=Paper=
{{Paper
|id=Vol-2500/paper_20
|storemode=property
|title=Development of Imitation-resistant Authentication Protocol for Low-orbital Space Satellite
Communication
System
|pdfUrl=https://ceur-ws.org/Vol-2500/paper_20.pdf
|volume=Vol-2500
|authors=Igor Kalmykov,Maria Lapina,Maxim Kalmykov,Igor Provornov,Evgeniy Voloshin
}}
==Development of Imitation-resistant Authentication Protocol for Low-orbital Space Satellite
Communication
System
==
https://ceur-ws.org/Vol-2500/paper_20.pdf
Development of Imitation-resistant Authentication
Protocol for Low-orbital Space Satellite Communication
System
Igor Kalmykov Maria Lapina Maxim Kalmykov
NorthCaucasus Federal NorthCaucasus Federal NorthCaucasus Federal
University University University
Stavropol, 355017 Stavropol, 355017 Stavropol, 355017
kia762@yandex.ru mlapina@ncfu.ru kim762@yandex.ru
Igor Provornov Evgeniy Voloshin
NorthCaucasus Federal NorthCaucasus Federal
University University
Stavropol, 355017 Stavropol, 355017
kia545@yandex.ru norra170@gmail.com
Abstract
In recent years, there has been a tendency to expand the use of low-
orbit satellite communication systems (LOSCS). A special role belongs
to the systems of remote monitoring, control and management of unat-
tended objects of environmentally hazardous technologies. To ensure
uninterrupted operation of the satellite communications system, a cer-
tain number of spacecrafts are combined into an orbital group. For
a low-orbit CAS, the group consists of 48-60 satellites. However, due
to the increase in the number of LOSCS, a situation may arise when
a ”foe” satellite gets in sight of a satellite communications receiver,
which is located at the subscriber terminal of an unattended facility,
attempts to impose a previously intercepted control command. This
can lead to failure of the control object and provoke an environmen-
tal disaster. In order to prevent such a situation, it is necessary to
increase the imitation resistance of the LOSCS. It is possible to solve
this problem by using the identification of the Identification-Friend-or-
Foe system (IFF system) of a spacecraft. Obviously, the effectiveness of
such a system is primarily determined by the authentication protocol.
Therefore, the goal of the research is to improve the imitability of the
LOSCS by a satellite identification system using the developed authen-
tication protocol, built on evidence with zero disclosure zeroknowledge
proof knowledge (ZKPK).
Keywords: satellite identification system, authentication protocols with
zero knowledge disclosure, algorithm of checking for session key reuse.
Copyright 2019 for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
In: S. Hölldobler, A. Malikov (eds.): Proceedings of the YSIP-3 Workshop, Stavropol and Arkhyz, Russian Federation,
17-09-2019–20-09-2019, published at http://ceur-ws.org
1
Introduction
Providing the communication services for global projects like the development of the Northern Sea Route, the
creation of information and telemetric systems for air and land transport in high latitudes is impossible without
the use of low-orbit satellite communication systems (LOSCS). Satellite communication systems such as Iridium
and Iridium NEXT are now widely used to solve these tasks [Iri18], [Iri18], [Wha18].
One of the most promising areas of application of the LOSCS is the exploitation of mineral resources in
the regions of the Far North. In this case, LOSCS are an important part of automated management, remote
monitoring and control systems, which are used to manage the maintenance-free hydrocarbon production and
transportation facilities located beyond the Polar Circle. To organize uninterrupted communications, the NSSS
group should contain 48 to 60 spacecraft. However, the increase in the number of countries participating in the
development of the natural resources of the Arctic, as well as the large spatial extent of communication lines
leads to an increase in the number of satellite constellations. Because of this, a situation may arise when a
satellite of the intruder may be in the visibility zone of the receiver, which may disrupt the operation of the
LOSCS. This can lead to the failure of maintenance-free control facility and provoke an environmental disaster.
It is possible to solve this problem by increasing the imitation resistance of the LOSCS. To counteract the
imposition of an intercepted command, it is advisable to use a satellite identification system (SIS). For efficient
work of the inquiry-response identification system friend-foe, it is necessary, foremost, to use an imitation-
resistant authentication protocol, and, secondly, encrypting algorithms should not be used when checking satellite
status. This problem can be solved only with the help of request-response type protocols, built on evidence with
zero disclosure of zero-knowledge proof knowledge (ZKPK) [Feg03], [Pas18], [Sta99], [Smi02]. Therefore, the
development of an imitation-resistant authentication protocol with zero disclosure and the minimum time spent
on checking the status of a satellite is a topical task.
1 Material and methods of research
1.1 Destructive effects on the low-orbit satellite communication system
An analysis of the following works was carried out to develop the most effective method of countering the
destructive effects of the intruder satellite [Mcd17], [Poi12], [Spe02], which allowed to discern three groups of
such effects. The basis of the first class of effects on the communication system consists of various methods
of electronic signal suppression. The main goal of the electronic signal suppression methods is blocking the
transmitted signal from the spacecraft to the control object and back. Usually, active or passive interference is
used for this, among which there are:
– harmonic continuous interference, which is determined by equation:
UGN P (t) = Umm cos(ωp2 t + φp2 (t)), (1)
where ωp2 ∈ [ω0 −π∆f2 ; ω0 +π∆f2 ] - angular interference frequency; Umm - amplitude of harmonic continuous
interference; φp2 - initial phase of harmonic continuous interference.
– quasi-white noise-like interference determined by
USHP (t) = Umm cos(ωp1 t + φp1 (t)), (2)
where Umm (t) - alteration in enveloping noise-like interference; φp1 (t) - phase change in noise-like interfer-
ence; ωp1 - average interference frequency; ω1 ≈ ω0 ; ω0 = 2πL; L - carrier wave frequency; - active amplitude
modulated noise interference
U (t) = UP [1 + Kα ∆UM OD (t)], (3)
where Kα - slope of modulation transmitter characteristic; ∆UM OD (t) - modeling direction coming from
the noise generator.
The basis of the second group is simulated interference. Such interference is called intelligent interference, as it
is able to adapt to the transmitted signal, thereby disrupting the effective operation of the radio communication
system. The most widespread are:
2
– targeted simulating disturbance, which is determined by the equation:
UP IP (t) = KUm Qι (t − td − ∆τ )sin[2π(L ± ∆f )(t − td − ∆τ ) + φ], (4)
where K - coefficient accounting the targeted simulating disturbance;
– tracking imitation interference,
USIM (t) = KUm Q(t − td − τ (t))sin[2π(L ± ∆f )(t − td − τ (t)) + φ], (5)
where τ (t) = r(t)/c - distance from satellite to the station.
A special place among the destructive effects on the satellite communication system is occupied by the relay
interference. In this case, the intruder satellite intercepts the control command, delays it, and then sends it.
Then the receiver located on the control object perceives the received signal as its own and transmits a command
to the control system of a maintenance-free object, which can lead to disruption and breakdown.
Studies have shown that in the conditions of the Far North, the method of setting relay interference is the
most effective method, while the use of active, passive and imitating interference is a difficult task. Therefore,
this paper will propose methods for countering relay interference.
In order to forbid the intruder satellite imposing an intercepted and delayed command on the subscriber
station, it is necessary to prevent data exchange between such a spacecraft and the receiver located at the
control object. To do this, it is advisable to determine the satellite status before starting a communication
session. Because of the use of the friend or foe identification, a satellite that fails authentication will not be able
to communicate with the receiver of the subscriber terminal of the remote control object. Currently, there are
many ”friend or foe” identification systems, which are widely used in many countries. The analysis of the basic
principles of building data of the friend or foe identification showed that they are unable to authenticate the
satellite and cannot be used in the LOSCS.
1.2 Authentication protocols
This problem can be solved by developing a new method for constructing the friend-foe identification system,
which would allow to authenticate the LOSCS satellite using a strong cryptographic protocol. Currently, cryp-
tographic authentication protocols can be divided into three groups. The first group is based on password
authentication protocols [Sta99], [Shr96], [Feg10], [Smi02].
Authentication protocols that make up the second group have higher cryptographic security. Such protocols
use a request-response method. As these works show, [Sta99], [Feg10], [Smi02] it is proposed to use both
symmetric and asymmetric cryptographic systems to increase the strength of such protocols. It should be noted
that the following this condition for a group of spacecrafts LOSCS is rather difficult. This is because not only
satellites, but the unattended control objects must have the secret keys.
Authentication protocols with zero knowledge proof lack this flaw. They make up the third group. These
works [Sha03], [Feg10] examine the Fiat-Shamir protocol.
In order to ensure the required level of probability of noticing an intruder, the authentication procedure is
performed repeatedly, where W = 20-40 rounds.
The Schnorr protocol, which is presented in these works, allows to reduce the time spent on authentication,
[Sch96], [Fer03]. Although this protocol allows one round authentication, it nevertheless has drawbacks:
– three data exchanges are required between applicant P and verifier V for authentication;
– periodically changing session keys Sj, j = 1, 2, ... are not used.
The developed authentication protocol built on evidence with zero knowledge disclosure and minimum number
of identification steps allows to eliminate these drawbacks [Gos15]. This protocol consists of the following steps:
At the preliminary stage of the protocol, the irreducible polynomial p(x) and the value of the secret key and
the random number S are chosen. The value of the secret key and S are used to calculate the session keys Sj ,
where j = 1, 2, ..., which satisfy the condition
K sek ≤ 2degp(x) − 1. (6)
3
S ≤ 2degp(x) − 1. (7)
where degp(x) - is the degree of p(x) polynomial.
The operation of the authentication protocol involves the transponder, which resides on board the satellite
and the interrogator that resides at the control site. First, the transponder, upon receiving the value Sj K sek of
the session key, calculates the true status of the satellite
sek
Mj (x) = X Sj X K modp(x) (8)
sek −1
where Sj (x) = x(Sj −1+K ) modp(x)- value of the j-th session key.
If during the calculation of the session key Sj the following condition is true,
Sj−1 + K sek = 0mod2degp(x)−1 , (9)
then this value is replaced by 2degp(x)−1 − 1.
The next step is to conduct the noise interference of the secret key values and Sj. To do this, the values that
change during each session are used. As a result, we get the following expressions
K̃jsek = (K sek + ∆K sek )mod2degp(x) − 1, (10)
S̃j = (Sj + ∆Sj )mod2degp(x) − 1, (11)
where K̃jsek , S̃j are noise-modified values.
Then the noise-modified satellite image will be determined based on the expression
sek
M̃j (x) = xS̃j xK̃ modp(x) (12)
True and noise-modified satellite images will be used to verify its authenticity. To perform such authentication,
the interrogator sends a question, which is a random number.
Upon receiving the dj query, the transponder must answer the question
rj (1) = (K̃jsek − dj K sek )mod2degp(x) − 1, (13)
rj (2) = (S̃j − dj Sj )mod2degp(x) − 1, (14)
The transponder sends (Mj (sek), M̃j (sek), rj (1), rj (2)) to the interrogator.
To verify the correctness of the received answers, the verifier V uses an expression in which the true Mj (sek),
noise-modified M̃j (sek) images of the satellite, two answers rj (1) and rj (2), and the question dj must be included.
The following expression is used to check the received answers:
Bj (x) = Mj (x)dj X rj (1) X rj (2) modp(x). (15)
If the Bj (x) = Mj (x) condidtion is true, then the satellite is assigned the status of ”friend”. Otherwise,
satellite status is foe.
Analysis of the developed authentication protocol indicated that it can conduct satellite identification at a
higher speed, since the authentication process consists of two stages. To assess the effectiveness of the developed
authentication protocol, a comparative analysis was conducted with the Fiat-Shamir and Schnorr protocols. The
analysis showed that the developed protocol allows the authentication procedure to be performed in two stages,
which is 30 times faster than the Fiat-Shamir protocol and 1.5 times faster than the Schnorr protocol.
It is obvious that the imitability of this authentication protocol will be determined by the session keys Sj ,
where j = 1, 2, ... If during the operation of the transponder the value of the session key does not change, it will
result in the signals transmitted to the interrogator during the j − thand(j + 1) − th session to overlap, since
Cj (x) = Cj+1 (x). In this case, the length of the L-bit response transmitted to the interrogator was reduced by
the degree of the selected polynomial p(x). This will lead to an increase in the probability of the answer being
guessed by the foe satellite, since
4
1 1
P = L
< P ∗ = L−degp(x) (16)
2 2
where p∗ - probability of guessing the answer when the session key is reused.
This means that double use of the session key reduces the protocol’s imitation resistance. In [Lap18] an
algorithm that allows to check the correctness of the generation of Sj and the additional parameter Tj is presented.
To do this, the verifying party sends the satellite a random query number r. After receiving the question r, the
spacecraft calculates the answers.
a∗j (S) = (aj (S) − r)modq, a∗j (T ) = (aj (T ) − r)modq, (17)
1 1
where aj (T ) = Πm m
ι=1 Tj +Kj modq; aj (S) = Πι=1 Sι +Kι modq.
The blurred values are then calculated.
∗ ∗
Sj∗ = g aj (S) modq, Tj∗ g aj (T ) modq. (18)
The spacecraft finds the product of the true values of Sj and Tj, as well as the blurred parameters. The results
are sent to the verifying party V, which checks the obtained values.
Sj Tj
A= = g 2r modq. (19)
Sj∗ Tj∗
If the calculated value, according to (19), satisfies
A‘ = (g r )2 modq = A, (20)
this suggests that the values of Sj and the corresponding parameter Tj are generated correctly. However, this
algorithm does not allow to determine the reuse of Sj . The developed algorithm for the dual session key reuse
check allows to eliminate this drawback.
The satellite and the operation support center (OSC), which controls the operation of the automated facility
monitoring system, are involved in the verification. In the developed protocol, an additional parameter Tj is
introduced, with which it would be possible to verify if the session key was reused
sek −1
Sj (x) = x(Sj−1 +K )
modp(x). (21)
sek
+T )−1
Tj (x) = x(Sj−1 +K modp(x). (22)
where S0 = S; j = 1, 2, ... If during the calculation of the session key Sj the condition is true, (23)
(Sj−1 + K sek + Tj−1 ) = 0mod2degp(x)−1 , (23)
degp(x)−1
then this value is replaced by 2 .
In the developed algorithm, Tj is used to calculate the test parameter Ej , with which the satellites public key
will be obtained when the condition Sj = Sj + 1 is true. In the course of the research, an equation was chosen
to determine
Y
Ej = K pub Tj j modp(x). (24)
where Yj - query number, that is set by OSC on j-th session; Yj < 2degp(x)−1 .
The developed algorithm for checking the reuse of the session key in the satellite identification system consists
of the following steps.
1 The transponder calculates the values of the session key Sj and Tj.
2 At the j-th session, the center makes a request for which a random number is used.
3 The trasponder, upon receiving this request, calculates the answer (24)
4 Ej , Yj are transmitted to the center.
5
5 In the next session, the responder calculates the values of the session key Sj+1 and Tj+1 .
6 At the j + 1-st session, the center makes a request for which a random number is used.
7 The trasponder, upon receiving this request, calculates the answer
Y
Ej+1 = K pub Tj+1
j+1
modp(x). (25)
8 Ej+1 , Yj+1 are transmitted to the center.
9 The center performs a session key reuse check in the spacecraft identification system
(Y(j+1) −Y(j) )−1 +
(Ej )Y(j+1)
W = (26)
(Ej+1 )Y(j)
2degp(x) −1
If the public key of the K pub spacecraft is obtained, this indicates that the satellite reused the session key Sj .
Consider the situation when the operation of the pseudo-random function generator that calculates session
keys Sj was disrupted. In this case, the values of the neighboring session keys Sj and Sj+1 will match
sek −1 sek −1
Sj (x) = x(Sj−1 +K )
modp(x) = x(Sj +K )
modp(x) = Sj+1 (x). (27)
Suppose we have the following equation Sj = Sj+1 = S. Parameters Tj andTj+1 are used in the algorithm for
checking the reuse of the session key
sek
+T )−1 sek
+T )−1
Tj (x) = x(Sj−1 +K modp(x) = x(Sj +K modp(x) = Tj+1 (x). (28)
Then on receiving Yj+1 < 2degp(x) − 1 query, the transponder sends to the center
Y
Ej = K pub Tj j mod2degp(x) − 1. (29)
And on receiving Yj+1 < sdegp(x) − 1 query, center gets the response:
Y
Ej+1 = K pub Tj j+1 mod2degp(x) − 1. (30)
Then the center gets the equation, where q = 2degp(x) − 1.
(Yj −Yj+1 )−1 + Y
!(Yj −Yj+1 )−1 +
Yj (K pub Tj j+1 )Yj
(Ej+1 )
W = = Y
= K pub (31)
(Ej )Yj+1 (K pub Tj j )Yj+1
q q
The calculated value of the Kpub satellite public key allows to determine the corresponding satellite and
restart the session key generator.
It is obvious that the use of the developed algorithm to verify the reuse of the session key will improve the
imitation resistance of the satellite communication system. Therefore, a modification of the developed protocol
was carried out. As a result, it consists of the following steps.
2 Preliminary stage of the protocol
For the operation of the satellite identification system built on the basis of the authentication protocol with zero
disclosure, an irreducible polynomial p(x) with a large degree degp(x) is chosen. The secret key
K sek < degp(x) − 1. (32)
To obtain the j-th session key Sj , where j = 1, 2, ..., a random number S that satisfies
S < degp(x) − 1. (33)
In order to verify the dual use of the session key, a random number T is chosen from the equation
6
T < degp(x) − 1. (34)
The selected parameters are stored in the satellite memory..
The working stage of the authentication protocol.
Stage 1. The transponder on the satellite board calculates the session key Sj (x) and the parameter Tj (x).
Stage 2. The transponder calculates the true status of the satellite
sek
Mj (x) = xSj xK xTj modp(x). (35)
Stage 3. The transponder produces noise-modified parameters using random variables ∆K̃jsek , ∆S̃j , ∆T̃j .
K̃jsek = (K sek + ∆K sek )mod2degp(x) − 1, (36)
S̃j = (Sj + ∆Sj ))mod2degp(x) − 1, (37)
T̃j = (Tj + ∆Tj )mod2degp(x) − 1, (38)
Stage 4. The transponder calculates the noise-modified satellite status
sek
M̃j (x) = xS̃j xK̃ xT̃j modp(x). (39)
Authentication process.
Stage 1. The transponder chooses a question number dj < 2degp(x)−1 , which it sends to the transponder.
Stage 2. The transponder calculates the answers to the query after receiving the number dj :
rj (1) = (K̃jsek + dj K sek )mod2degp(x) − 1, (40)
rj (2) = (S̃j + dj Sj )mod2degp(x) − 1, (41)
rj (3) = (T̃j + dj Tj )mod2degp(x) − 1, (42)
Transponder sends (Mj (x), M̃j (x), rj (1), rj (2), rj (3)). to transponder.
The transponder verifies the correctness of the response:
Bj (x) = Mj (x)dj xrj (1) xrj (2) xrj (3) modp(x). (43)
If the equation Bj (x) = M̃j (x) is true, then the satellite gets the friend status.
3 Results and Discussion
Consider the work of the developed authentication protocol. Let an irreducible polynomial be given:p(x) =
x5 + x2 + 1. Then the parameters are chosen: K sek = 14, S = 14, T = 18. Let us assume j = 1.
Step 1. The transponder calculates the session key S1 (x) and T1 (x), where S0 = S; j = 1.
+ 1 + +
sek −1 1 +
S1 (x) = x(S1−1 +K )
= x S0 +K sek = x 14+14 = x10 x5 +x2 +1 = x4 + 1 = 10001.
P (x) x5 +x2 +1 x5 +x2 +1
+ + + +
sek
+T )−1 −1 −1
T1 (x) = x(S0 +K = x(14+14+18) = x(15) = x29 x5 +x2 +1 = x3 + 1 = 01001.
P (x) x5 +x2 +1 x5 +x2 +1
Step 2. The transponder calculates the true satellite status according to (8).
sek
M̃j (x) = xS̃j xK̃ modp(x) = x17 x14 x9 modx5 + x2 + 1 = x4 + x3 + x = 11010
7
Step 3. The transponder calculates the encrypted secret parameters. K̃1sek , S̃1 , T̃1 . If {∆K̃1sek = 4, ∆S̃1 =
10, ∆T̃1 = 2} < 25 − 1. Then,
K̃1sek = (K sek + ∆K1sek )mod25 − 1 = 18, S̃1 = (S1 + ∆S1 ))mod25 − 1 = 27,
T̃1 = (T1 + ∆T1 )mod25 − 1 = 11.
Step 4. The transponder calculates the encrypted status of the satellite
sek +
M̃j (x) = xS̃j xK̃j xT̃j = x27 + x18 + x11 modx5 + x2 + 1 = x25 = x4 + x3 + 1 = 11001
p(x)
4 Authentication process
Step 1. The interrogator chose a d1 = 11 number and sent it to the transponder.
Step 2. The respondent, upon receiving d1 = 11, calculates the answers to the question (13) - (15)
+ =
r1 (1) = K̃1sek + d1 K sek = 19, r1 (2) = S̃1 + d1 S1 26,
31 31
+
+
r1 (3) = T̃1 + d1 T1 = |11 − 11 · 9|31 = 5
31
Transponder responds to interrogator:
(M1 (x), M̃1 (x), r1 (1), r1 (2), r1 (3)) = (11010, 11001, 10011, 11010, 00101).
Satellite status is checked. Transponder calculates the following:
+
B1 (x) = M1 (x)d1 xr1 (1) xr1 (2) xr1 (3) = (x4 + x3 + x)11 x19 x26 x5 x5 +x2 +1 = x4 + x3 + 1.
p(x)
Since the B1 (x) = M̃1 (x) condition is met, then the satellite is assigned the status of ”friend”. To evaluate the
imitation resistance of the developed authentication protocol, the Matlab R2017b application software package
was used. As a criterion for assessing the level of imitation resistance, the probability of a satellites omission by
the identification system was chosen. The probability of missing is determined according to the equation:
N (ι)
PP C = PP O (ι), (44)
Nmax
where PP O (ι) = 1/2Lι - probability of selecting the answer; N (ι), - the number of identification steps in the
ι − th protocol; N (max) = 60 - the maximum number of steps in the protocol; Lι - the number of bits in the
answer to the question.
Figure 1 shows the dependence of the probability of a satellites omission by the identification system on the
bit depth of the answer to the question posed.
8
Figure 1 - Dependence of the probability of satellite omission from bit depth of the answer to the question:
1- Fiat-Shamir protocol is used;
2- the developed protocol is used.
Analysis of the graph shows that with a bit depth of L = 72 bits, the probability of a satellite passing by an
identification system based on the Fiat-Shamir protocol will be PP S (1) = 2.1 · 10−22 . Whereas, when using the
developed protocol, defined by expressions (6) - (15), the probability of a satellite passing by the identification
system will be PP S (2) = 6.7 · 10−24 . Thus, the use of the developed protocol makes it possible to increase
the imitation resistance of a satellite communication system by 3.19 · 102 in comparison with the Fiat-Shamir
protocol.
Consider an example of applying a session key reuse check algorithm. We use the data given in the previous
sek + +
example. Then the public key will be equal to K pub = xK = x14 x5 +x2 +1 = x4 + x3 + x2 + 1 = 11101.
p(x)
During the first communication session, the following parameters were obtained. S1 (x) = x4 +1 = 10001, T1 (x) =
x3 + 1 = 01001.
Let the satellite receive a Y1 = 5 question from the OSC. Then, using expressions (18), the test parameter is
E1 = k pub T1Y1 modp(x) = (x4 + x3 + x2 + 1)(x3 + 1)5 x5 +x2 +1 = x4 = 10000..
The calculated value of E1 is transmitted to the OSC. During the second communication session,
+ + + +
sek
+T )−1 −1 −1
T2 (x) = x(S1 +K = x(17+14+18) = x(18) = x19 x5 +x2 +1 = x2 + x = 00110.
P (x) x5 +x2 +1 x5 +x2 +1
In the second session the satellite received an Y2 = 17 question from OSC. Then
E2 = k pub T2Y2 modp(x) = (x4 + x3 + x2 + 1)(x2 + x)17 x3 +x+1 = x4 = 01011.
The calculated value of E2 is transmitted to the OSC, which checks the answers according to (21)
(Y2 −Y1 )−1 + (Y2 −Y1 )−1 +
(E1 )Y2 (x4 )17
W = = = x4 + x2 + x.
(E1 )Y1 (x3 + x + 1)5
p(x) x5 +x2 +1
Since the calculated value does not match the public key of the satellite, this means that the session keys
change in a timely manner.
9
5 Conclusion
The article presents an imitation-resistant authentication protocol based on proof with zero knowledge disclosure,
which allows to determine the status of a spacecraft with minimal time costs. A comparative analysis showed
that with a response depth of L = 72 bits, the probability of a satellite passing by an identification system based
on the Fiat-Shamir protocol will be PP S (1) = 2.1 · 10−22 , and using the developed protocol, the probability of a
satellite passing by an identification system will be PP S (2) = 6.7 · 10−24 . Thus, the use of the developed protocol
makes it possible to increase the simulated resistance of a satellite communication system in comparison with
the Fiat-Shamir protocol.
6 Acknowledgments
This work was supported by the Russian Foundation for Basic Research, project No. 18-07-01020.
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