   The following analysis shows that the criterion used for determining the spoofing result is reasonable. If the spoofing sweep velocity is very low or if the bandwidth is very high in the spoofing simulation, there will be sufficient time or control input for the DLL to track the peak point of the ACF. In this case, the discriminator output would become zero and XP would be located at the point where XE equals XL. Figure 8 shows a series of snapshots where the discriminator output is zero with respect to the ACF. The τ value for the case, shown in Figure 8a, can be derived as follows: = −(a s + 1)x + a s + a s D + 1 = −(a s + 1)(τ + 1 2 ) + a s + a s D + 1 XE = XL (a s + 1)(τ − 1 2 ) + a s − a s D + 1 = −(a s + 1)(τ + 1 2 ) + a s + a s D + 1 (a s + 1)τ − 1 2 (a s + 1) + a s − a s D + 1 = −(a s + 1)τ − 1 2 (a s + 1) + a s + a s D + 1 2(a s + 1)τ = 2a s D τ = a s D a s +1 The following analysis shows that the criterion used for determining the spoofing result is reasonable. If the spoofing sweep velocity is very low or if the bandwidth is very high in the spoofing simulation, there will be sufficient time or control input for the DLL to track the peak point of the ACF. In this case, the discriminator output would become zero and XP would be located at the point where XE equals XL. Figure 8 shows a series of snapshots where the discriminator output is zero with respect to the ACF. The τ value for the case, shown in Figure 8a, can be derived as follows: Moreover, the τ values for the cases, shown in Figure 8b,c, can be derived in a similar manner as follows: Figure 9 shows the summary of Equations (8) to (10). For any ACF, shown in Figure 8, τ can be estimated using a s and D when the spoofing sweep velocity is very low or when the bandwidth is considerable. Moreover, it is possible to calculate D corresponding to the different equations of τ Figure 10 shows the ACF change with respect to the spoofing signal when the spoofing signal strength is lower than the authentic signal strength. We can derive an equation to calculate τ in the same manner as above. The τ value for the cases, shown in Figure 10a,b,c can be derived as follows:      Figure 11 shows the summary of Equations (11) to (13). Figure 12 shows a graphical representation of Equations (8) to (13). The blue lines indicate spoofing attack success, whereas the red lines indicate spoofing attack failure. If a s is greater than 1, τ follows the blue line, and if a s is lower than 1, it follows the red line. When a s is 1, D at the time of transition, from (b) to (c) in Figure 8, is −1, and the absolute value of τ becomes 0.5. In case that the spoofing sweep velocity is infinitely slow or the DLL bandwidth is infinitely large, the condition for a successful spoofing attack is that the power of the spoofing signal sets in greater than that of the authentic signal. If the spoofing signal power is a little greater than the authentic signal power, the replica code phase is lower than −0.5 at D is −1. If the spoofing signal power is a little smaller than the authentic signal power, the replica code phase is higher than −0.5 at D is −1. Thus, we could regard that the τ value is a boundary value when D is −1. This implies that the absolute value of τ should exceed 0.5 chip before D approaches −1 for a successful spoofing attack. Figure 13a shows the τ histories with respect the spoofing signal power. ε indicates a very small positive value. Slope of each line is differently changed after D exceeds −1. Figure 13b shows the τ value change with respect the spoofing parameters in the real scenario. (13) Figure 11. Equation of τ and D in case that the spoofing signal strength is lower than authentic signal and XE is same with XL. Figure 11 shows the summary of Equations (11) to (13). Figure 12 shows a graphical representation of Equations (8) to (13). The blue lines indicate spoofing attack success, whereas the red lines indicate spoofing attack failure. If s a is greater than 1, τ follows the blue line, and if s a is lower than 1, it follows the red line. When s a is 1, D at the time of transition, from (b) to (c) in Figure 8, is −1, and the absolute value of τ becomes 0.5. In case that the spoofing sweep velocity is infinitely slow or the DLL bandwidth is infinitely large, the condition for a successful spoofing attack is that the power of the spoofing signal sets in greater than that of the authentic signal. If the spoofing signal power is a little greater than the authentic signal power, the replica code phase is lower than

Derivation of SPE
Assuming that the spoofing signal shifts from right to left, the spoofing signal affects the DLL discriminator when D approaches within 1.5 chip. Moreover, to determine the spoofing results, we only need to calculate τ until D reaches −1. Therefore, if CI is 0.125 chip, it is possible to determine the spoofing attack results by a total of 19 calculations. In Equation (4) determine whether the spoofing attack is a successful one or not by only one calculation. When setting the CI to 0.125, the ACF variation could be known. However, it is not possible to specify the previous τ values. This is because τ value at a certain D changes according to the spoofing parameters. However, the range of τ can be defined for each D value for τ to be close to −0.5 chip when D is −1.
The spoofing attack results can be determined by checking whether the absolute value of τ at D = −1 exceeds 0.5 chip or not. In our case, τ [19] is the final τ value. For τ [19] to be close to −0.5, τ [18] must be in a specific range. Moreover, τ [17] must be in a specific range for τ [18] to be in the defined range. Thus, we can define each range according to the D value of the entire process.

Derivation of SPE
Assuming that the spoofing signal shifts from right to left, the spoofing signal affects the DLL discriminator when D approaches within 1.5 chip. Moreover, to determine the spoofing results, we only need to calculate τ until D reaches −1. Therefore, if CI is 0.125 chip, it is possible to determine the spoofing attack results by a total of 19 calculations. In Equation (4) determine whether the spoofing attack is a successful one or not by only one calculation. When setting the CI to 0.125, the ACF variation could be known. However, it is not possible to specify the previous τ values. This is because τ value at a certain D changes according to the spoofing parameters. However, the range of τ can be defined for each D value for τ to be close to −0.5 chip when D is −1.
The spoofing attack results can be determined by checking whether the absolute value of τ at D = −1 exceeds 0.5 chip or not. In our case, τ [19] is the final τ value. For τ [19] to be close to −0.5, τ [18] must be in a specific range. Moreover, τ [17] must be in a specific range for τ [18] to be in the defined range. Thus, we can define each range according to the D value of the entire process.

Derivation of SPE
Assuming that the spoofing signal shifts from right to left, the spoofing signal affects the DLL discriminator when D approaches within 1.5 chip. Moreover, to determine the spoofing results, we only need to calculate τ until D reaches −1. Therefore, if CI is 0.125 chip, it is possible to determine the spoofing attack results by a total of 19 calculations. In Equation (4), all the previous τ values and ACF are required to calculate τ [n]. The SPE can be used to calculate τ at the point where D is −1 to determine whether the spoofing attack is a successful one or not by only one calculation. When setting the CI to 0.125, the ACF variation could be known. However, it is not possible to specify the previous τ values. This is because τ value at a certain D changes according to the spoofing parameters. However, the range of τ can be defined for each D value for τ to be close to −0.5 chip when D is −1. The spoofing attack results can be determined by checking whether the absolute value of τ at D = −1 exceeds 0.5 chip or not. In our case, τ [19] is the final τ value. For τ [19] to be close to −0.5, τ [18] must be in a specific range. Moreover, τ [17] must be in a specific range for τ [18] to be in the defined range. Thus, we can define each range according to the D value of the entire process.  [19] will be calculated close to −0.5. The details of the SPE derivation are given in appendix A. Finally, SPE has the following form like: τ [19] = f (a s , Vs, B).
(14) The inputs to the SPE are the spoofing signal strength, spoofing sweep velocity and DLL bandwidth. When the calculation of SPE is performed, τ [19] is calculated for D = −1 by just one calculation. The success or failure of the spoofing attack is determined by the absolute value of τ.

SPE Performance Analysis
To verify the performance of the SPE, we compared the estimated SPE results with the original DLL results. Table 5 presents the various spoofing signal parameters and τ results in the cases of using the original DLL and SPE at D = −1. τ 1ms indicates the estimated replica code phase obtained using the original DLL, the integration time of which is 1ms. τ SPE is the estimated replica code phase obtained using the SPE. The calculation time required by the SPE is significantly lower than that required by the original DLL, because τ SPE can be calculated in just one calculation using the SPE. We can see that the replica code phase values estimated using the two methods are very similar. Figure 14 shows the τ SPE errors with respect to CI. The τ SPE errors decrease with the decrease in CI. Thus, the SPE error is due to the reduction in the number of DLL calculations during the spoofing attack process. τ can be calculated in just one calculation using the SPE. We can see that the replica code phase values estimated using the two methods are very similar. Figure 14 shows the SPE τ errors with respect to CI. The SPE τ errors decrease with the decrease in CI. Thus, the SPE error is due to the reduction in the number of DLL calculations during the spoofing attack process.       Figure 14 shows the SPE τ errors with respect to CI. The SPE τ errors decrease with the decrease in CI. Thus, the SPE error is due to the reduction in the number of DLL calculations during the spoofing attack process.    In other words, τ SPE of the spoofing parameters on the boundary line is −0.5. τ SPE error increases as the distance from the boundary value and spoofing parameter increases. A large error indicates that the previous τ values are outside the defined range in the τ SPE calculation process. We define the range of previous replica code phase range at every D[i] like Table 4 in order that the final code phase is calculated around the boundary line at D is −1. If the previous code phases, τ [1] ∼ τ [18], are deviated from the defined range, SPE would provide inaccurate result.
Also, it could be explained that the ACF functions used to calculate XE and XL vary with respect to the already defined XE and XL. Figure 16 shows the SPE error distribution in two dimensions. The SPE error is lowest around the boundary line. The spoofing parameters are divided using different colors with respect to the SPE error size. We define the range of previous replica code phase range at every D[i] like Table 4 in order that the final code phase is calculated around the boundary line at D is −1. If the previous code phases, [1]~ [18] τ τ , are deviated from the defined range, SPE would provide inaccurate result. Also, it could be explained that the ACF functions used to calculate XE and XL vary with respect to the already defined XE and XL. Figure 16 shows the SPE error distribution in two dimensions. The SPE error is lowest around the boundary line. The spoofing parameters are divided using different colors with respect to the SPE error size.

Determination of Boundary Line and Surface Using SPE
The boundary line and surface that divide the spoofing attack success and failure can be estimated using the SPE. The input parameters of the SPE are the spoofing signal strength, spoofing signal sweep velocity, and DLL bandwidth. In Equation (15), if we set each variable as follows:

Determination of Boundary Line and Surface Using SPE
The boundary line and surface that divide the spoofing attack success and failure can be estimated using the SPE. The input parameters of the SPE are the spoofing signal strength, spoofing signal sweep velocity, and DLL bandwidth. In Equation (15), if we set each variable as follows: Only one variable, i.e., as, remains, and the SPE becomes an equation to calculate as. We use MATLAB solver to obtain a s which is an estimated value of a s obtained using Equation (15). This means that the SPE result of the spoofing parameters, ( a s , 40, 2), becomes −0.5. Therefore, the spoofing parameter, ( a s + ε, 40, 2), will result in a successful spoofing attack. The other spoofing parameter, ( a s − ε, 40, 2), will result in a failed spoofing attack. ε is a small positive value. Figure 17a shows the boundary line dividing the spoofing attack success and failure zones. We obtain the spoofing signal strength values by fixing the other spoofing parameters and τ. Table 6 presents the estimated a s values with respect to the spoofing parameters. The red line in Figure 17a indicates the boundary line. The boundary can be estimated using the spoofing parameters listed in Table 6 as follows: where f bl is the function of the boundary line for the DLL bandwidth of 2 Hz. p 1 , p 2 , p 3 , and p 4 are the coefficients at the boundary line. Moreover, this line expresses the correlation between two parameters for a successful spoofing attack. From Equation (16), we find that as the spoofing signal strength increases, the spoofing attack becomes successful even with a higher sweep velocity. Only one variable, i.e., as, remains, and the SPE becomes an equation to calculate as. We use MATLAB solver to obtain s a  which is an estimated value of s a obtained using Equation (15). This means that the SPE result of the spoofing parameters, ( s a  , 40, 2), becomes −0.5. Therefore, the spoofing parameter, ( s a ε +  , 40, 2), will result in a successful spoofing attack. The other spoofing parameter, ( s a ε −  , 40, 2), will result in a failed spoofing attack. ε is a small positive value. Figure   17a shows the boundary line dividing the spoofing attack success and failure zones. We obtain the spoofing signal strength values by fixing the other spoofing parameters and τ . Table 6 presents the estimated s a  values with respect to the spoofing parameters. The red line in Figure 17a indicates the boundary line. The boundary can be estimated using the spoofing parameters listed in Table 6 as follows: where bl f is the function of the boundary line for the DLL bandwidth of 2 Hz. 1 p , 2 p , 3 p , and 4 p are the coefficients at the boundary line. Moreover, this line expresses the correlation between two parameters for a successful spoofing attack. From Equation (16), we find that as the spoofing signal strength increases, the spoofing attack becomes successful even with a higher sweep velocity. The spoofing attack success and failure zone is divided based on the boundary line, as shown in Figure 17b. The zone above the boundary line indicates spoofing attack failure, whereas the zone below indicates spoofing attack success. Figure 18a shows the boundary lines for various DLL bandwidths. We find that the receiver becomes more vulnerable to spoofing attacks as its DLL bandwidth increases. With the increase in the DLL bandwidth, the spoofing attack becomes successful even for a low spoofing signal strength when using a fixed spoofing sweep velocity. Figure  18b shows the boundary lines in three dimensions.  The spoofing attack success and failure zone is divided based on the boundary line, as shown in Figure 17b. The zone above the boundary line indicates spoofing attack failure, whereas the zone below indicates spoofing attack success. Figure 18a shows the boundary lines for various DLL bandwidths. We find that the receiver becomes more vulnerable to spoofing attacks as its DLL bandwidth increases. With the increase in the DLL bandwidth, the spoofing attack becomes successful even for a low spoofing signal strength when using a fixed spoofing sweep velocity. Figure 18b shows the boundary lines in three dimensions. Table 7 presents the computational complexities of the conventional DLL and SPE in terms of the number of iteration and computational time with respect to the spoofing signal velocity. We can see that the proposed method has much lower computational complexity than that of the conventional DLL. The simulation were performed on a personal computer with Intel Core i7-4790k. The boundary surface can be estimated using the boundary lines, as shown in Figure 19. The boundary surface can be expressed as follows: where f s f indicates the function of the boundary surface. c 1 ∼ c 11 are coefficients of f s f . For specific spoofing parameters, the spoofing attack results can be determined using f s f . Equation (18) is the case for spoofing attack failure, whereas Equation (19) is the case for spoofing attack success: B < f s f (a s , V s ).  The details of SPE derivation are as follows. Figure A1 shows the ACF snapshots for i ranging from 1 to 4. Also, [1] τ , [2] τ , [3] τ , and [4] τ can be expressed as follows:

Appendix B
To derive the SPE, each DLL order could be expressed as follows: Equations (A11) to (A13) are the DLL in case of 1st, 2nd, and 3rd order, respectively. ω 0 is the 1st order filter value. ω 1 and a 2 are the 2nd order filter values. ω 2 , a 3 and b 3 are the 3rd order filter values. K 1 , K 2 and K 3 are the SPE coefficients with respect to the DLL order, respectively. SPE coefficients are determined through the filter setting. Also, τ 1 [n] in Equation (A12) can be derived using Equation (A11). And τ 2 [n] and τ 3 [n] in Equation (A13) can be derived using Equation (A11) and Equation (A12), respectively. This derivation shows that the SPE could be derived from any DLL order.