RU2325039C2 - Method of encoding and encryption keys transmission - Google Patents

Abstract

FIELD: information encryption.

SUBSTANCE: method includes forming quantum photonic status sequence on the transmitting station to encode encryption keys and transmission of these statuses via open space to receiving station; at that, the distance between the stations is previously measured and clock on both station is synchronised; then, transmitting station converts quantum photonic statuses to one- or multiphoton orthogonal statuses and measures their sending time, which is sent to receiving station; the receiving station measures receiving time of the one- or multiphotonic quantised statuses, determines delay value, using which decoding and eavesdropping are performed.

EFFECT: provision of encryption key security during its long-distance transmission via open space, provision of long-term stability and reducing error stream in transmitted encryption keys on receiving station.

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Description

The invention relates to the field of coding and transmission of cryptographic keys through open space.

The prior art in the art knows a method for encoding and transmitting cryptographic keys, which includes generating a sequence of quantum photon states at a transmitting station and transmitting these states through open space to a receiving station (see US Pat. No. 6,784,083 B1, class MKI H04K 001/08; G06F 001/02; G06F 007/58, published June 8, 2004).

The fundamental disadvantages of this method include the inability to technically implement a source that generates strictly single-photon states. With non-strict single-photon source and attenuation in a space exceeding a certain critical value, the known method does not guarantee the secrecy of transmitted cryptographic keys. In addition, the known method requires precise and consistent control over the polarization degrees of freedom both in coding and in the detection of single-photon states at the receiving and transmitting stations. Due to these fundamental shortcomings, the known method does not allow secrecy of transmitted cryptographic keys over long distances, long-term stability of the system and minimize the flow of errors at the receiving station.

The technical result to which the invention is directed is to ensure the secrecy of cryptographic keys when transmitting the latter through open space over long distances, including transmission between ground objects and low-orbit satellites (up to altitudes of 1000 km) in the case when the source of quantum states is not is strictly single-photon, and during communication in space there is arbitrary attenuation, as well as to achieve long-term stability and reduce the flow of errors in front of Vai primary cryptographic keys at the receiving station.

Using such a fundamental principle as the principle of relativistic causality in quantum cryptography allows us to formulate a fundamentally new approach to ensuring the secrecy of transmitted cryptographic keys. In this method of encoding and transmitting cryptographic keys, the fundamental difficulties existing in the previously known method are removed (see US patent No. 6748083 B1, CL MKI H04K 001/08; G06F 001/02; G06F 007/58, published on 08.06.2004, ) The proposed method for encoding and transmitting cryptographic keys can naturally be called relativistic quantum cryptography. Relativistic quantum cryptography provides secrecy in the transmission of cryptographic keys: 1) during any attenuation during communication (attenuation only reduces the transmission speed of keys, but does not affect their secrecy); 2) does not require a strictly single-photon source of quantum states (the presence of a single-photon component in states with only a certain probability is sufficient. The scheme remains secret even with an arbitrarily small fraction (probability) of a single-photon component. This means that the quantum state can have an arbitrarily large average number photons. The fraction of a single-photon component affects only the transmission speed of keys, but not their secrecy).

The specified technical result is achieved by the fact that in the method of encoding and transmitting cryptographic keys, which includes generating a sequence of quantum photon states at the transmitting station and transmitting these states through open space to the receiving station, the distance between the stations is preliminarily measured and the clock is synchronized on them, then at the transmitting station transform sequences of quantum photon states into single or multiphoton orthogonal states and measure the time of their sending, and n the receiving station measures the time of receiving and determining a delay value at which produce decoding to detect eavesdropping.

The invention is illustrated by drawings. In FIG. 1a), b), c), and 2 show spatial-temporal diagrams of quantum states with a finite length for encoding cryptographic keys, explaining the inevitability of a time delay at the receiving station if there were attempts to eavesdrop and extract information during the transmission of keys. Figure 3 shows an example implementation of the method.

In figure 1:

a) a spatio-temporal diagram explaining the process of copying one of the orthogonal quantum states of a photon with a spatio-temporal amplitude of length L by converting it to a localized at the spatio-temporal point G, measuring the latter, then converting back to a quantum state with the same spatial time amplitude as the initial state, but with a time delay equal to the length of the state;

b) a space-time diagram explaining the process of copying an extended quantum state of a photon by unitary conversion to atomic degrees of freedom, measuring the quantum state of an atomic system, then converting the quantum state of an atomic system to a state of a photon with the same spatio-temporal form as the original, but with a time delay equal to the extent of the condition;

c) a spatio-temporal diagram explaining the violation of the principle of relativistic causality if one of the orthogonal quantum states is copied faster than prescribed by diagrams a) and b).

On figa) adopted the following notation:

1 - the initial quantum state with a spatio-temporal extent L at the time slice 5;

2 - an intermediate quantum state with a smaller spatio-temporal extent than the initial one at a later time slice 6;

3 is a copy of the initial quantum state at the time slice 8, but with a time delay compared to the initial state 4, which experiences free propagation at the speed of light;

7 - a time slice at which the initial quantum state is converted to a localized one at G. L / c is the minimum time necessary between slices 5 and 7, which is dictated by the principle of relativistic causality, for converting the initial quantum state to a localized one. Similarly, L / c is the minimum time required between slices 7 and 8 to convert a localized quantum state at point Г to an extended one with the same spatio-temporal amplitude as the original, but delayed in time compared to quantum state 4 with free spread with the speed of light.

On figb) adopted the following notation:

9 - the initial quantum state with a spatio-temporal extent L at the time slice 13;

10 - quantum state at the time slice 14, the dashed line shows the fraction of the normalization of the state, which is transformed into atomic degrees of freedom, localized at the point (spatial slice 12);

11 is the amplitude of the freely propagating quantum state of a photon in a time slice 15.

On figv) adopted the following notation.

16 - the initial quantum state with a spatio-temporal extent L;

17 - a hypothetical transformation of the initial state into a quantum state with a smaller spatio-temporal extent, violating the principle of relativistic causality.

Any quantum states have some finite spatio-temporal extent (respectively, a finite frequency band). In this case, when formulating a real-time exchange protocol, to distinguish between the number of photons in a line as for any other measurements, an inevitably finite time is required, which leads to a delay in the measurement results at the receiving station, which makes it possible to detect any attempts to eavesdrop.

For orthogonal quantum states, there is no prohibition on reliable discrimination without disturbing them if the state is immediately available as a whole.

Access to a quantum state immediately as a whole inevitably implies access to that part of space-time where the amplitude (wave function) of state 1 is nonzero (see Fig. 1a)). If only a part of the space is available where the state amplitude is nonzero, then even orthogonal states cannot be reliably copied or distinguished. The latter is more or less obvious, since no process, including copying or distinguishing, can have a probability of an outcome greater than the fraction of the normalization of states, which is typed in the available space-time domain, and thus automatically in the accessible part of the Hilbert space. That is, in order to reliably copy or distinguish between orthogonal states, they are needed immediately and in their entirety.

In nonrelativistic quantum mechanics, where there are no restrictions on the limiting speed of propagation of signals, access to any finite region can be obtained instantly. In quantum field theory, where there are restrictions on the limiting speed, access to the entire quantum state can be obtained only if the extended quantum state is previously transformed to a quantum state with an amplitude that is nonzero only in an arbitrarily small spatial region. Due to the principle of relativistic causality, such a transformation of a quantum state defined in a finite space-time region into a quantum state localized in an arbitrarily small spatial region can be realized only in a finite time. The minimum necessary time is determined from the condition that the previous part of the light cone with the vertex in G (see figa) is covered with the original spatial region where the amplitude of the quantum state 1 was different (see figa)). The top of this cone is located in an arbitrarily strongly localized region (point Г), into which the initial amplitude of the quantum state 1, 2 is converted (see Fig. 1a)). Each of a pair of orthogonal quantum states transformed (“collected”) in a localized region can then be reliably copied or distinguished. Since we are talking about massless states of a quantized field (photons) that propagate at the maximum permissible speed, such a conversion and further copying will lead to a shift (delay) of states in space-time compared to the initial free evolution (propagation) of states.

This circumstance makes it possible to detect any eavesdropping attempts by measuring the delays of the measurement results at the receiving station.

In other words, for the orthogonal states of a massless quantized field (photons), the copy prohibition theorem is as follows. Orthogonal states can be copied with a probability arbitrarily close to unity. But at the same time, as a result of copying, states with the same form of amplitudes are obtained, but shifted (translated in space-time). Those. a weaker process is allowed compared to the nonrelativistic case

U _L the translation operator in space-time along the branch of the light cone, the quantity L = Δ (xt) is the size of the region where the amplitude of the states | φ ₀ , ₁ > is non-zero (we assume, for brevity, that both states are non-zero in the same spatial -time domain, but differ in the shape of the amplitudes

| φ _0,1 (xt)>).

The theorem on distinguishing orthogonal states is similarly modified; only a weaker process is allowed in comparison with the nonrelativistic case

The foregoing is convenient to explain using figa), b).

Since the amplitude of the states of a massless quantized field (photons) propagating in the same direction of the x axis depends only on the difference xt, we can fix the time and consider the coordinate as a variable, or vice versa. We will do this for both cases. These two cases exhaust all situations. Let one of the orthogonal states with an amplitude | φ _0,1 (xt)> propagating with the speed of light be given (a system of units is chosen when c = 1, the state index 0 or 1 is omitted for brevity for now). Let state 1.9 be concentrated in region L (Fig. 1a), b)).

In order to have all the values of the state amplitude for all x at once, at the moment t ₀ in the region where it is nonzero, it is necessary to perform a unitary transformation immediately over the entire state. Let the unitary transformation over the state amplitude be U φ _0,1 (xt ₀ ) = φ _0,1 (xt) (t> t ₀ ), the amplitude of the new state φ _0,1 (x'-t) can be nonzero already in a smaller spatial area. Essentially, the minimum size of the region in x 'at time t is dictated by the relativistic causality principle. By the time instant not earlier than L / c, the amplitude of the initial state can be unitarily converted to a state with an arbitrarily strongly localized amplitude in the vicinity of Γ (see Fig. 1a)). It is crucial that this will be a different state than the original φ _0,1 (xt _0). By the moment Г, the values of the state amplitude are available for all x at once (instantly). Now you can instantly get the outcome of the measurement and have complete (with probability one) information about the state. If the pair of initial states was orthogonal, then by a unitary transformation it is also possible to obtain a pair of orthogonal states by the time Г and, accordingly, reliably distinguish one from the other. We emphasize once again that these will be other orthogonal states that are different from the initial ones. The “restoration” or copying of a state can also be realized by an inverse unitary transformation “directed” forward in time. State 3 with the same form of amplitude as the initial one can be obtained by the time no earlier than this is dictated by relativistic causality (see figa). The amplitude of the state, with the same shape as the original, is in front of the light cone released from point G. The resulting state 3 (figa) is also different from the original 4 (figa)), in the sense that it is delayed in time with respect to the initial state, which would have propagated forward in x by the time L / c, just by the value of L / c, if there were no attempts to copy or obtain information about it. While it was a question of obtaining information about the states in the channel with a probability of one. The same reasoning is suitable for obtaining information with a probability of less than one. Moreover, the time delay introduced by the eavesdropper will be less than L / c (figa), b)), but it will inevitably be non-zero.

Similarly, it can be shown when the state is converted in a unitary way to the state of an auxiliary localized system (see fig. 1b)) An example of such a unitary transformation takes place when the light "stops". This unitary transformation translates the state of the photon field into a vacuum state due to its masslessness and the inability to have a zero propagation velocity, and the state of the atomic system to a new state. The conversion, being unitary, also requires access to all values of the amplitude of the photon packet at the localization point of the atomic system 12 (see Fig. 1b)). Such access is achieved naturally as the packet propagates at the speed of light and reaches a localized atomic system (the "entry" of the packet as a whole into the atomic system). This process, when it comes to obtaining a result with a probability of one, also requires L / c time (a single-photon packet must completely “enter” the atomic system). The process of "entry" is shown in fig.1b) - an intermediate state 10. In this case, the photon field is in another - vacuum state, and the auxiliary system - in a new state, depending on the input photon state. By the time L / c, with probability one it is possible to find out what kind of condition it is and prepare the same, but with the inevitable delay on L / c, which will take place in comparison with the free distribution of the original package 11 (Fig. 1b)).

Thus, any receipt of information about one of the orthogonal states leads to their inevitable modification - broadcast in space-time (time delay).

It is also important that no evolution of a massless quantized field interacting with the environment (other quantum and classical degrees of freedom in the channel) can lead to a “compression” of the state, in the sense that the normalization of the state will accumulate in a smaller spatial region beyond the light cone compared with free distribution (see figv)). As a rule, such an interaction will lead to the state being mixed, but the carrier of the density matrix in space-time cannot be “squeezed” and led out of the light cone (Fig. 1b) - state 16). Otherwise, this would make it possible to transmit information using quantum states faster than the speed of light. Indeed, let there be one of a pair of orthogonal quantum states (Fig.1c) - 16 - a state of length L is covered by the last part of the light cone released from point A. At point A, classical information can be extracted from the quantum state no earlier than at the time instant determined condition for covering the state amplitude with the last part of the light cone. After that, he can transfer already classical information to point B (see figv)). Such a transmission cannot be made faster than with the speed of light (observers are connected by a branch of the light cone of FIG. 1 c)). If, as a result of the evolution of the quantum state in the channel, it could “shrink” in such a way that when the state was covered with the last part of the light cone, the vertex of this cone would appear in a spatially similar region with respect to the light cone with the vertex at point A, one of whose branches passes through point B (see figv)). In this case, at point B, classical information can be extracted from the quantum state earlier than it could be transmitted at the speed of light from point A, since the apex of the light cone covering the “squeezed” quantum state enters a spatially similar region 17 (see Fig. .1c)).

For cryptography, this means that the noise in the channel prevents the eavesdropper from either copying or receiving state information earlier than dictated by the diagrams in Fig. 1a), b) (the value of the eavesdropper error under the condition of passing a temporary delay test cannot be less, than L / c).

This circumstance is fundamentally important for the secrecy of relativistic quantum cryptography, which remains secret at any attenuation. Since in our case secrecy is based on the relativistic principle of causality for the evolution of quantum states, the damping, whatever it may be, cannot cancel the restrictions dictated by relativistic causality.

This method of encoding and transmitting secret cryptographic keys through open space guarantees the detection of an eavesdropper and the receipt of secret keys even if the quantum photon state of the radiation is not strictly single-photon. The value that is responsible for detecting the eavesdropper is δ ^OK _E - the probability for the eavesdropper to know the transmitted bit and pass the delay test at the receiving station, depending on the delay p (δT), which the eavesdropper can choose arbitrarily, and the average number of photons in the quantum condition. There is an optimal delay for the eavesdropper, which depends on the average number of photons in the state.

For a small average number of photons μ = 0.1, the value of δ ^OK _E , as can be seen from table 1, is δ ^OK _E = 0.565, which is only 0.065 higher than the probability of a simple guessing by the eavesdropper of a transmitted bit of information (recall that the probability of simple guessing is 0.5 and is itself disadvantageous for the eavesdropper). For what follows, it will be more convenient to use the minimum probability of error for the eavesdropper δ _E = 1-δ ^OK _E. This value has the following meaning.

Since at the receiving station at point B only those messages are left that gave an outcome in the desired time window, for the eavesdropper this means that the probability of an error with which he knows the bits in the received positions at the point B that passed the test is δ _E. Even with an average number of photons in a coherent state up to μ = 5, the error probability for the eavesdropper is 30%, which means that the eavesdropper knows the bits accepted by legitimate users in about 70% of the positions. With a long accepted sequence of N legitimate participants at points A and B, 0.3 N secret ones can be extracted.

With a large average number of photons, the value of δ ^OK _E tends to unity (respectively, the probability of error δ _E tends to 0, but never becomes zero). The optimal eavesdropper delay also tends to zero, but never becomes zero. Although the fraction of normalization p (δT), which accumulates in the time window for measuring δT, tends to zero (at μ = 100, while p (δT) = 0.04), nevertheless, this smallness is compensated by the large number of photons that can give a count in the window δT. For large μ >> 1, the classical situation takes place, two states of the electromagnetic field with non-overlapping frequency bands can be reliably identified instantly without delay.

The fact that it is possible to use coherent states with an average μ = 1-3, where the quantity δ _e still large, for experimental implementation means that do not require laser attenuation level to μ = 0.1-0.3, as is usually the case. With an average number of photons μ = 3, there are practically no empty packages when only the vacuum component is present in the channel, which allows increasing the key generation rate by 10 times (see table 1).

Table 1 μ mean pure photons p (δТ) the proportion of the normalization of the state in the delay window δT δ ^OK _{E is the} maximum probability for the eavesdropper to find out the state and pass the toast to a delay 0.1 0.25 0.565 0.3 0.25 0.569 1.0 0.25 0.587 3.0 0.23 0.642 5.0 0.2 0.691 10.0 0.14 0.771 20.0 0.11 0.847 50.0 0.06 0.918 100.0 0.04 0.952

The answer for the probability δ _E , essentially, does not depend on the initial width of the frequency band δk, but depends only on the dimensionless parameter δk δT. Therefore, for any frequency band, you can always select the required value of the time window into which the entire state of the photon field is "placed". The quantity δ _E depends only on the structure of states and is maximum for a pure state (not a statistical mixture of states with different numbers of photons) of a single-photon packet propagating at the speed of light.

table 2 μ average number of photons η optimal beam splitter maximum probability of undetectable state discrimination 0.1 0.0001 0.502 one 0.20 0.515 3.0 0.40 0.746 5.0 0.45 0.893 10.0 0.5 0.997

The state of a single-photon packet corresponds to the extremely quantum relativistic situation, and in this case the quantity δ _E is universal in nature and is, in a certain sense, a world constant, since nothing was used to normalize the amplitude of the quantum state and the fact of the propagation of the state at the speed of light .

The probability δ ^0K _E for an attack with splitting of the multiphoton state of laser radiation depending on the beam splitter η and the average number of photons μ is given in Table 2. Table 2 shows the maximum value of δ ^OK _E for optimal η.

For small average photon numbers μ = 0.1 for the eavesdropper, the probability of finding out the state and remaining undetected is close to 0.5 (with δ ^OK _E = 0.502, which only slightly exceeds the probability of the eavesdropper simply guessing the transmitted bit of information). This result is qualitatively understandable, since at μ = 0.1 (minus the probability of the vacuum component), there is mainly one photon in the channel. The probability of occurrence of two or more photons is significantly less. Therefore, after a beam splitter, a single-photon state can be registered either by an eavesdropper or by a legitimate user at a receiving station, but never by both.

For large occupation numbers μ> 10 photons, the state is close to the classical one; therefore, half of the state can be “split off” (η = 0.5), and the probability of simultaneous readings from the eavesdropper and the legitimate user at the receiving one will be close to unity.

The temporal evolution of states is illustrated in the space-time diagram of FIG. 2.

Figure 2 shows a spatio-temporal diagram explaining a method for encoding and transmitting cryptographic keys.

The process of propagation of quantum states with clock synchronization at the receiving and transmitting stations. Quantum states are prepared at the transmitting station (A), measurements are taken at the receiving station (B) at a specific time window.

In figure 2, the following notation:

18 - quantum photon state at transmitting station A;

19 is the length of the communication channel;

20 - evolution of a quantum state in space-time (state transfer from transmitting station A to receiving station B;

21 - quantum photon state at the receiving station;

22 is a time window of detection at the receiving station. Only photocounts in this time window contribute to the cryptographic key.

Figure 3 shows an experimental implementation of a method of encoding and transmitting cryptographic keys through an open space. Extended orthogonal quantum states, into which information about the transmitted keys is encoded, are formed by “cutting out” states of optical filters with narrow, non-overlapping frequency bands centered around wavelengths λ ₁ and λ ₂ .

In figure 3, the following notation:

23 - lasers generating coherent quantum states;

24 - an optical filter with a central wavelength λ ₁ and a frequency passband δk;

25 is an optical filter with a central wavelength λ ₂ and a frequency passband δk;

26 - optical prisms;

27 - optical photodetectors;

28 - distance between stations;

29 - direction of signal propagation.

The method is as follows (see figure 3). Preliminarily measure the distance 28 (see figure 3) between the transmitting and receiving stations and synchronize the clock.

Then, at the transmitting station, sequences of quantum photon states are converted into single or multiphoton orthogonal states and the time of their departure is measured.

At the transmitting station, two lasers 23 are shown (see FIG. 3). Turning on a laser with a carrier wavelength λ ₁ corresponds to transmitting a logical bit of 0 to the receiving station, respectively, turning on a laser with a carrier wavelength λ ₂ corresponds to transmitting a logical bit of 1.

At some point in time of the start of the protocol, one of the lasers is accidentally launched, its power is preliminarily fixed, and one- or multiphoton quantum states are obtained. The laser outputs an optical pulse with a broad spectrum δω ₁ and, respectively, a short time δt ~ 1 / δω ₁ $. The moment the laser is turned on is the time of departure of the states. Subsequently, the state departure time is reported via a public (open) classic communication channel to the receiving station. The clock synchronizes with an accuracy of ~ δt. Next, the broadband state is sent to the input of the filters 24, 25 (Fig.3). Each filter cuts out orthogonal (non-overlapping in frequency) states with a narrow frequency band δk from the wide spectrum of δω quantum photon states. The power at the laser output is controlled (the average number of photons μ is set), and thereby single- or multiphoton states are formed. Next, the states are sent to an optical prism 26 (Fig. 3), which provides a deviation of the photon states centered around different wavelengths, and then sent through the open space to the receiving station.

Cutting out a narrow frequency band with a width of δk leads to the formation of quantum states with a spatial extent cδТ ~ c / δk. The accuracy of the time of sending an extended quantum state to open space is known with high accuracy δt << cδT. Accordingly, the duration of the photon state with a narrow frequency spectrum δk is much longer than the time δt with which the clocks are synchronized at the transmitting and receiving stations. Using the known distance between the stations, the sending time, and the length of the photon states, the arrival time of the leading edge of the state at the receiving station is calculated. Based on the known length of the photon states and the arrival time of the leading edge of the photon state, the time window at the receiving station 22 (FIG. 2) is calculated, in which counts in photodetectors 27 (FIG. 3) from undetected states should take place.

At the receiving station, exactly the same optical filters 24, 25 (Fig. 3) as at the transmitting station pass photon states to the detectors 27 (Fig. 3) only with a narrow frequency band δk, respectively, with a spatial length of no less than cδТ ~ c / δk. Detectors are in standby mode. It is required that the time constant of the avalanche photodiode τ _D be much less than the duration of the state (τD << δТ). The user at the receiving station performs decoding, which boils down to the fact that he leaves only those parcels in which the response was in time window 22 (figure 2). Depending on which of the photodetectors worked, the user at the receiving station writes a logical 0 or 1 and generates a cryptographic key.

Filters 24, 25 (FIG. 3) at the receiving station in front of the detectors 27 (FIG. 3) are needed in order to prevent the eavesdropper from sending photonic states shorter in time than δT (respectively, photonic states with a wide frequency spectrum), which could compensate for the delay introduced by the eavesdropper when measuring an extended photon state of duration δT.

If the eavesdropper measures the transmitted photon states (respectively, logical bits 0 or 1), then it inevitably causes a delay in the photon states. Delayed photon states will result in photocounts outside the time window 22 (FIG. 2). According to the photocounts of the timeless window 22 (FIG. 2), an eavesdropper is detected at the receiving station. All parcels in which there were photocounts outside the time window 22 (Fig. 2) at the receiving station are discarded and not taken into account when generating the cryptographic key.

Claims (1)

A method of encoding and transmitting cryptographic keys, comprising generating a sequence of quantum photon states at a transmitting station for encoding cryptographic keys and transmitting a sequence of quantum photon states through open space to a receiving station, characterized in that the distance between the transmitting and receiving stations is preliminarily measured and the clock is synchronized on them , then at the transmitting station, sequences of quantum photon states are converted into one or many orthogonal orthogonal quantum states and measure the time of their sending, which is reported to the receiving station, and at the receiving station, they measure the time of reception of a sequence of single or multiphoton orthogonal quantum states, determine the delay by which they are decoded and the eavesdropper is detected.

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