587
1 INTRODUCTION
Satellitenavigationisoneofahandfuloftechnologies
thatliethefoundationsofmoderncivilisation.Being
technology‐driven, every Global Navigation Satellite
System (GNSS) shortcoming or vulnerability
immediately affects a growing number of GNSS‐
enabledtechnology andsocio‐economicsystemsand
their services (HM Government Office for
Science,
2018).GNSSusesPseudo‐RandomNoise(PRN)Gold
codesinsignalstransmittedbysatellitesastheyallow
for (GPS Directorate, 2013): (i) precise GNSS
pseudorangemeasurements,used inposition
estimationprocess,(ii)unambiguousidentificationof
relatedsatellitesignals(everyGNSSsatelliteusesits
ownuniqueGoldcode),(iii)efficient
radiospectrum
management through the application of spread‐
spectrum Code Division Multiple Access (CDMA),
achieved through lack of cross‐correlation between
theorthogonalGoldcode.AbilityofGNSSreceiverto
replicate the PRN codeʹs waveform is one of the
essentialrequirementsforasuccessfulGNSSposition
estimation (Petrovski, Tsujii, 2012).
As an example,
theUSGlobalPositioningSystem,oneoftheGNSSs,
utilisestheCoarseAcquisition(C/A)PRNcodeaimed
for civil users of the 1023 bits length. Additionally,
implementation of PRN codes ensures essentially a
level of information security through hiding data
fromtheplanesight,theencryptionembedded
inthat
manner is considered an intrinsic feature of a PRN
code(Roeck,2009),(Yang,Xiao‐Jun,2012).PRNGold
codes are generated deterministically and in a
transparent manner using shift‐register arithmetic
(GPSDirectorate,2013),anapproachsuitableforearly
GPSreceiverswithlimitedcomputingcapacity.
Recentstudiesrevealeda
rangeofpotentialissues
withGoldcodeimplementation.Increasingdemand
fora limitedspectrumrequires moreefficientcodes,
with autocorrelation effects minimised even better
than in the case of Gold code. A tendency has
emerged tominimise the exposure of code
generation, as an information security measure.
Finally, a security
threat has been identified in
Logistic Map-encrypted Chaotic Ranging Code as a
Proposed Alternative to GNSS PRN Pseudorange Code
M.Filić&F.Dimc
UniversityofLjubljana,Ljubljana,Slovenia
ABSTRACT: Pseudo‐Random Noise (PRN) Gold code was selected for utilisation as the Global Navigation
SatelliteSystem(GNSS)pseudo‐rangemeasurementcodesequence.Recentstudiesrevealedapotentialsecurity
vulnerability issue due to the Gold PRN code utilisation in a GNSS‐related cyber‐attack known as
GNSS
spoofing. Here a PRN code construction method based on chaotic‐form logistic map is proposed as an
alternativetotheexistingGoldcodepractice.DubbedChaoticRangingCode(CRC),isaPRNcodegeneration
methodthatgeneratesrangingcodewithorthogonalpropertiesasgoodas,ifnotbetter,then
thoseoftheGold
PRN code, while assuming the encryption embedded in the proposed CRC code provides improved GNSS
informationsecurity.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 13
Number 3
September 2019
DOI:10.12716/1001.13.03.14
588
utilisationofPRNGoldcodesforGNSSpseudorange
measurements, an essential input for GNSS position
estimation process. It has been shown that the flaw
may be exploited in cyber‐attacks against GNSS,
known as GNSS spoofing (Tippenha uer et al, 2011),
(Filić,2018).
A novel approach in GNSS ranging PRN
code
generation has been proposed as a counter‐measure
totherisingGNSSspoofingthreats.Chaoticprocesses
have been identified as a potential class of ranging
signalsourcesthatmaytameGNSSspoofingproblem
(Petrovski, Tsujii, 2012), (Yang, Xiao‐Jun, 2012). In
addition, chaotically‐driven ranging PRN codes
allows for
fast synchronisation as an advantage for
GNSS receivers, due to embedded chaotic system
properties(Pecora,Carroll,2015).
Logisticmaphasbeenasimpleandpopularmodel
of biological population growth by Pierre Francois
Verhulst in 1838, and has gained even more
popularity after (May, 1976) it was disclosed its
unusual behaviour
for specifically configured map,
capable of extending a chaotic behaviour (Kanso,
Smaoui, 2009). Research revealed the onset of chaos
for its control parameter in the (3.6, 4) interval of
values. Since then, the chaotic behaviour of the
logisticmaphasbeenusedinnumerousapplications,
including communications‐related encryption
methods
(Yang,2004),(Mitra,2007),(Kanso,Smaoui,
2009).
Here utilisation of Chaotic Ranging Code (CRC),
an alternative ranging PRN code for GNSS
pseudorange measurement, is proposed, based on
utilisation of the chaotic‐behaving configuration of
the logistic map. Methodology for the CRC
construction is presented in Section 2, along with
descriptionof
codeperformance examination
methodology. Practical realisationresults are
presented in Section 3. Characteristics and related
performance of the proposed alternative GNSS
logistic map‐encrypted PRN code are discussed in
Section 4. Manuscripts concludes with the research
results,CRCperformanceandpotentialshortcomings
summary,andaproposalforfurtherresearchsubjects
inSection5.
2 METHODFORCHAOTICRANGINGCODE
GENERATION
2.1 Alternativechaotically‐behavinglogisticmap‐based
CRCconstruction
Logisticmapisasimplemathematicalexpressionthat
may generate a chaotically behaving dynamics,
dependingon itscontrolparameterr, andthe initial
conditionx
0(Figures 1,2,3).Logisticmapisdefined
asinEq(1).


1: 1 0,1,xk r xk xk forx r
(1)
Thechaotically‐behavinglogisticmap‐basedPRN
codealternativeproposedherewasconstructedusing
the logistic PRN binary data LOGMAP1 algorithm
suggested by (Kanso, Smaoui, 2009). The algorithm
readsasfollows(Algorithm1).
Figure1. Bifurcation diagram of logistic map (May, 1976)
forcontrolparameterrin[‐2,4]
Figure2. Bifurcation diagram of logistic map (May, 1976)
forcontrolparameterrin[3,4]
Figure3. Bifurcation diagram of logistic map (May, 1976)
forcontrolparameterrin[3.99996,4]
2.2 Performanceassessment
(Kanso, Smaoui, 2009) proved randomness of the
PRN sequence a, resulting from implementation of
Algorithm1,andexaminedotherstatisticalproperties
oftheirproposedmethod,butdidnotexamineeither
orthogonality or potential communications
applications.Theypresentedtherationaleforsetting
thresholdlevelatthe0.5value,and
selectionofther
control parameter from the interval (3.99996, 4)
(Figure 3) in order to obtain pseudo‐random noise
codes.
(Mitra, 2007) presented a methodology for the
assessment of the PRN code orthogonality through
theexaminationofauto‐correlationfunction,givenas
inEq(2).
 
1
,
1
1
N
ij i j
N
rcncn
N

(2)
589
_______________________________________________
Algorithm1:LOGMAP1(Kanso,Smaoui,2009)
_______________________________________________
Data:Vectorofchaotically‐behavinglogisticmapʹsinitial
conditions(encryptionsecretkey)ic=(x0,r),withx0asthe
initialvalue,andrasthelogisticmapcontrolparameter;
requestedPRNcodelength(N);thresholdparameter(t)
Result:Abinarysequenceofchaotically‐encryptedPRN
codea
_______________________________________________
1setx[0]:=x0,y[0]:=0;t :=0.5;a[0]:=0;
2fori:=1toNdobegin
3

11
x
irxixi
;
4


0
11
i
k
yi xk mod

;
5 if(x[i+1]<0.5){z[i+1]:=0}else{z[i+1]:=0};
6 if(y[i+1]<0.5){w[i+1]:=0}else{w[i+1]:=0};
7 a[i+1]:=w[i+1]XORz[i+1];
8end;
_______________________________________________
(Mitra, 2007) proposed a Figure of Merit (FoM)
parameter, as a quantitative measure of frequency
suitabilityofthePRNcodeforCDMA(Eq(3)),FoMis
definedforasequencex
i(n),ofthelengthN,andwith
theauto‐correlationfunctionr
i(τ)given.(Mitra,2007)
also suggested the rule of thumb for estimation of
encryption strength from FoM, when larger FoM
values refer to larger bandwidth and stronger
encryption using the PRN code constructed. Finally,
(Mitra, 2007) presented the Gold code assessment
auto‐correlation results, used here for reference, but
didnotdisclosenumericalresultsoftheGoldcodein
thestudy.



2
,
2
2
,
0
0
ii
ii
r
r
FoM x
r
(3)
(Huangetal,1998)gaveanindependentandmore
generaldiscussiononorthogonalityassessment.
3 PRACTICALREALISATION
The CRC code generation was performed based on
the methodology and approach given by (Mitre,
2007). Practical realisation of the proposed CRC,
based on chaotically‐behaving logistic map, was
conducted in the
R open‐source programming
framework for statistical computing. Sets of length
100and1023wereconstructedforeveryscenariowith
different logistic map parameters, as presented in
Table1.
Table1.Scenariodescription
_______________________________________________
Scenariorvaluex0PRN
(logisticmap(initial code
controlparameter) value) length
_______________________________________________
A3.999970.4100
B3.999970.41023
C3.999980.4100
D3.999980.41023
_______________________________________________
Table2FoMsforscenariosconsidered
_______________________________________________
Scenario FoM
_______________________________________________
A‐2.66685
B‐2.582446
C0.5498525
D14.86747
_______________________________________________
The CRC sequences constructed within Scenarios
A,B,C,andD,respectively,wereexaminedfortheir
auto‐correlation functions to address the spectrum
utilisationefficiencyandorthogonality.Thisresearch
did not examined the encryption strength in more
detailsthat was givenin(Mitra,2007)methodology,
adoptingthehypothesisofFoM
asa singledescriptor
ofbothbandwidthandencryptionstrength.
ResultsoftheanalysisaredepictedinFigures4,5,
6,and7,respectively.Figure8presentsthePRNcode
resulted from Scenario D. The cross‐correlation
function of two independent PRN codes with the
samecodelength,constructedwithin
ScenariosBand
D, respectively, were examined from the inter‐
operabilityperspective,anddepictedinFigure9.
Figure4. Auto‐correlation function of the PRN code
constructedunderScenarioA
Figure5. Auto‐correlation function of the PRN code
constructedunderScenarioB
Figure6. Auto‐correlation function of the PRN code
constructedunderScenarioC
Figure7. Auto‐correlation function of the PRN code
constructedunderScenarioD
590
Figure8. A PRN code sequence a, of length 1023,
constructedwithinScenarioD
Figure9. Cross‐correlation function of twoalternative
PRN codes, constructed within Scenarios B and D,
respectively
4 DISCUSSIONANDCONCLUSION
ShortcomingsofthecurrentlydeployedGoldcodefor
GNSS pseudorange measurements are addressed in
researchpresentedinthismanuscript.Constructionof
Chaotic Ranging Code (CRC), an alternative logistic
map‐basedrangingPRNcodeforsatellitepositioning,
was considered in this research, and resulting code
assessedforits
performance,asapotentialcandidate
for the GNSS pseudorange measurement code. The
alternative CRC code performance was compared
with Gold code ones, based on methodology and
Gold code assessment presented by (Mitra, 2007).
Research was focused on spectrum efficiency and
code orthogonality issues, as well as as with
transparency of
the process, with the quality of
encryptionmarginallyconcerned in accordance with
theguidingmethodologyadopted.
Scenarios A, B, C, and D (Table 1) presented the
CRC sequences with different performance levels.
Their auto‐correlation functions performed well,
allowingforacceptableorthogonality, andthereforea
proper inter‐operability, allowing for simultaneous
utilisationofthesamespectrumbyaconstellationof
GNSS satellites. Lattices are suppressed sufficiently,
with several scenarios performing even better than
the standard GNSS Gold code PRN (Mitra, 2007).
However,FoMvaluesvarylargely.Asitisexpected,
larger codes and wider spread brings more robust
encryption and wider
bandwidth. Still, the selection
ofparticularconfigurationofalogisticmapmayhave
significant impact on the FoM for particular logistic
map‐encryptedPRNcode.
In summary, a framework for construction of a
ranging code alternativeto the GNSS Gold code for
pseudorange measurement was assembled, in
compliancewithrequirementsfor
GNSSrangingcode
performance and based on utilisationof chaotically‐
behavinglogisticmap.TheCRCgenerationalgorithm
developedby(Kanso,Smaoui,2009) was configured
and assessed for performance using methodology
developed by (Mitra, 2007) to compare its quality
withtheexistingGNSSPRNGoldcode.Fourvariants
oftheCRCs
weregenerated,andtheirperformances
assessed. The interpretation of research results
revealed variations in encryption robustness
depending on logistic map configuration, and
confirmed importance of construction of long PRN
codes. Future research will address a wider set of
logistic map configurations across the range of
configuration parameters, development and
utilisation of
more exact measures of encryption
strength for PRN code assessment based on
methodological approach outlined in (Roeck, 2009)
and(Yang,Xiao‐Jun,2012),andCRCfieldvalidation
in simulated scenarios of GNSS spoofing cyber‐
attacks.
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