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NEW CLASS OF BINARY PSEUDO-RANDOM SEQUENCES WITH A NONLINEAR GENERATION ALGORITHM FOR COMMUNICATION SYSTEMS WITH CDMA
Evgeny A. Golubev,
MTUCI, Moscow, Russia
Evgeny M. Lobov,
MTUCI, Moscow, Russia, lobov@srd.mtuci.ru Nikolay A. Kandaurov,
MTUCI, Moscow, Russia, kandaurov@srd.mtuci.ru Dmitriy N. Shubin,
MTUCI, Moscow, Russia, steputin@l234G.ru
DOI 10.24411/2072-8735-2018-10035
Keywords: pseudorandom binary sequence, Gold, de Bruijn, CDMA.
The article deals with the problem of developing a new class of binary pseudo-random sequences with a nonlinear generation algorithm for use in communication systems with Code Division Multiple Access (CDMA). An analysis of existing ensembles and an assessment of their main characteristics are carried out. Namely, the size of the ensemble, which influences the cryptography of the communication system, and the level of maximum emissions of the correlation function. Existing ensembles of binary pseudo-random sequences do not simultaneously possess satisfactory values of the considered indicators. Therefore, the actual task is to develop an ensemble that finds a compromise between these two characteristics and has a nonlinear generation algorithm. The essence of the algorithm for the formation of the proposed new class of binary pseudo-random sequences consists in modulo addition of linear sequences, for example, Gold or Kasami, with good correlation properties, and de Bruijn sequences having a large ensemble and a nonlinear algorithm of formation. Comparative analysis of the proposed method of forming a class of PN- sequences shows the advantage of its use. Improvement of the correlation characteristics of new sequences in comparison with de Bruijn sequences and a significant increase in size in comparison with the Gold and Kasami sequences. These properties can be used to increase crypto protection and anonymity of transmission in high-speed wireless information networks based on CDMA technology in comparison with known methods of multichannel information transmission.
Information about authors:
Evgeny A. Golubev, Doct. Tech. Sciences, professor, chief researcher at SRD MTUCI, Moscow, Russia Evgeny M. Lobov, Cand. Tech. Sciences, senior researcher at SRD MTUCI, Moscow, Russia Nikolay A. Kandaurov, postgraduate student MTUCI, minor researcher at SRD MTUCI, Moscow, Russia Dmitriy N. Shubin, SRD MTUCI, Moscow, Russia
Для цитирования:
Голубев Е.А., Лобов Е.М., Кандауров Н.А., Шубин Д.Н. Новый класс двоичных псевдослучайных последовательностей с нелинейным алгоритмом формирования для систем связи с кодовым разделением абонентов // T-Comm: Телекоммуникации и транспорт. 2018. Том 12. №2. С. 76-80.
For citation:
Golubev E.A., Lobov E.M., Kandaurov N.A., Shubin D.N. (2018). New class of binary pseudo-random sequences with a nonlinear generation algorithm for communication systems with CDMA. T-Comm, vol. 12, no.2, pр. 76-80.
Introduction
In modem telecommunication systems, such as mobile communication networks, wireless computer networks, satellite communication lines, etc., code division multiple access (CDMA) technology has been widely used. This technology is based on expanding the spectrum of subscriber signals using pseudo-random binary sequences (PRBS) with their further separation on the receiving side. Separation of subscriber signals in CDMA communication systems is made possible by assigning each subscriber its own pseudo-random binary sequences. Subscriber signals generated on the basis of the pseudo-random binary sequences should have a large central peak of the autocorrelation function (ACF) as compared to the side lobes, as well as the minimum emissions of the cross-correlation function (CCF). An important property of modern telecommunication systems is the ability to provide cryptographic protection and confidentiality of data transmission. These properties directly depend on the properties of the PRBS and the algorithm of their formation. To increase these indicators, it is required to use large ensembles of PRBS with a complex generation algorithm. There are many different pscudo-random binary sequences used for spectrum expansion. For example, M-sequences, Walsh, Gold, Kasami, de Brain and others.
PRBS is divided according to the generation algorithm, namely, there are linear algorithms for the formation of PRBS, in which linear operations are used to obtain PRBS elements (linear PRBS). Examples of linear sequences are M-sequences, Golda, Kasami, etc. The advantage of the linear algorithm of formation is its simplicity, which depends little on the length ofthe PRBS. An essential disadvantage ofthe linear algorithm of formation is the relative simplicity of its "decoding" from the limited realization of the signal. This allows to restore the whole structure of linear PRBS, observing only a small part of it, which affects the cryptoprotection ofthe system. A significant complication ofthe generation algorithm can be achieved by introducing nonlinear operations into it. Such algorithms have a significant advantage in their resistance to their disclosure in comparison with algorithms that have only linear operations. PRBS generated by nonlinear algorithms are called nonlinear PRBS (NPRBS). An example of an NPRBS is the PRBS de Bruijn, which have a significantly larger ensemble than the known linear PRBS, hut at the same time, the non-uniformity cifthe correlation characteristics within the ensemble.
Thus, an important task is to develop ensembles of pseudorandom sequences that have a nonlinear algorithm of formation and a large ensemble, but at the same time acceptable emissions of cross-correlations function (CCF).
Analysis of existing ensembles
Gold's sequences are characterized by a large ensemble size and homogeneous and limited values of the cross-correlation function. Any Gold sequence is formed by modulo 2 addition of two M-sequences that form the preferred pair [Ij. In the Gold set, there are only 2" +1 sequences including the two original M-sequences, and n is the length ofthe generating shift register. A large set of Kasami sequences also includes a small Kasami set. They differ in the size ofthe ensemble and in the values of the lateral peaks of the correlation functions. The sequences in these sets have length 2" -1. The number of sequences of a small Kasami set is equal to 2"n. The side ejections of the PACF
and the maximum values of the CCF Kasami sequences of the small set belong to the three-level class and assume the values {-1,±2""} [2]. The size of an ensemble of sequences of a large
Kasami set for n mod 4 = 2 is equal to 2:(2n + l) and for n mod
4 = 0 is equal to| +2= (1 + 1(2"-I))- The lateral emissions ofthe
PACF and the maximum values of the CCF sequences of the Kasami of the lanie set are five-level ones and can take on the
values
■1 in i г., 1 .,,
■ -I,(-2- -1),(2- -1),(22-1),(-2г -1) И*
Pseudo-random de Bruijn sequences are sequences of maximum length with a nonlinear formation algorithm [4]. The length ofthe de Bruijn sequence is 2", where 11 is an integer. A distinctive feature of de Brain's sequences is the large size of the ensemble. De Bruijn sequences arc also known as de Bruijn eyclcs. One way of forming De Bruijn cycles is to use the de Bruijn graph. The cycles of de Bruijn are named after the Dutch mathematician Nicholas de Bruijn, who examined them in 1946 [5], although they were studied earlier [6].
The number of de Bruijn sequences of a given length
k"
where k is the base of the number system; n - logk(N), N - length sequences.
For binary de Brain sequences k = 2, we obtain
Q=2
(1)
Table 1
Dimensions ofthe Gold and dc Brain ensembles with different length ofthe PRBS
Length PRBS (Gold / de Bruijn) Size of an ensemble Gold, О Size of an ensemble small Kasami Size of an ensemble big Kasami Size of an ensemble де dc Bruijn. Q
31/32 33 2" = 2048
63/64 65 8 520 2"" = 67108864
127/128 129 2"
511/512 513 2rîH"
1023/1024 1025 32 32800 2Ш
2047/2048 2049 2 502
As can be seen from Table I, the de Bruijn sequences greatly exceed the volume oflinear PRBS.
The maximum emission values of the Gold CCF sequences (length 63) and de Bruijn sequences (length 64) are given in Table 2. In studying the cross- correlation properties of de Bruijn sequences, pairs of sequences within an ensemble of sequences ofthe same length were considered. The values of CCF pairs of sequences and their number are given in Table 3. The ensemble of sequences of length 64 contains 67108864 sequences, as a result of the CCF study it was obtained ( = 210" ) pairs of sequences and their values of the maximum emission of CCF.
7ТЛ
Table 5
The maximum emissions CCF of PRBS obtained as a resuit of XOR of the Gold PRBS without a parity bit and a truncated PRBS de Brnijn depending on the PRBS de Bruijn index ( A' = Q— 1 max index PRBS de Bruijn,
Q-size of the ensemble de Bruijn (1) )
l.ength PRBS Index by PRBS de Bruijrt
0 1 10' N -1Û5 max N -1 max AU
31 0,612903 0,741935 - - 0,612903 0,483871
63 0.492063 0,555556 0,428571 0,428571 0,492063 0,428571
127 0,370079 0,370079 0,338583 0,370079 0,370079 0,370079
511 0,232877 0,225049 0,201566 0,201566 0,201566 0,201566
1023 0,155425 0,155425 0,155425 0.155425 0,139785 0,143695
2047 OJ 1871 0.120664 0.114802 0,105032 0,106986 0.106986
From the results of the study of the correlation function presented in the tables above, it can be concluded that for the expansion of an individual Gold PRBS ensemble, it is possible to use all available de Bruijn PRBS. But in order to ensure the satisfactory correlation properties resulting from the addition of the Gold PRBS XOR the PRBS de Bruijn, it is necessary to select the PRBS de Bruijn with satisfactory correlation properties,
nepHoatwecKaa BsaKMOKoppejwurtONHaft (tvuKuufi
0 8
-0.4 -0.6
.1---i---i---i---1---1---1---1---1---1---iJ
100 200 300 100 500 600 700 800 900 1000
Fig. 3. CCF new PRBS with maximum emissions of 0.12
nepHQmrjeCPOH eiawwoKOpperaurtoHHaa CP^MKUHB
0-8 0 6
■o.el -
20 40 60 80 100 120
Fig. 4. CCr new PRBS with maximum emissions ofO.2
To ensure satisfactory correlation properties of the PRBS ensembles resulting from the XOR of the Gold PRBS ensemble and
the de Bruijn PRBS, an investigation was made of the maximum and minimum ejection of the correlation function by the example of the Gold's PRBS cross correlation with the index 0 and 2 and the Bruijn PRBS with different indexes taken from beginning and from the end of the ensemble of PRBS de Bruijn. Figures 3 and 4 show the CCF of the new ensemble with maximum emissions ofO. 12 and 0.2.
Table 6
Cross correlation of sequences formed by XOR of the Gold sequence without a parity bit with index 2 and de Bruijn sequence with different index. In brackets - the index of the PRBS de Bruijn
Length PRBS Min CCF{index) Msx CCF (index)
63 0,238095 (OWjV -1) % max 0,809524 (0)-(l)
127 0,149606 (OMA^-IO") 0,968504 (0)-(l)
511 0,115460 (lO^fJV -10s) ' v max ' 0,992172 (0)<1)
1023 0,096774 OWN ) % max 0,992180 (0)-(l)
2047 0,085491 (10'HA^) 0,998046 (0}-(0
On the basis of the results of this study, it can be concluded that the most suitable from the point of view of obtaining the lowest values of the maximum outlier of the correlation function of the sequences obtained by XOR of the Gold PRBS and de Bruijn's PRBS are the de Bruijn sequences with indices far apart from each other, m .e. with a minimum number of graph intersections. The nature of the unevenness persists, but within the framework of Gold's ensemble CCF good. This pattern can also be observed when using a Gold PRBS with a parity bit.
In the same way, a study was made of ensembles ofNPRBSs obtained by combining ensembles of Kasami and de Bruijn in the same way as above. The results are shown in Table 7.
Table 7
Maximum emissions of the CCF size of the ensemble when different types of PRBS are combined with Ihe NPRBS de Bruijn (in parentheses the size of the ensemble)
Length PRBS Gold (=10" ) Kasami small <*10M) Kasami big {«10* )
1023 (without a parity bit) 0,155425 0,135875 0,182796
1024 (with a parity bit) 0,183594 0,132813 0,21875
As can be seen from the table, with an increase in the size of the ensemble, the level of maximum emissions of the CCF also increases. Using Gold's PRBS for expansion allows to increase the size of the ensemble by two orders of magnitude in comparison with the use of the small ensemble Kasami, the loss in the maximum level of emissions of CCF is about 2%. It is possible to further increase the size of the ensemble by an order of magnitude due to the use of a big Kasami, but in this ease it increases by 3% the maximum ejection in comparison with the use of Gold.
Thus, the optimal use case is an ensemble obtained by XOR the Gold PSP without the parity bit and the NPRBS de Bruijn.
References
Conclusion
A new class of binary pseudo-random sequences with a nonlinear generation algorithm is proposed for use in code division communication systems. Increasing the size of the ensemble makes it possible to increase the cryptographic protection and confidentiality of data transmission in comparison with the Gold ensemble. At the same time, to improve the correlation properties in comparison with the de Bruijn ensemble, namely, to reduce the emissions of CCF.
The use of a new class ofNPRBS s makes it possible to increase the size of the ensemble (for the length of the sequence 1023) by a factor of time compared to the Gold PRBS ensemble, with the maximum emission level of the CCF reaching 10% of the ACF maximum. To reduce cross-correlation links it is necessary to take dc Bruijn's sequences with indices far behind each other, i.e. with minimal intersections of graphs.
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НОВЫЙ КЛАСС ДВОИЧНЫХ ПСЕВДОСЛУЧАЙНЫХ ПОСЛЕДОВАТЕЛЬНОСТЕЙ С НЕЛИНЕЙНЫМ АЛГОРИТМОМ ФОРМИРОВАНИЯ ДЛЯ СИСТЕМ СВЯЗИ С КОДОВЫМ РАЗДЕЛЕНИЕМ АБОНЕНТОВ
Голубев Евгений Аркадьевич, Московский технический университет связи и информатики, Москва, Россия Лобов Евгений Михайлович, Московский технический университет связи и информатики, Москва, Россия, lobov@srd.mtuci.ru Кандауров Николай Александрович, Московский технический университет связи и информатики, Москва, Россия,
kandaurov@srd.mtuci.ru
Шубин Дмитрий Николаевич, Московский технический университет связи и информатики, Москва, Россия,
shubin@srd.mtuci.ru
Aннотация
Рассматривается задача разработки нового класса двоичных псевдослучайных последовательностей с нелинейным алгоритмом формирования для применения в системах связи с технологией кодового разделения абонентов. Проводится анализ существующих ансамблей и оценка их основных характеристик. А именно размера ансамбля, влияющего на криптозащищенность системы связи, и уровня максимальных выбросов взаимокорреляционной функции. Существующие ансамбли двоичных псевдослучайных последовательностей не обладают одновременно удовлетворительными значениями рассматриваемых показателей. Поэтому актуальной задачей является разработка ансамбля, находящего компромисс между двумя этими характеристиками и обладающего при этом нелинейным алгоритмом генерации. Сущность алгоритма формирования предлагаемого нового класса двоичных псевдослучайных последовательностей заключается в сложении по модулю линейных последовательностей, например, Голда или Ка-сами, обладающих хорошими корреляционными свойствами, и последовательностей де Брейна, обладающих большим ансамблем и нелинейным алгоритмом формирования. Сравнительный анализ предлагаемого способа формирования класса ПСП показывает преимущество его использования. Улучшение взаимокорреляционных характеристик новых последовательностей по сравнению с последовательностями де Брейна и значительным увеличением размера по сравнению с последовательностями Голда и Касами. Указанные свойства могут быть использованы для повышения криптозащиты и анонимности передачи в высокоскоростных беспроводных информационных сетях на основе технологии CDMA по сравнению с известными способами многоканальной передачи информации.
Ключевые слова: CDMA, ПСП, нелинейные последовательности, де Брейн, Голд, криптозащита, системы связи. Литература
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Голубев Евгений Аркадьевич, Московский технический университет связи и информатики, г.н.с., д.т.н., Москва, Россия Лобов Евгений Михайлович, Московский технический университет связи и информатики, с.н.с., к.т.н., Москва, Россия Кандауров Николай Александрович, Московский технический университет связи и информатики, м.н.с., Москва, Россия Шубин Дмитрий Николаевич, Московский технический университет связи и информатики, инженер, Москва, Россия
