CC BY
32
MODELLING OF PROCESS OF USE OF MEANS OF QUANTUM CRYPTOLOGY WITH A SATELLITE LIAISON CHANNEL
HARBARCHUK V.I., KUZMICH O.I._____________________
Ukraine
Introduction
The quantum cryptography is one of the newest directions in problems of protection of the information. As against other methods of cryptography, the quantum cryptography has reliable theoretical base as quantum mechanics. Though the same base have cryptoanalysts.
One of technical problems on introduction of systems of quantum cryptography is a problem of a communication facility. Here the fibre-optical cable can be used while only. But fibre-optical communication lines demand assembly connections.. And on these connections of fibre-optical lines because of the known physical reasons all positive effects of quantum cryptography are lost. Therefore researches as on overcoming problems of fibre-optical communication lines, so on search of other ways which will allow to do without fibre-optical lines are conducted.
It is obvious, that the ideal decision would be use of a wireless liaison channel for systems of quantum cryptography. The most suitable variant for this purpose is use of a satellite liaison channel.
Then the satellite would play a role of a retransmitter in system of quantum cryptography for a confidential liaison channel.
Researches spent by us on this problem and show results of computer modelling, that the decision of this task probably, but at a number of essential conditions.
In the report some approaches and methods of the decision of problems of quantum cryptography with the wireless communication line are stated.
Problematics
So, use wireless, first of all satellite liaison channels solves a problem of range and speed of transfer of the information. But here there are the problems connected to preservation of polarization, reduction of absorption and dispersion of information quantums [1].
Atmosphere - a strong absorber energetics particles [1-3].
Energy of quantums of an optical range is proportional with energy of connection of electrons in a atom. At collision of a photon with a molecule or atom depending on energy a photon probably the phenomenon photoeffect at which the photon is absorbed, letting out thus electron.
Therefore already at several collisions with atoms and molecules of an atmo sphere the probability of full loss energy a photon strongly grows. In this connection it is offered to choose overenergy particles with energy from 0,1 up to 5 MeV for minimization of absorption of information photons by an
atmosphere. Process not coherent or Compton dispersion at which the photon dissipates nuclear electron here takes place, losing some share of the energy, sufficient to transfer ^neKTpoH from the connected condition in untied. The photon can repeatedly scatter on particles of an atmosphere, thus continuing the way to the satellite [1,2].
At modelling phenomena of transfer of information photons of radiation with initial energy 5 MeV the method of Monte Carlo - a numerical method of the decision of tasks has been chosenby means of modelling random variables. Phenomena of movements of quantum in a direction from station of transfer on station of reception through an atmosphere it is composed from independent “histories” ofa particle. [5,6]. In area T of phase space of coordinates function of distribution of sources f (r, Q, E, t) is set. Modelling this distribution at the moment oftime to , we shall receive initial coordinates ro, Q.0, Eo,to quantum. Further the photon goes rectilinearly and in regular intervals before collision with electron of atom. The probability of collision on a wayd/ is equal E(r,E)dl. Size E(r, E) is defined under the formula
m
2(r,E) = ZP i(r)c i(E), (1)
i=1
Where p i (r) there is a density of element with number i, c i (E) - microscopic section of interaction for element with number i.
The distance l which the particle passes not cooperating with substance at the set direction of movement Q is distributed under the indicative law with parameter E(r, E):
H(l) = 1 - exp(-J E(ro + Ql, E)dl). (2)
The probability of dispersion is equal £ pacc/ £. Factor £ pacc - the microscopic section of dispersion is defined to similarly full section ofinteraction £. It represents the sum ofsections of every possible types of dispersion. For each concrete type of dispersion the density of probability is set to change directions of movement Q on Q' and energy E onE' -function g(Q, E E'), which refers to function of
dispersion. Modelling of this density enables to define a new direction of movement and new energy of a particle. Further the way gone by a particle according to (2), i. e. calculation of a history is again defined will consist of the same operations. It terminates in an output ofa particle from area. At dispersion the direction of movement of quantum and its energy changes. At absorption the particle is absorbed by an atmosphere and its history stops [6].
The developed computer program models a traj ectory of each information photon with registration him at station of reception that allows to lead the detailed analysis. Behind results of work of the program the small percent of particles gets on the block of reception of pulses, therefore the system of transfer requires system engineering devices of focussing of absent-minded pulses on the satellite. Energy photons on at output essentially decreases.
Method of realization of a satellite liaison channel for systems of quantum cryptography
The research project of system issues the ciphered messages with use of satellite lines of transfer is developed. Such system will consist of three main parts (fig. 1):
1. Ground station of transfer 1 (fig. 1);
2. Ground station of reception 3 (fig. 1);
R&I, 2003, Ns 3
144
3. A satellite - retransmitter 2 (fig. 1), consisting of the block of reception 4, the block of transfer 5, the block of storing of the information 13.
Transfer of the information is carried out in some stages.
At the first stage there is a transfer of a confidential key to use of the quantum-cryptographic channel. The basic advantage of the mentioned kind of the channel of transfer is an opportunity to define presence in the channel unauthorized the user as at attempt of measurement of polarizing conditions - data carriers malicious penetrator brings distortions in the channel which are found out by legal users.
So, the station oftransfer 1 (fig. 1) generates a confidential key with quantum data carriers and transfers it to the block of reception 4 satellites 2 (fig. 1). The satellite at the specified stage of transfer is the subscriber of reception.
The station of transfer 1 (fig. 1) will consist of the block of generation ofa key 9 (it is in details represented on fig. 4), the block of the analysis safety (fig. 3), coder 7 and the block of transfer code text 8.
In the block of generation of a confidential key (fig. 4) radiation from a source gets on the block offormation of weak pulses 21 where photon pulses are weakened. In the block of activation 23 at connection of the generator of random numbers 22 polarizing modulators are activated under the casual law, resulting polarization of each concrete photon to one of four kinds of polarization.
At data transmission polarization of photons is supervised. Polarization canbe orthogonal (horizontal orvertical), circular (left or right) and diagonal (450 or 1350).
Measurements are interpreted according to the binary circuit: diagonal polarization 450 or horizontal - 0, diagonal 1350 or vertical - 1. So with the help of polarizing modulators as which Pockels cells can act occurs formation of polarizing conditions of quantums with their subsequent registration and storing
on a memory 24. Thus for each quantum there is a storing bases of polarizing conditions for subsequent their comparison to bases of the transmitter (the block 25). Further the polarized pulses get on a set of filters with the purpose of their easing up to concentration ~1 photon on a pulse (the block ofa filtration 26). After that the generated polarized one-photon pulses of radiation, which each pulse is the carrier of information zero or unit are transferred to the satellite on a liaison channel (the block 27).
On the device 28 there is a preparation of possible bits of a key according to the generated polarizing conditions.
In the block of the analysis of safety 29 there is an analysis of a quantum liaison channel on presence in it of the nonauthorized subscriber by dialogue comparison of bases of reception and transfer without publicity of values of polarization and the subsequent analysis of mistakes. The mentioned block 29 which will be in details represented on fig. 3 is considered below.
If at an estimation the level of mistakes exceeds certain theoretically set factor it specifies presence in the channel malicious penetrator. In this case the result of transfer is nonacceptance of a key that demands its repeated generation until the quantity of mistakes becomes allowable.
If by transfer of a key intrusion was not revealed - the key is accepted for further use. In this case the confidential key and the confidential message is transferred on mn^parop 7 (fig. 1) for formation ciphertext. Further ciphertext by transfer (the block 8) through the usual classical channel gets on the satellite 2. As data carriers will act here electromagnetic waves. At the specified stage it is possible to guarantee maximal reliability of transfer as information leakage is not found out, a key absolutely random, its length unlimited (or it is equal to length of the message), and behind theorem Shennon [10] specifies it absolute stability ciphertext.
4 Units of a method of the mformatio n
10 The unit ofa method ofa key
11 The unit of decoding
12 The unit of a method of a
cipher text
2 Satellites
13 The unit of-storege of the information
9 The unit of generate! g ofa key
Station oftransfer
6 The unit of a safety
Outc o me
I
The key is not
adopted
I
The key is
16 The unit of a safety assessment outcome
14
Encrypt or
The key
The key is not
17 The unit of generati ngof a ksy
8 The unit of transfer of a cipher text
7 The unit of encoding
Input the secret report
1 5 The unit of transfer □fa cipher text
3 Sta ion of a method
18 Blocks of tile acceptance cipher text
19 Block of the receiving the key
20 The unit of decoding
Output
The secret report
Fig. 1. A General Arrangement
i\«i, zw:>, j
Momentums from station of transfer
System of reception on the satellite
At passage by photons of an atmosphere depending on quantity diffusings the condition of polarization of quantums can change. Besides owing to Kompton effect energy quantums will decrease, some photons will not get on the block of reception of the information 4 satellites (fig. 1) owing to change of directions of movement. Because of it it is necessary to inform on station of transfer on acceptance, or nonacceptance of a concrete information photon.
We shall consider the block of reception of a key on the satellite (fig. 2). In system of reception on the satellite photons at first get on the polarizing controller 31 for detection of a degree of change of polarizing conditions, further - on the device for restoration of initial polarizing conditions 32 as which the set of wave plates can act.
At passage of each wave plate the direction of polarization of a photon varies on 450. Further the signal acts on the block of a choice ofbases ofvertical / horizontal polarization 33 into which structure enters coupler by means of which the casual choice of basis of detecting is carried out. The choice ofbases is fixed for each photon in the block of storing of bases 34 then the given information acts on the block of the analysis of safety of transfer of a key 35 for the subsequent dialogue comparison to bases of transfer.
On the block 33 photons which pass through coupler will choose orthogonal basis and then will get on the analyzer 38. Polarization reflected in coupler photons turns a wave plate on 45 ° (with-45°up to 0°)in the block of change of polarization 3 7 with the purpose of change of basis with orthogonal on diagonal. After that photons are analyzed in the block by 38 second set from polarizing coupler and counters of photons. Thus there are measurements in diagonal basis. With the purpose of amplification of a signal in the block 3 9 photon gets on the photo multiplier then in the block 36 formation of possible bits of a key is carried out. Further preparatory bats
of a key are transferred to the block of the analysis of safety 3 5 for the dialogue analysis on presence of mistakes.
For an evident illustration we shall look after a way of the photon polarized on +45°. After it leaves the transmitter, its polarization by casual image will be transformed in an atmosphere. In system of reception the polarizing controller should be established so that to return polarization back to +45°. If the photon will choose an output coupler, corresponding to orthogonal basis of horizontal / vertical polarization, at it equal odds will get in one of detectors that will lead to casual result. On the other hand, if it he will choose diagonal basis, its polarization will be revolved on 90°. Then polarizing coupler will reflect it with individual probability that will lead to the certain result.
Instead of use of two polarizing coupler in blocks 33 and 38 by accepting party, applicationof active polarizing modulators, such as cells Pockels is possible also.
For each pulse of light the modulator becomes more active under the casual law in the casual order rotating polarization of half of accepted pulses on 45°.
At an analysis stage of polarizing conditions registration and storing of information bats as possible bits of a required confidential key and also storing ofbases of reception of each concrete pulse for the subsequent comparison to bases of the transmitter is carried out with the purpose of detection unauthorized user.
Formation of a confidential key
So, at the current stage information photons are transferred on the satellite and each subscriber owns some sequence of information bats each of which it will be possible bit a key.
The problem will consist in elimination of those bits for which the analyzer of reception station has chosen basis which does not coincide with basis of the transmitter as the choice of incorrect basis results in incorrect casual result.
146
R&I, 2003, N 3
For this purpose not disclosing value ofpolarizing conditions it is spent dialogue comparison of bases of reception and transfer on classically protected channel, having thus in view of an opportunity of information leakage which is useful for melicious penetrator only in case of it undetecting.
But in the quantum-cryptographic channel almost 100-percent probability of detection of the non-authorized user just takes place.
So the information on bases of transfer for him is small even if it to him will get
But at availability malicious penetrator to the classical channel of negotiations of the extremely important there will be other problem - an opportunity to deform dialogue comparison of bases that can lower essentially efficiency revealings of the malefactor. It, certainly demands sufficient security not only transfer of a confidential key, but also the channel of negotiations.
After comparison of bases in the block of dialogue 40 it is possible to form a primary crude key on the basis of polarizing conditions (the block 41) which bases of reception have coincided with bases of transfer. Values of bats of station of transfer at absence of interception will coincide with bats of station of reception. Presence malicious penetrator will specify essential handicapes in the channel, discrepancies of bats which were accepted in identical bases. (fig. 3)
Further having two sequences of bits at stations of reception and transfer it is required to execute check on identity. We carry out sorting bats behind beforehand known identical algorithm (the block 42).
Brought mistakes can be found out and eliminated with the help of calculation of parity, thus one of bats from each block is rejected (Bennet’s report). The sender and the addressee agree about any rearrangement bits in lines to make positions of mistakes casual. Lines share on blocks (the block 43) the
size k (k gets out so that the probability of a mistake in the block was small). For eachblock the sender and the addressee calculate and openly notify each other on the received results (the block 44). Last bits of each block leaves. The addressee and the sender make iterative search and correction for each block where parity appeared different, incorrect bits (the proof-reader of mistakes 45). To exclude multiple mistakes which can be not noticed, operations of items 1-4 repeat for great value k (the block of repeated checks 46). To define, have remained whether or not the undetected mistakes, the addressee and the sender repeat pseudo-casual checks (the amplifier ofprivacy 47). The addressee and the sender openly declare casual hashing positions of half of bits in their lines. The addressee and the sender openly compare parity. If lines differ, parity should not coincide with probability 1/2. If difference takes place, the addressee and the sender, uses binary search and removal incorrect bits. If differences are not present, after m iterations the addressee and the sender receive identical lines. At the given stage oftransfer information leakage as the estimation of mistakes is carried out in dialogue on the classical channel is possible.
The quantity of the found out mistakes is analyzed (the analyzer of mistakes 48) and compared to its theoretically allowable value (the analyzer 49). These are the mistakes caused insufficiency of work of the equipment and also incorrect detecting of polarizing conditions. This quantity is approximately equal 3-5 %. The quantity of mistakes at presence ofthe malefactor makes 25 %. On the basis ofthese data it is judged acceptance of a key to use (the Block of the decision on keys 50). If the decision positive the key is accepted to use, otherwise it is generated repeatedly.
Further it is necessary as reliably to transfer the message of station of reception on the Earth. For maintenance of stability ciphertext for Shennon [10] we do not use repeatedly confidential key, and we generate new absolutely casual confidential key equal to length of the message and we use
Bases from the
Bases from the repeater
To an encoder
Fig. 3. The Unit of a safety assessment
it one time. So on the satellite the message is necessary for deciphering completely.
We come back to consideration fig. 1. At the current stage on the block of reception of the information 4 satellites 2 confidential key is transferred on the safe quantum-cryptographic channel (the block of reception of a key 10). Ciphertext from the block of reception Ciphertext 12 and a key from the block 10 are transferred to the block of decoding 11 then the confidential message gets on the block of storing of the classified information 13.
At the following stage the satellite acts as the transmitter of the information (the block oftransfer 5, fig. 1). The confidential key between the satellite and terrestrial station N°2 - the final subscriber is at first generated. The block of generation 17, is in details considered above and submitted on fig. 4. Further the key is analyzed on presence of mistakes (the block of the analysis 16, is considered above, represented on rice.. 3), their quantity which is compared to theoretically allowable value then it is judged acceptance of a key for transfer of the message is estimated. If the key is not accepted, there is its repeated generation. If the key is accepted that it use for encoding the message (the block of enciphering 14) with the subsequent transfer mn^porpaMMbi through the classical channel (the block of transfer 15).
At ground station ofreception 3 (fig. 1) cipher text is accepted on the block of reception 18, then on the block of decoding together with generated on the block of reception by 19 key. The block of reception works similarly to the block of reception of a key 10 on the above described algorithm. In details the block is represented on fig. 2.
Conclusion
According to the developed algorithm of transfer number and the information on has each let out photon, whether it has reached the detector on the satellite is transferred the ground subscriber before release of the following photon. Only those photons which have reached the detector form a confidential key. Hence, we judge, that the atmosphere as the physical
channel of transfer in any measure does not deform a confidential key, and influences only quantity of transmitted information photons, so for speed of transfer of confidential messages. The choice as data carriers high energy quantums essentially raises probability of their registration at reception station. Technical problems is development of the corresponding equipment for functioning blocks. The real quantum cryptographic system demands also creation of the mechanism of active indemnification of polarizing changes. Though creations of such mechanism essentially probably, obviously, that its practical realization is rather complicated. Besides despite of active design development in a corresponding directionthe primary goal is designing sources of quantums, filters for allocation of single quantums, diodes. As has shown computer modelling process of passage of quantums through an atmosphere it is necessary to solve a problem and focussings of quantums in system of reception because of dispersion.
References: 1. ShpolskyE.V. Nuclear physics. Introduction in nuclear physics, that 1, “Science”, 1974. 575p. (in Russian). 2. Vavilov V.S. Action of radiations on semiconductors. M.Fizmatgiz, 1963.264 p. (in Russian). 3. Hajakava S. Physics of space beams. Astrophysical aspect // Moscow: World, 1974. Part2. 347p. (inRussian). 4. Turygin I A. Applied optics. M.: Mechanical engineering, 1966. 432p. (in Russian). 5. Ermakov S.M. The Method of Monte Carlo and adjacent questions. M.: Science, 1975. 474p. (in Russian). 6. The Decision of regional problemsby method of Monte Carlo. M.: Science, 1980. 178p. (in Russian). 7. Quantum calculations and cryptography in Los Alamos - http: // qso.lanl.gov/qc. 8. http: // www.irepolusgroup.com/ErbLasers/EFLGuide.htm.
9. Kadomtsev B.B. Dynamics and the information. M.: J. Successes of physical sciences. 1997. (in Russian). 10. Shennon K. Theory of communication in confidential systems // In : Shennon K.Raboty under the theory of the information and cybernetics. M: SILT, 1963. P.333-402. 11. Agrawal G.P. Fiber-Optic Communication Systems. John Wiley and Sons, 1997. P. 255-260. 12. Reference Manual, MillimeterResolution OTDR System, Opto-Electronics, Inc. Oakville, Canada. 1994.
13. Lomonaco S. A Quick Glance at Quantum Cryptography, quant-ph/9811056. 1998.
Input
Fig. 4. The unit of generating of a key
