Hiding Signal Strength Interference from Outside Adversaries

Wifi is the dominant last leg of connecting devices to the internet, and is ubiquitous in households and businesses. However, recent work [22, 17, 13] has shown that these wifi signals can leak the location of occupants in those homes and businesses, even if the occupants are passive and do not have devices sending signals. The presence of a person can reduce the Wifi signal strength, and an adversary outside of a home or business can detect this reduction, inferring the presence and even location of the person.

This paper examines if and when it is possible to prevent the leaking of the presence of a person using only passive reduced signal strength of Wifi and other wireless transmissions. Different from, for instance, [8, 7], we assume the messages can be encrypted, and for high security settings equipment can even continually fill the channel with messages in a regular (or random) pattern to avoid detection of the presence of signals. The only information leaked is the magnitude of the signals.

This is a challenging domain with many possible strategies an adversary could use [12, 6, 23, 18]. As such, this paper starts with a very simple model where a formal analysis can be developed. Then it builds on these basic ideas to create a more comprehensive array of possible modeling assumptions. The first model is in 1 dimension where the sender has full information, then when the model does not know if a person interferes or not. We ultimately consider 2-dimensional (spatial) models, and if the sender knows the state of potentially interfering people or adversaries. We empirically demonstrate the effectiveness of our models in the simple controlled scenarios. While these models do not reach the specification of the transmission protocol and hardware devices, they develop a characterization of which factors are essential to models, and which are less pertinent.

Hence, the main contribution of this paper is the formalization of how to protect against signal strength inference attacks, which outcomes are possible, and a series of modeling assumptions and corresponding protocols and analysis to protect against inference. Our methods are based on information theoretic and statistical information arguments, and given modeling assumptions are impervious to any adversarial attack, or show a certain amount of transmissions must be made before the presensence of the person can be identified with sufficient (e.g., $95\%$) confidence.

We begin with some basic properties about distributions which will guide our characterization of various scenarios and the corresponding protocols. The “signal” is a bit $b$, if a person interferes with a signal ($b=1$) or not ($b=0$). We identify three cases: when the scenarios are completely indistinguishable, when it must be one of the scenarios and not the other, and when it is not immediately clear, but the adversary gains information favoring one or the other. In this last scenario, we quantify how much information the adversary gains, and then if $n$ readings are made in an iid fashion, when the adversary can reach a certain amount of confidence about one scenario or the other. This reduces to the expectation under a distribution $\mu$, denoted ${\bf{\mathsf{E}}}_{\mu}$, of the Kullback-Liebler (KL) divergence between certain distributions, denoted $\mathrm{KL}(\cdot\mid\mid\cdot)$.

We first consider a case with a single sender $x\in\mathbb{R}$ and a single adversary $a\in\mathbb{R}$ in the one-dimensional domain along the line of sight between them. The sender will be communicating (e.g., via wifi) and may encrypt signal content, but can not hide the signal strength, and the adversary can measure this amplitude.

The unimpeded signal presents a larger amplitude the closer it is to the sender, and so the amplitude measured by an adversary $a$ is a function of the initial magnitude $\alpha$ used by the sender, and the distance between them $|x-a|$. We restrict that the sender can produce signals in range $[0,M]$, with maximum signal strength $M$. While our protocols can be adapted to any known (or learned) decay function, for simplicity in this model we will assume a linear decay of the observed signal

Here $c$ is the fixed linear decay rate, and $b=0$ (no person) indicates no interference by a person. We next explore a few models and associated protocols so if $b=1$ (exists person), which can prevent an adversary from knowing the bit. Here we assume the sender knows this bit $b$.

The Basic Shift Models allowed for Case 1 (perfect hiding) in Lemma II.1. This notably require that the sender knows if a person is interfering. In this section we focus on when the sender does not know the $b$ bit, indicating interference. In this setting, we are not able to achieve perfect hiding, and instead settle for noisy hiding (Case 2 in Lemma II.1).

For simplicity, we analyze the constant offset interference model with reduction of $\delta$, with a linear decay with constant $c$. Although, this can again be generalized to other decay and interference models.

The approach will be to inject noise into signal strength chosen by the sender. We will investigate a few types of noise, and how much is required to achieve various guarantees.

The situation in 2 dimensions with 1 person and 1 adversary can be reduced to the 1-dimensional setting. If the location of the person and adversary is known ($b$ is known), this maps to the Basic Shift models in Section III. If the location of the person or adversary is not known ($b$ is not known), this maps to the Random Noise models in Section III-B.

When there are 2 adversaries and 1 person, the setting is more challenging as one may be interfered with and the other not. Assuming the adversaries collude, this allows them to immediately detect the interference, if the sent signal strength is always the same in all directions. This immediate detection holds even for the Random Noise model as long as the effect of the noise is the same for both adversaries.

To circumvent this obstacle to hiding, we consider using broadcast equipment that can control directional signal strengths (as is commonplace in cell phone towers, and emerging in wifi routers). We consider two models for directional signal strength where hiding results can exist in this setting: very narrow band, and gradual decay.

Very narrow band. In the very narrow band model, the angle in which signal is emitted from the sender is defined by an direction $\theta$, and only is detectable within a very narrow set of angles $(\theta-\tau,\theta+\tau)$, for some parameter $\tau$. If the angle between the two adversaries from the perspective of the sender is greater than $2\tau$, then hiding protocols exist. At each time point, the sender chooses a random direction to send its signal. Then this can be observed by at most one adversary, and it again reduces to the 1 adversary case.

Gradual decay. A different model does not enforce a very narrow band, but instead assumes that for a fixed direction $\theta$, the signal strength decays symmetrically in both directions as the angle of the observer becomes further from $\theta$. Then again if the angle between the adversaries is large enough, a protocol can be designed to hide the interference of a person. In particular, this requires the sender to know if an adversary is interfered with, and that the difference in signal strength received by the two adversaries (because of the angular decay) to be larger than the effect of the interference of the person.

The protocol then is as follows. If there is no interference, direct the signal so its highest signal direction $\theta$ is directly between the two adversaries; use less than the maximum possible strength. If a person interferes with one adversary, then direct the signal closer towards that interfered with adversary and increase the signal strength. If these two parameters (direction and signal strength) are chosen correctly, they can ensure both adversaries receive the same signal as without interference.

Note that in both cases, the most challenging case, which neither can overcome, is when the two adversaries are in a very similar direction from the sender, but are separated enough so a person can interfere with one but not the other. From one perspective, this provides a perhaps surprisingly useful setup for a pair of colluding adversaries. From another perspective, this model may be problematic since a person’s interference with an adversary may not be so binary, and may be diffuse within a range. If this contributes to a random distribution, some version of the noisy hiding analysis may be applicable.

Handling more than 2 adversaries is challenging in the gradual decay model because trying to solve for a setting that retains the same signal strength for 3 adversaries when there are only 2 degrees of freedom (signal strength and angular direction) is, in general, not possible.

We perform simulations on the noisy hiding model under Laplace distribution and normal distribution for different values of the ratio between standard deviation $\sigma$ and the strength of the interference effect $\delta$. For each value of the ratio ($\frac{\sigma}{\delta}=3,6,9$) we draw $1000$ random numbers coming from the specified distribution ($\mathsf{Lap}(\mu,\sigma)$ or $\mathsf{Normal}(\mu,\sigma)$). Then we calculate the likelihoods of the random sample drawn from the true distribution $\mathsf{Lap}(\mu,\sigma)$ and another hypothetical distribution $\mathsf{Lap}(\mu-\delta,\sigma)$, and then the logarithm of the likelihood ratio between these two likelihoods. Our theoretical results indicates that the confidence to determine if the samples come from the true distribution rather than hypothetical distribution would be increased as the sample size increases. We perform $10$ trials for each value of the ratio, and compute the resulting confidence (via confidence = $1/(\exp(\textsf{LLR})-1)$); and report its average and standard deviation. For each ratio $\frac{\sigma}{\delta}$, we vary the number of readings from $100$ to $1000$, and plot the mean and error bars (showing one standard deviation) in Figure 3. For instance when $\sigma/\delta=6$, then it usually requires between $500$ and $800$ readings to reach $95\%$ confidence. Although the Laplace distribution has a better theoretical bound (because of convenient mathematical properties), the normal empirically requires fewer trials to reach high confidence.

We propose new models and protocols to prevent an adversary snooping on the strength of a signal and detecting the presence of a person. Depending on if the interference is known, the presence can either be perfectly hidden, or hidden for a number of rounds with a statistical confidence bound. If the potentially interfering person collaborates with sender, and knows the protocol, it could potentially determine if it interferes with high confidence before a non-collaborating adversary, and shift to a perfect hiding strategy.

The models proposed in the paper are quite general. We omitted some potential specifications – they become tedious quickly – but results will not deviate too much from the general theorems/properties we produce in Section II. For instance, in observing real signal strength decay, they do not decay at a linear rate, the effect of a person blocking does decrease it but the effect is not binary (block or not block), and the fixed structures (like walls) in the environment play a role. When these can be modeled (or learned), very similar statements can be made as simple corollaries of our main results. When they cannot be modeled effectively or precisely, the fix is essentially to add more noise that the sender is not aware of, and the extensions appear much like Theorem IV.3.

For instance, some spatial (2-dimensional) settings with several colluding adversaries may seem hopeless, resulting in immediate detection. But this assumes perfect information and noiseless sensing among adversaries. Noise in the process may lead to something like the noisy hiding scenario in practice.

These methods reduce communication rates, since a lower than maximum signal strength is used. In the noisy hiding, we bound the probability an adversary can determine the bit $b$ indicating if a person is interfering with the signal or not. Hence we must set up the mean of the distribution of the signal strength, say $\mu$, which is lower than the maximal signal strength, say $M$, and the power utilization is roughly $\mu/M$. As discussed, how large $\mu$ can be set depending on $\sigma$ (i.e. variance of the noise), since we desire that $M-\mu>2\sigma$ or $>3\sigma$ so that the truncation of a normal (or Laplace) distribution insignificantly changes the KL divergence. In turn, this value of $\sigma$ should be roughly proportional to $\delta$ (i.e. person effect). Since again, it is common that the interference effect of a person $\delta$ is much less than $M$, this change in power utilization should generally be tolerable.