Secure Communication in the Twin Paradox

Abstract

The amount of information that can be transmitted through a noisy channel is affected by relativistic effects. Under the presence of a fixed noise at the receiver, there appears an asymmetry between “slowly aging” and “fast aging” observers which can be used to have private information transmission. We discuss some models for users inside gravitational wells and in the twin paradox scenario.

Appendix

1.1 Average Capacity

In the twin paradox scenario, the channel changes with time. In this case, we can no longer use Shannon’s noisy channel formula directly. We will use instead an average capacity $\bar{C}$. We define the average capacity in terms of the maximum number of bits that can be sent with our signal of length $T$ (for time and other parameters measured in the transmitter’s frame).

The average capacity is chosen so that $N=T\bar{C}$, where $N$ is the maximum number of decodable bits that can be sent through the channel. We assume that the whole length of the transmission $T$ is divided into $n$ sections $T_i$ so that $T=\sum _{i=1}^{n}T_i$. During each segment of duration $T_i$ the channel is constant and we can assign it a capacity $C_i=W\log _2 \left( 1+\alpha _i SNR \right)$. We suppose that each $T_i$ is long enough for the asymptotic results of Shannon’s theorem to hold.

In each of these segments the transmitter can find codes to send up to $N_i=T_iC_i$ bits. If Alice uses $n$ separate signals (one for each segment), she can send a total of $N=\sum _{i=1}^{n}N_i$ bits. This is not necessarily the best she can do. There might be, for instance, better encoding schemes using blocks of length $T$. However, for $n$ concatenated blocks of length $T_i$, for $i=1\ldots n$, she cannot send more information.

With all these details in consideration, we define the average capacity as

$$\begin{aligned} \bar{C}=\frac{1}{T}\sum _{i=1}^{n}T_iC_i. \end{aligned}$$

(5)

We can define the capacity of the channels going from Alice to different frames using the same formula. In this case, the $C_i$ would have a different Doppler factor $\alpha _i$ and would give a different maximum number of bits. We could even have a different number of segments in each frame. For different receiver frames the passage of time at the destination is also different. However, in the relativistic capacity formula of Jarett and Cover [2] the relevant frame is that of the transmitter and only $\alpha _i$ changes with the receiver. When calculating the average capacity, we only need to calculate the time segments at the transmitter for which the signals will be transformed by a different Doppler factor when they arrive at the receiver.

If Alice sends the same signal to two identical receivers, Bob and Eve, in different frames, the channel from Alice to Bob and from Alice to Eve will, in general, have different average capacities $\bar{C}_{AB}$ and $\bar{C}_{AE}$. There is an important difference between two channels with different capacities $C_{AB}>C_{AE}$ and two channels with different average capacities $\bar{C}_{AB}>\bar{C}_{AE}$. In the first case we can choose a rate $R$ so that $C_{AB}>R>C_{AE}$ to have a direct secure channel. At that rate, Bob can decode all the information and Eve can only get a vanishing amount of bits.

In the second case, when $\bar{C}_{AB}>\bar{C}_{AE}$, we can only guarantee that Eve will get, at most, $\bar{C}_{AE}T$ bits. It could even be the case that in various of the $n$ sections $\bar{C}_{AB_{i}}<\bar{C}_{AE_i}$ allowing Eve to read all the information. Different eavesdroppers could read different segments. However, if the average capacity of the channel between Alice and Bob is greater than the capacity of the channels to any eavesdropper, Alice and Bob can share more bits than Eve is able to capture. As long as Alice and Bob are sure they have more common information than a potential eavesdropper, they can use privacy amplification schemes to distill a completely secret key [11]. This key, when used in conjunction with a one-time pad [13] allows them to have a perfectly secure channel.

1.2 Definitions and Useful Identities

In the analysis of the average capacity and the proofs of security it will be useful to have some intermediate results which relate different parameters that appear in the twin paradox. In this section, we give some of these identities.

1.2.1 Composition of Doppler Factors

For a transmitter $A$ and a receiver $B$, the Doppler factor $\alpha _{AB}=\frac{f_B}{f_A}$ gives the ratio between the frequencies that are measured in each frame. In our average capacity results it is useful to write the parameters in various frames in terms of the Doppler factors between two particular reference systems.

In the twin paradox scenario we are working on, all the given Doppler factors are between frames in uniform motion in the $x$ direction. For two such systems $\alpha _{AB}=\alpha _{BA}$ (there are no preferred inertial frames). When we have more than two systems, we can compose the Doppler factors by simply multiplying them.

We can see an example in Fig. 4 (left). We have three frames $A$, $B$ and $C$, and suppose $A$ is sending a signal at frequency $f_A$. Upon reception, $B$ sees a frequency $f_B=\alpha _{AB}f_A$ and $C$ sees $f_C=\alpha _{AC}f_A$. It is easy to see that

$$\begin{aligned} \alpha _{AC}=\frac{f_C}{f_A}=\frac{f_B}{f_A}\frac{f_C}{f_B}=\alpha _{AB}\alpha _{BC}. \end{aligned}$$

(6)

There is one particular instance of this identity which will be useful in our calculations. We consider the Doppler factor of a system moving with a constant velocity $\beta$ either going away ($\alpha _{-}$) or approaching ($\alpha _{+}$) a reference frame we consider stationary. The space-time diagram of Fig. 4 (right) shows a scenario with two stationary observers, $A$ and $C$, and a travelling observer $B$ moving at velocity $\beta$ from $A$ to $C$. This is a particular case in which Eq. (6) can be applied. The factor $\alpha _{AC}$ is clearly $1$ ($A$ and $C$ are the same reference frame) and $\alpha _{AB}=\alpha _{-}$ and $\alpha _{BC}=\alpha _{CB}=\alpha _{+}$. From Eq. (6) we see that

$$\begin{aligned} \alpha _{+}\alpha _{-}=1. \end{aligned}$$

(7)

Fig. 4

Space-time diagrams for the Doppler factor identities

1.2.2 Useful Identities

The Doppler factors $\alpha _{-}$ and $\alpha _{+}$ of the frames going away or approaching a stationary observer at a constant velocity $\beta$ are of a particular interest to us. They have different properties which will be useful when calculating the average capacities in the twin paradox communication scenario.

We will also use the Lorentz factor $\gamma$ corresponding to a velocity $\beta$

$$\begin{aligned} \gamma =\frac{1}{\sqrt{1-\beta ^2}}. \end{aligned}$$

(8)

Our velocities are normalized to the speed of light ($0\le |\beta | \le 1$) and, therefore, $\gamma \ge 1$. For $|\beta |\rightarrow 0$, $\gamma \rightarrow 1$ and for $|\beta |\rightarrow 1$, $\gamma \rightarrow \infty$. The function $\gamma (\beta )$ is increasing with $\beta$. If we have two velocities $\beta$ and $\beta '$ with Lorentz factors $\gamma$ and $\gamma '$ and $\beta >\beta '$, $\gamma >\gamma '$. The aging ratio between the stationary and the travelling observers in the twin paradox is the $\gamma$ corresponding to the velocity $\beta$ of the journey.

There are different ways to write $\alpha _{-}$ and $\alpha _{+}$ in terms of $\beta$ and $\gamma$:

$$\begin{aligned} \alpha _{+}=\sqrt{\frac{1+\beta }{1-\beta }},&\quad \alpha _{-}=\sqrt{\frac{1-\beta }{1+\beta }}, \end{aligned}$$

(9)

$$\begin{aligned} \alpha _{+}=\gamma (1+\beta ),&\quad \alpha _{-}=\gamma (1-\beta ). \end{aligned}$$

(10)

In the twin paradox, the description is given in terms of a positive velocity $\beta$ greater than zero (so that there is an aging asymmetry). For these $0<\beta \le 1$, we can see from (9) that

$$\begin{aligned} \alpha _{+}> 1,&\quad \alpha _{-}< 1. \end{aligned}$$

(11)

Factors $\alpha _{+}$ and $\alpha _{-}$ could only be equal if $\beta =0$ (when there is no aging difference).

Another relevant quantity is the sum $\alpha _{+}+\alpha _{-}$. From (10), we have

$$\begin{aligned} \alpha _{+}+\alpha _{-}=\gamma (1+\beta )+\gamma (1-\beta )=2\gamma . \end{aligned}$$

(12)

All the given expressions are valid for the Doppler factors $\alpha _{+}$ and $\alpha _{-}$ of any $\beta$.

1.3 Average Capacities in the Twin Paradox

1.3.1 Travelling Twin to Stationary Eavesdropper

First we will analyse the channel from a travelling Alice to a stay-at-home Eve. We have two equal segments of length $\frac{T_A}{2}$ and two Doppler factors, one for the away journey $\alpha _{-}=\sqrt{\frac{1-\beta }{1+\beta }}$ and one for the return journey $\alpha _{+}=\sqrt{\frac{1+\beta }{1-\beta }}$ (see Fig. 5).

Fig. 5

Space-time diagram for the twin paradox channel going from a travelling Alice to a stay-at-home Eve

From (5), we see that the average capacity is

$$\begin{aligned} \bar{C}_{AE}=\bar{C}_{BE}=\frac{W}{2}\left[ \log _2\left( 1+\alpha _{-}SNR\right) +\log _2\left( 1+\alpha _{+}SNR\right) \right] . \end{aligned}$$

(13)

This capacity is independent from the total journey time in Alice’s frame, $T_A$. The expression is also valid for $\bar{C}_{BE}$. From the point of view of Eve, the journeys of Alice and Bob are symmetric.

1.3.2 Channel Between Travelling Twins

The second interesting channel is that connecting the travelling twins Alice and Bob. This channel is slightly more complicated. Now we can define three time segments in the frame of the transmitter. In the first part of the communication, Alice and Bob are going away from Eve at velocity $\beta$. Their Doppler factor with respect to Eve is $\alpha _{-}$. They are also receding from each other with a Doppler factor $\alpha _{-}'=\alpha _{-}^2$ between their frames. We can use Eq. (6) and the reference factor with Eve to show that. Additionally, it is easy to see that $\alpha _{-}^2$ is the Doppler factor corresponding to two frames going away at speed $\beta '=\frac{2\beta }{1+\beta ^2}$, which is the relativistic addition of the velocities of the frames of Alice and Bob. In the second part of the communication the frames are both moving to the right at velocity $\beta$ and there is no Doppler shift ($\alpha _{AB}=1$). Finally, in the third segment, both Alice and Bob approach Eve with velocity $\beta$. Using similar arguments to those of the outgoing journey, we can show that $\alpha _{AB}=\alpha _{+}^2$ in this final phase. Now we can compute the relevant $C_i$s. We only need the length of each segment, which can be obtained from the space-time diagram of the journey (Fig. 6).

Fig. 6

Space-time diagram for two communicating travelling twins

We will do all the calculations from Alice’s frame, but everything is symmetric with respect to a transmitting Bob and a receiving Alice. The easiest segment is the third, which comprises the whole return journey of Alice ($T_3=\frac{T_A}{2}$). The first segment corresponds to the signals that arrive to Bob during the first half of his journey. In his frame, the interval lasts $\frac{T_B}{2}=\frac{T_A}{2}$ seconds.¹ Time segments in different frames $a$ and $b$ are related by $T_b=\frac{T_a}{\alpha _{ab}}$. Therefore, $T_1=\alpha _{-}^2\frac{T_A}{2}$. The remaining time can be deduced by noticing that the two first communication segments take the first half of Alice’s journey ($T_1+T_2=\frac{T_A}{2}$).

Now we have $T_1=\alpha _{-}^2\frac{T_A}{2}$, $T_2=(1-\alpha _{-}^2)\frac{T_A}{2}$ and $T_3=\frac{T_A}{2}$, and the corresponding Doppler factors $\alpha _1=\alpha _{-}^2$, $\alpha _2=1$ and $\alpha _3=\alpha _{+}^2$ which define the capacity. The average capacity from Alice to Bob is

$$\begin{aligned} \bar{C}_{AB}= & {} \bar{C}_{BA}=\frac{W}{2}\nonumber \\&\times \left[ \alpha _{-}^2\log _2(1+\alpha _{-}^2SNR)+(1-\alpha _{-}^2)\log _2 (1+SNR)+\log _2(1+\alpha _{+}^2SNR)\right] .\nonumber \\ \end{aligned}$$

(14)

This capacity is also independent of $T_A$.

1.3.3 Travelling Twin to an Inertial Eavesdropper

A third relevant channel is the one between Alice and an inertial eavesdropper $I$ moving at a constant speed $\beta _{I}$ in the $x$ axis (Fig. 7). We characterize the inertial observer by its Doppler factor $\alpha _{I}$ with respect to the stationary Eve.

Fig. 7

Space-time diagram for the communication between Alice and an inertial eavesdropper

The analysis is exactly the same as in the case with a stationary observer. In Eq. (13) we only need to correct the Doppler factors multiplying them by $\alpha _{I}$. This is a direct consequence of the Doppler factor composition law of Eq. (6). The average capacity is

$$\begin{aligned} \bar{C}_{AI}=\frac{W}{2}\left[ \log _2\left( 1+\alpha _{-}\alpha _I SNR\right) +\log _2\left( 1+\alpha _{+}\alpha _I SNR\right) \right] . \end{aligned}$$

(15)

Again, the capacity does not depend on $T_A$. For $\alpha _{I}=1$ (stationary observers), we recover capacity $\bar{C}_{AE}$.

For certain gravitational fields, the Doppler factors coming from gravitation and motion can be factored [10]. In these cases, capacity $\bar{C}_{AI}$ is also valid for any observer in the $x$ axis with the same $\alpha _I$ as the inertial observer. The Doppler factor can come from a gravitational shift (either red or blue) or a combination of motion and gravitation.

1.4 Proof of Security

1.4.1 Security Against a Stationary Eavesdropper

The travelling twins can establish a private channel if they can share more information between them than with the frame of the eavesdropper. There is a secure channel if we can prove that $\bar{C}_{AB}>\bar{C}_{AE}$ (or that $\bar{C}_{AB}>\bar{C}_{AI}$ for inertial eavesdroppers). Equations (13) and (14) tell us that privacy is possible if,

$$\begin{aligned}&\frac{W}{2}\bigl [\alpha _{-}^2\log _2(1+\alpha _{-}^2SNR)+(1-\alpha _{-}^2)\log _2(1+SNR)+\log _2(1+\alpha _{+}^2SNR)\bigr ] \nonumber \\&\quad >\,\frac{W}{2}[\log _2(1+\alpha _{-}SNR)+\log _2(1+\alpha _{+}SNR)]. \end{aligned}$$

(16)

We can simplify the expression extracting common factors and noticing that the logarithm is an increasing function. Proving that the channel between Alice and Bob is secure is equivalent to showing that

$$\begin{aligned}&\alpha _{-}^2\log _2\left( 1+\alpha _{-}^2SNR\right) +\left( 1-\alpha _{-}^2\right) \log _2\left( 1+SNR\right) +\log _2\left( 1+\alpha _{+}^2SNR\right) \nonumber \\&\quad >\,\log _2\left( 1+\alpha _{-}SNR\right) +\log _2\left( 1+\alpha _{+}SNR\right) , \end{aligned}$$

(17)

or

$$\begin{aligned}&\log _2((1+\alpha _{-}^2SNR)^{\alpha _{-}^2}(1+SNR)^{1-\alpha _{-}^2}(1+\alpha _{+}^2SNR))\nonumber \\&\quad > \log _2((1+\alpha _{-}SNR)(1+\alpha _{+}SNR)), \end{aligned}$$

(18)

or

$$\begin{aligned}&\left( 1+\alpha _{-}^2SNR\right) ^{\alpha _{-}^2}\left( 1+SNR\right) ^{1-\alpha _{-}^2}\left( 1+\alpha _{+}^2SNR\right) \nonumber \\&\quad > \left( 1+\alpha _{-}SNR\right) \left( 1+\alpha _{+}SNR\right) . \end{aligned}$$

(19)

As $\alpha _{-}<1$, $1+SNR>1+\alpha _{-}^2SNR$. Using (7), we can give a lower bound to the left hand side of (19):

$$\begin{aligned}&\left( 1+\alpha _{-}^2SNR\right) ^{\alpha _{-}^2}\left( 1+SNR\right) ^{1-\alpha _{-}^2}\left( 1+\alpha _{+}^2SNR\right) \nonumber \\&\quad \ge \,\left( 1+\alpha _{-}^2SNR\right) \left( 1+\alpha _{+}^2SNR\right) =1+\alpha _{+}^2SNR+\alpha _{-}^2SNR+SNR^2.\qquad \end{aligned}$$

(20)

Expanding also the right hand side of (19), we can prove $\bar{C}_{AB}>\bar{C}_{AE}$ by showing

$$\begin{aligned} 1+\alpha _{+}^2SNR+\alpha _{-}^2SNR+SNR^2 > 1+ \alpha _{+}SNR+\alpha _{-}SNR+SNR^2. \end{aligned}$$

(21)

Taking out common factors, this reduces to proving

$$\begin{aligned} \alpha _{+}^2+\alpha _{-}^2 > \alpha _{+}+\alpha _{-}, \end{aligned}$$

(22)

which, from (12), is equivalent to

$$\begin{aligned} \gamma ' > \gamma , \end{aligned}$$

(23)

for the $\gamma '$ corresponding to $\alpha _{+}'=\alpha _{+}^2$ and $\alpha _{-}'=\alpha _{-}^2$ (a velocity $\beta '=\frac{2\beta }{1+\beta ^2})$. For $0<\beta <1$, $\beta '>\beta$. As $\gamma (\beta )$ is increasing, (23) is true for all the $\beta$s we have in the twin paradox. $\square$

1.4.2 Security Against Inertial Eavesdroppers

The channel between Alice and Bob is also secure against a family of inertial eavesdroppers with an $\alpha _I$ below a certain threshold. If we take the expressions for $\bar{C}_{AB}$ and $\bar{C}_{AI}$, we can give a security condition similar to (19). Alice and Bob can share a secure channel if

$$\begin{aligned}&\left( 1+\alpha _{-}^2SNR\right) ^{\alpha _{-}^2}\left( 1+SNR\right) ^{1-\alpha _{-}^2}\left( 1+\alpha _{+}^2SNR\right) \nonumber \\&\quad > \left( 1+\alpha _{I}\alpha _{-}SNR\right) \left( 1+\alpha _{I}\alpha _{+}SNR\right) . \end{aligned}$$

(24)

For any given $SNR$ we can evaluate (24) numerically. Figure 8, which corresponds to Fig. 3 in the paper, gives the secure regions (shaded) for $SNR=1$. In this figure, we have also included the velocity values $\beta _I$ which give a Doppler factor $\alpha _I$ between the inertial observer and our reference stationary Eve. The factor $\alpha _{+}$ for the twin velocity $\beta$ is included for comparison.

Fig. 8

Security regions for inertial eavesdroppers: The limiting values, $SNR\rightarrow 0$ and $SNR\rightarrow \infty$, are derived in the section “Limiting Case” of this Appendix

These maps of the secure communication regions against inertial eavesdroppers are a good starting point for many interesting generalizations. In the following sections, we will extend the security results to different signal-to-noise ratios and to eavesdroppers moving in any direction and with arbitrary noise levels.

First, we will derive the analytic expressions for the frontiers of the secure regions for different signal-to-noise ratio regimes. Then, we will give a worst-case bound on the security against arbitrary inertial observers not moving in the $x$ direction. Finally, we will show how to extend the security scheme to situations where the eavesdropper has a different local noise level.

1.5 Limiting Cases

The secure regions give the speeds $\beta$ which guarantee the existence of a private channel for Alice and Bob for a given family of inertial observers with an $\alpha _I$ below a certain value. For $\alpha _I<1$ Alice and Bob always have a secure channel. The average capacity $\bar{C}_{AI}$ is bounded by $\bar{C}_{AE}$ which is always smaller than $\bar{C}_{AB}$.

On the other hand, for a fixed twin velocity $\beta$ there is always an eavesdropper with $\alpha _{I}>1$ for which $\bar{C}_{AI}\ge \bar{C}_{AB}$. She can listen to the channel between Alice and Bob. Below that $\alpha _{I}$ the channel is secure. Above it, the eavesdropper might decode the transmission. The frontier depends on the value of the signal-to-noise ratio. Both $\bar{C}_{AB}$ and $\bar{C}_{AI}$ grow with $SNR$. However, for our range of interest with $\alpha _{I}>1$, smaller values of $SNR$ will reduce the amplifying effect of $\alpha _{I}$.

An inertial eavesdropper with a Doppler factor $\alpha _{I}$ is equivalent to a stationary eavesdropper with signal-to-noise ratio $SNR'=\alpha _{I}SNR$. For the same increment in $\alpha _I$, a greater $SNR$ will give a greater increment in $SNR'$. This gives two important limiting cases. The first is the strong signal (low noise) limit, with $SNR\rightarrow \infty$. The second is the weak signal (high noise) limit with $SNR\rightarrow 0$. This last limit gives the maximum attainable secure region. Alice and Bob should try to work with small values of $SNR$ whenever possible.

1.5.1 $SNR\rightarrow \infty$:

We first study the worst case against inertial observers. In the strong signal limit we suppose that, even for high velocities (very small $\alpha _{-}$), $\alpha _{-}^2 SNR\gg 1$. In that case, we can approximate the security condition (24) by

$$\begin{aligned} \left( \alpha _{-}^2SNR\right) ^{\alpha _{-}^2}\left( SNR\right) ^{1-\alpha _{-}^2}\left( \alpha _{+}^2SNR\right) >\left( \alpha _{I}\alpha _{-}SNR\right) \left( \alpha _{I}\alpha _{+}SNR\right) , \end{aligned}$$

(25)

which is equivalent to asking that

$$\begin{aligned} {\alpha _{-}^2}^{\alpha _{-}^2}\alpha _{+}^2>\alpha _{I}^2. \end{aligned}$$

(26)

If we use (7), we can write everything in terms of $\alpha _{-}$ only. The condition for security in the strong signal limit becomes

$$\begin{aligned} \alpha _{-}^{\alpha _{-}^2-1}>\alpha _{I}. \end{aligned}$$

(27)

This condition gives the limiting upper curve of Fig. 8. When $\beta \rightarrow 1$ ($\alpha _{-}\rightarrow 0$) the condition can be approximated by

$$\begin{aligned} \alpha _{+}>\alpha _{I}. \end{aligned}$$

(28)

For high $\beta$s, an inertial eavesdropper must, at least, match the twins’ velocity. For smaller values of $\beta$ a smaller $\beta _I$ is enough to compromise security.

1.5.2 $SNR\rightarrow 0$:

Similarly to the previous case, with this condition we mean that even for a high $\beta$ we have an $SNR$ so that $\alpha _{+}^2 SNR\ll 1$. In that case we can use the approximation $\displaystyle \lim _{x\rightarrow 0} \log _2(1+x)=\frac{1}{\ln 2}x$. We start from security condition (17) adapted to the inertial observer case,

$$\begin{aligned}&\alpha _{-}^2\log _2\left( 1+\alpha _{-}^2SNR\right) +\left( 1-\alpha _{-}^2\right) \log _2\left( 1+SNR\right) +\log _2\left( 1+\alpha _{+}^2SNR\right) \nonumber \\&\quad >\,\log _2\left( 1+\alpha _{I}\alpha _{-}SNR\right) +\log _2\left( 1+\alpha _{I}\alpha _{+}SNR\right) . \end{aligned}$$

(29)

In the weak signal limit, the condition becomes

$$\begin{aligned} \alpha _{-}^4SNR+\left( 1-\alpha _{-}^2\right) SNR+\alpha _{+}^2SNR>\alpha _{I}\alpha _{-}SNR+\alpha _{I}\alpha _{+}SNR. \end{aligned}$$

(30)

Taking out common factors we have the security condition

$$\begin{aligned} \frac{\alpha _{-}^4+1-\alpha _{-}^2+\alpha _{+}^2}{\alpha _{+}+\alpha _{-}}>\alpha _{I}. \end{aligned}$$

(31)

This formula could also be written only in terms of $\alpha _{-}$ or $\alpha _{+}$, but that does not provide any additional insight. Once again, in the $\beta \rightarrow 1$ limit ($\alpha _{-}\rightarrow 0$ and $\alpha _{+}\rightarrow \infty$) we recover the condition

$$\begin{aligned} \alpha _{+}>\alpha _{I}. \end{aligned}$$

(32)

When $\beta$ is close to the speed of light, the results for high and low $SNR$ converge. If $\alpha _I$ comes only from motion in the $x$ direction, $\beta >\beta _I$ guarantees security.

1.6 Inertial Eavesdroppers in Arbitrary Directions

The security regions we have derived are valid for eavesdroppers in the $x$ axis with a Doppler factor $\alpha _{I}$ (caused either by a uniform movement or a gravitational field). In this section, we show how to generalize the security conditions to an inertial eavesdropper $I'$ moving in an arbitrary direction. If $I'$ has a velocity of absolute value of, at most, $|\beta _{I}'|=\beta _I$, her Doppler factor $\alpha _{I}'$ will be bounded by $\alpha _{+}\alpha _{I}$ (for the $\alpha _{+}$ and $\alpha _I$ of the previous sections). This bound limits the capacity of the eavesdropper’s channels with Alice and Bob. We can adapt the security conditions so that $\bar{C}_{AB}$ is always greater than both $\bar{C}_{AI'}$ and $\bar{C}_{BI'}$.

To prove $\alpha _{I}'<\alpha _{+}\alpha _I$, we look into the worst case situation (highest $\alpha _{I}'$). The Doppler factor resulting from any uniform motion can be written as [4]

$$\begin{aligned} \alpha =\frac{1-\beta '\cos \varPhi }{\sqrt{1-\beta '^2}}. \end{aligned}$$

(33)

Here, $\beta '$ is the relative velocity between the transmitter at the moment of transmission and the receiver at the moment of reception. $\varPhi$ gives the angle between the direction of the velocity of the receiver and the direction of propagation of the signal, both measured at the moment of reception. This $\alpha$ is maximized for $\varPhi =\pi$ (a receiver approaching an incoming wave).

We can study the relevant Doppler factors in this new situation from the relativistic composition of two velocities. We will discuss the communication with Alice, but everything will also hold for Bob. Suppose we know the velocities of the two observers in the frame of Eve. Observer $I'$ has a velocity $\beta _I$ in a direction given by the unit vector $\hat{u}_I$ and observer $A$ has a velocity $\beta$ in the direction $\hat{u}$. For these conditions, the relative velocity between the observers in the resulting direction is [4]

$$\begin{aligned} \beta '=\frac{\sqrt{\beta _{I}^2+\beta ^2+2\beta \beta _I\cos \varPsi -\beta _I^2\beta ^2\sin ^{2}\varPsi }}{1+\beta _I\beta \cos \varPsi }, \end{aligned}$$

(34)

where $\varPsi$ is the angle between $\hat{u}_I$ and $\hat{u}$. We are interested in the maximum relative velocity between $A$ and $I'$. This velocity can help us to give a limit to the maximum attainable Doppler factor of a general inertial eavesdropper in the twin scenario. The dependence of $\varPsi$ means that there are angles for which the same velocities $\beta$ and $\beta _I$ give a higher relative velocity. We can easily find the optimal values. Taking the first derivative of $\beta '$ with respect to $\varPsi$ we see that $\beta '$ becomes extremal if:

• 1. Either $\beta _I$ or $\beta$ are equal to $1$, for any value of $\varPsi$ and the other velocity. In this case, $\beta '=1$. This is a consequence of the constant value of $c$: a photon will be seen to move at the speed of light in any other inertial frame.

• 2. Either $\beta _I$ or $\beta$ are equal to $0$, for any value of $\varPsi$ and the other velocity. Here, $\beta '$ is the other velocity. This case and the previous one are saddle points.

• 3. $\sin \varPsi =0$, for any combination of velocities $\beta _I$ and $\beta$. From the second derivative, we see that $\varPsi =0$ gives a minimum and $\varPsi =\pi$ a maximum. Velocity $\beta '$ is the one-dimensional relativistic addition of velocities. The maximum happens for two frames approaching in the same direction, as expected. This means that an inertial observer with velocity $\beta _I'\le \beta _I$ has a Doppler factor $\alpha _I'$ smaller than $\alpha _{+}\alpha _{I}$ (the factor for the addition of velocities $\beta$ and $\beta _I$ in the same direction).

The limiting case is an inertial observer approaching Eve in the $x$ direction. For such an eavesdropper, both Eqs. (33) and (34) are maximized. For our $\alpha _{I'}<\alpha _{+}\alpha _{I}$, both $\bar{C}_{AI'}$ and $\bar{C}_{BI'}$ are bounded by $W\log _2\left( 1+\alpha _{+}\alpha _{I}SNR\right)$. We can now adapt condition (16) to a general observer and write it as

$$\begin{aligned}&\frac{W}{2}\biggl [\alpha _{-}^2\log _2\left( 1+\alpha _{-}^2SNR\right) +\left( 1-\alpha _{-}^2\right) \log _2\left( 1+SNR\right) +\log _2\left( 1+\alpha _{+}^2SNR\right) \biggr ] \nonumber \\&\quad >\,W\log _2\left( 1+\alpha _{+}\alpha _{I}SNR\right) . \end{aligned}$$

(35)

We can see this condition in the limit of weak signal, $SNR\rightarrow 0$, which was the regime which offered a greater security for the twin channel. The twins choose their transmitting power and can always place themselves in this situation. Under this approximation, we can derive a condition similar to inequality (30), which in our new scenario becomes

$$\begin{aligned} \alpha _{-}^4SNR+\left( 1-\alpha _{-}^2\right) SNR+\alpha _{+}^2SNR>2\alpha _{I}\alpha _{+}SNR. \end{aligned}$$

(36)

We can group all the terms relative to the twins’ movement in the left to obtain

$$\begin{aligned} \alpha _{-}^5+\alpha _{-}-\alpha _{-}^3+\alpha _{+}>2\alpha _{I}. \end{aligned}$$

(37)

As $\alpha _{-}<1$, $\alpha _{-}-\alpha _{-}^3>0$. We can guarantee security if

$$\begin{aligned} \alpha _{+}>2\alpha _{I}. \end{aligned}$$

(38)

This bound is not optimal. The channel capacity will be different for the outgoing and return journeys. We can use that to our advantage to give a better bound. In a general case, we need to pay attention to the combination of velocities, directions and initial position of the eavesdroppers in order to compute the real capacity. This simpler crude bound suffices for our purposes. We can also include the effects of gravitation. For any particular case, we can find a maximum $\alpha _I'=\alpha _{+}\alpha _I$ and give the corresponding security condition.

1.7 Eavesdroppers with an Arbitrary Local Noise

In our security conditions we have assumed the eavesdroppers have the same local noise as the legitimate users of the channel. In this section, we show that, as long as the eavesdropper has a receiver with a local noise greater than zero, there are velocities $\beta$ which allow Alice and Bob to preserve privacy.

Imagine Alice and Bob have receivers with a local white noise of power spectral density $N_0$ and a stationary eavesdropper with a white noise of power spectral density $N_0'$ so that $0<N_0'<N_0$. The assumption of a non-zero gaussian noise is reasonable, as any receiver with a temperature greater than absolute zero will suffer some thermal noise. This stationary eavesdropper is equivalent to an inertial eavesdropper of $\alpha _{I}=\frac{N_0}{N_0'}$. Both observers have the same capacity with Alice and Bob.

We can use the security conditions for inertial observers to deduce the twin velocity which gives a secure channel for any noise asymmetry. We can even include the effect of motion or a gravitational Doppler shift in addition to a smaller noise. For an eavesdropper with a maximum Doppler factor $\alpha _{I}$ and a local noise $N_0'$, we just need to consider an inertial observer with $\alpha _{I}'=\alpha _I\frac{N_0}{N_0'}$. We will study the twin channel, but the same substitution can be done for the redshift channel.

Figure 9 shows the secure regions for an example with a weak signal. If we can bound the noise reduction factor $\frac{N_0}{N_0'}$ of the eavesdropper, we can deduce the velocity $\beta$ which gives us an advantage in the average capacity. We have not considered the case where $N_0'>N_0$, which will only improve security.

Fig. 9

Security conditions for low noise eavesdroppers: Even if the eavesdropper has a noise $N_0'$ below the local noise $N_0$ of the twins, the twin channel is still secure for certain velocities. For a noise reduction factor $\frac{N_0}{N_0'}$, the graph shows all the velocities which permit a private channel. For high noise differences, the velocity $\beta$ is close to 1

We can also give an approximate bound. We will consider the weak signal limit and suppose that $\alpha _{I}=\frac{N_0}{N_0'}$ is big so that $\beta$ must be close to 1 and $\alpha _{+}\gg 1$ and $\alpha _{-}\ll 1$. In that case, from (9), we can see $\beta =\frac{1-\alpha _{-}^2}{1+\alpha _{-}^2}\approx (1-\alpha _{-}^2)(1-\alpha _{-}^2)\approx 1-2\alpha _{-}^2=1-\frac{2}{\alpha _{+}^2}$. For these conditions, we know from (32) that we have a secure channel if $\alpha _{+}>\alpha _{I}$. For a noise reduction factor $\frac{N_0}{N_0'}$ we can have a secure channel for velocities

$$\begin{aligned} \beta \ge 1-2\left( \frac{N_0'}{N_0}\right) ^2. \end{aligned}$$

(39)