Proceed
dings of the Asia-Pacific
A
Advanced N
Network 20155 v. 40, p. 500-56.
Networkk Research Workshop
W
htttp://dx.doi.oorg/10.7125//APAN.40.8
ISSN 222
27-3026
A Novel
N
Error
E
Correc
C
ction S
Scheme in Quantum
m
Key Disttributio
on (QK
KD) Prrotocool
Siao Ping Lee,
L Chee Ky
yun Ng and M
Makhfudzah M
Mokhtar
Dep
partment of Computer and Communicattion Systems Engineering,
Faculty of En
ngineering, Universiti
U
Putrra Malaysia, M
Malaysia
Emails: xquire_v@ho
x
otmail.com; mpnck@upm.
m
edu.my; fudzzah@upm.eduu.my

antum key disstribution (QK
KD)
Abstract— Ideally, in any qua
mmunication sy
ystem, each sifted key is expectted to be receiv
ved
com
witthout error. However
H
in prractice, due to
o infeasibility of
gen
nerating pure single
s
photon and
a device impa
airment problem,
som
me of the sifted
d key may experrience errors. This
T
results to the
t
inccrement of quan
ntum bit error rate (QBER) th
hat requires errror
recconciliation forr correcting errror. The main concept in errror
recconciliation is very much relateed to the capab
bility of correctiing
all errors while minimizing eavessdrop informatiion. The quantu
um
errror correcting code such as Hamming cod
de which used in
Wiinnow protocoll is found to bee more attractive. However the
t
Wiinnow protocol can only correct one error ou
ut of seven bits. In
thiis paper, a mo
odified Hammin
ng encoder/deccoder to impro
ove
Wiinnow protocol by correcting tw
wo errors out of seven bits whiich
leaads to reducing the QBER is presented.
p
Thiss design utilizess a
paiir of forward and
a reverse ord
der syndromes for error patteern
reccognition. A new
w reconciliation
n protocol has been
b
developed to
enh
hance the error correcting ca
apability in BB
B84 protocol. Itt is
carrried out in a siimple structuree which can corrrect up to doub
ble
errroneous bits and
d detect four errroneous bits for each seven bitts.
IIndex Terms—
— Cryptograp
phy, Hammin
ng code, errror
corrrection, QKD
D, reconciliation protocol, BB84 protoccol,
Wiinnow Protocol.
I. INTRO
ODUCTION
A
of cryptograaphy are mosstly designed to
h computationaal hardness asssumption and not
n
comply with
witthstanding im
mminent threat imposed by computationallly
effficient device. Permitting su
uperposition off binary states,, a
quantum computter which execu
utes operation on quantum bits
b
wn as qubits) is believed to
o be capable of
(coommonly know
speeeding up ted
dious computaations tremend
dously once the
t
asssociated techn
nologies are in place. Th
hus, it rendeers
staate-of-the-art asymmetric-key
a
y cryptograph
hy compromissed
andd endangers computation
nally secure symmetric-k
key
cryyptography [1]. Thereforee, a vintagee technique of
sym
mmetric-key crryptography kn
nown as one-tiime pad (OTP)) is
heeeded as the ulttimate solution
n because it has been proven to
be
information
theoreticaally
impreg
gnable
again
nst
50
LGORITHMS
cryptannalysis, if a pperfectly randoom secret keyy of infinite
length is employed onnly once and nnever reused [1]-[4]. Owing
mplementationn, it was not muuch attended
to lackk of practical im
until noow.
As uunguarded dellivery of secreet key may jeopardize the
plausibble scheme, qquantum key ddistribution (Q
QKD) which
escortss key through quantum chaannel using quuantum state
encodinng, i.e., photoon polarizationn, is suggestedd to facilitate
OTP inn order to set uup a secure com
mmunication fo
for secret key
sharingg [5]. Having its security aascertained byy Heisenberg
uncertaainty principle [6] and no-clooning theorem
m of quantum
mechannics, QKD guaarantees deliverry of key in succh a way that
possiblle eavesdroppiing can be coonfidently deteected during
error raate estimation [7]. The renow
wned QKD prootocol, which
has bbeen proven unconditionaally secure aagainst any
eavesddropping and practically viable, was built upon
inspiraation from quaantum realizattion of unforgeable bank
notes [ 8] and promullgated by devellopers Charles Bennett and
Brassard in 19984, typically kknown as Bennnett-Brassard
Gilles B
1984 (B
BB84) protocool [9]. In fact, the joint ventture between
OTP annd QKD is connsistent with K
Kerckhoffs's prinnciple which
enunciaates that key’s secrecy shouldd be the one annd only pivot
leveragging security oof a cryptosysttem [10]. How
wever, errors
attributted to imperfecctions in the phhysical implem
mentation are
prevaleent, with or without eaveesdropping. C
Consequently,
reconciiliation is vitall for secret keyy distillation, w
which serves
as pprerequisite ffor informattion-theoreticaally secure
cryptoggraphy. Reconnciliation is ccarried out inn (noiseless)
public yet authenticatted classical chhannel to correect undesired
ween sender’s aand receiver's
errors ssuch that discrrepancies betw
secret key can be fixed for suuccessful enccryption and
decryp tion respectiveely. It can be acccomplished bby employing
either simple classiccal error corrrecting code oor advanced
quantuum error correccting code [1].
Winnnow protocoll decreases tthe disclosuree of partial
mation to eaveesdropper by ttaking advantage of both
inform
parity bbit and Hamm
ming code for single-bit erroor correction.
Neverttheless, the need of several iteratioons is still
indispeensable becausse Winnow prrotocol tends to correct a
block oof sifted secret key that is interspersed with tthree or more
naccurately wh
hile abandonin
ng
odd multiple biits of error in
n multiple bits of error [11]. If convolution
nal
dettection of even
codde takes the pllace of Hammiing code, the Winnow
W
protoccol
cann correct any odd number of erroneous
e
bit(ss) with the loss of
operational simpllicity [12]. Chaaracterized to allow
a
multiple-b
bit
errror correction, Bose-Chaudh
huri-Hocquengh
hen (BCH) co
ode
suiitably becomees a sound alternative
a
forr reconciliatio
on.
Noonetheless, its error
e
correcting
g capability is rather limiting
g if
cloosely examineed [13]. Afterr all, there iss a very stron
ng
mootivation to dev
velop a reconciiliation protoco
ol that minimizzes
pubblic communiccation between
n legitimate com
mmunicants wiith
im
mproved error co
orrecting capab
bility. Thus, th
his paper is aim
med
to enhance reliab
bility of QKD by proposing an efficient an
nd
ol that rectifiess errors in sing
gle
efffective reconciliation protoco
passs with maxim
mum of doublee-bit error correcting capabiliity
intto BB84 protoccol. This design
n utilizes a paiir of forward an
nd
revverse order syn
ndromes for error pattern recognition. The neew
recconciliation protocol has beeen developed and
a evaluated in
terrms of amountt of disclosed bit and quantu
um bit error raate
(QBER)..
II. THE PROPOSED RECONCILIA
ATION SYSTEM
M ARCHITECTUR
RE
In the BB84 protocol
p
as sh
hown in Fig. 1,
1 Alice sendss a
m key through quantum chan
nnel to Bob aft
fter
stream of random
n state of each key element. The
T key is firsttly
reccording photon
codded in bits then further enccoded in conju
ugative quantu
um
staates, constituted by rectilineaar and diagonaal polarization of
photon conventtionally. Map
pping of bitt to respectiive
polarization is indicated at the bottom of
o Fig. 1. Bob
otons and measure them using
ga
ackknowledge his receipt of pho
stream of random
m rectilinear an
nd diagonal baases, independeent
froom those of Alice. Wheneverr the photon staate is a subset of
bassis of measurem
ment, he gets correlated
c
resullt. His choices of
bassis with corressponding measurement resultts, known as raaw
keyy, are recorded
d. After transfeerence of the random
r
key, Bob
infforms Alice about
a
the streeam of basis being used for
f
meeasurement through (noiseleess) public yet
y authenticatted
claassical channel, which is accessible to passiv
ve eavesdroppin
ng
sollely.
Fig. 1. Schematics of the BB
B84 protocol for id
deal case.
Alice notifies Bob which of his measuremeent is compatib
ble
d should have the photon staate
witth the photonss delivered and
dettected correctly
y, enabling theem to disregard
d those result th
hat
suscepttible to disruuptive measurrement. Afterr discarding
anomallies in respectiive raw key, thhey deduce ideentical sifted
keys inn secret, whichh can be used for cryptograpphic purpose.
Obviouusly, their secret key is nnot predeterm
mined but is
developped in conjuncction of their raandom choicess, with an aid
of guidded investigatioon [9].
Resuultantly, Bob’ss sifted key ssuffers from 225% [14] of
QBER in respect to A
Alice’s sifted kkey. Thus, afteer the sifting
processs, reconciliaation is neecessitated too ascertain
identic alness of the siifted keys pair.. Grueling prooofs of QKD’s
wcase correspoonding noise
securityy were presennted to show
resistannt threshold [114],[15]. In thhe earliest atteempts, BB84
protocool was proven secure againstt all attacks peermissible by
laws off quantum mecchanics wheneever the QBER
R is less than
7.4% [[16] and up to 7.56% [17] inn two independdent research
studiess. Once reconcciliation is initiated, error ddetection and
correcttion make conccerted effort to mitigate inconnsistencies in
the siffted keys paiir using interractive or noon-interactive
protocool. An interaactive reconciiliation protoccol requires
repetitiive exchange oof parity bit beetween Alice annd Bob via a
two-waay communicaation channel too detect and coorrect errors.
On thee contrary, a non-interactivve reconciliatiion protocol
appliess concept off one-way soource coding with side
inform
mation to eliminnate the interaactivity betweeen Alice and
wn in Fig. 2.
Bob whhen performingg error correctiion [2], as show
In a conceptual maanner, Alice’s sifted key is ffirst encoded
into reespective syndrrome. The synndrome is thenn transmitted
over a (noiseless) pubblic yet authennticated classicaal channel to
Bob annd fed into a deecoder togetherr with his own sifted key to
restoree Alice’s siftedd key with higgh probability. In this way,
sifted kkey with flaw aat receiving endd is mended allegedly [12].
The noon-interactive reconciliationn protocol is a preferable
techniqque since it caan catalyze effficiency of erroor correction
and miinimize public communicatioon concomitanttly.
At thhe beginning of Winnow prrotocol, after sshuffling the
bits off sifted keys pair in the sam
me way, Alice’’s and Bob’s
string of sifted key are also diviided into bloccks and then
subjectted to parity ccheck correspoondingly. Onee bit in each
block iis then discardeed because of tthe parity check. After that,
non-intteractive reconnciliation beginns. First of all, syndrome is
calculaated and sent fr
from Alice to B
Bob, for each oof the blocks
exhibitting odd resullt in preliminnary test. It iss noted that
syndro me is primitiveely an indicatoor implying corrrectness of a
receiveed codeword duuring error dettection, but herre is where it
fits intoo reconciliationn.
F
Fig. 2. Source codding with side infoormation in reconciiliation.
At reeceiving end, syndrome meaasurement is caarried out by
Bob ussing received ssyndrome in taandem with hiis own sifted
key’s syndrome too compute ddifference between their
d
asso
ociated correctaable error patteern
synndromes, and determine
of his sifted key such that the most
m probable error
e
can then be
m independently
y. Normally, th
he assigned errror
corrrected by him
corrrecting code is Hamming code,
c
the firstt effective lineear
bloock code inven
nted to be able to correct onee bit of error in
na
vallid codeword.
ode’s limited error correctin
ng
Confined by Hamming co
cappability, this method
m
will hav
ve a block of siffted key deducced
by Bob that is interspersed with
h three or more odd multiple bits
b
ous
of error incompleetely corrected, i.e., only one of the erroneo
bitts is corrected, not
n corrected or
o worse yet, wrrongly correcteed,
cauusing an extraa erroneous bit.
b Furthermo
ore, this method
cannnot detect even
e
multiple bits of errorr, leaving theem
uncorrected. Hen
nce, iterations that independeent of each oth
her
ning bits of siftted
aree still a must during reconciliiation. Remain
keyy in each block that equivaalent to redun
ndancy bits off a
Haamming code’s codeword, are also discarded
d
befo
ore
com
mmencement of new rou
und of recon
nciliation. Som
me
errroneous bits that
t
fall amon
ng the removeed bits are th
hus
disscarded withou
ut undergoing error
e
correction
n [11],[18].
H
Hence, in our proposed reco
onciliation prottocol, in order to
dettect any Hamm
ming (7, 4, 3) co
odeword that iss interjected wiith
up to two bits of error, codeworrds with weigh
ht of two in eveery
ndard array aree collectively gathered
g
as exttra
cosset of the stan
corrrectable error patterns associiated with respeective syndrom
me.
Reesultantly, there is a mix of single-bit and double-bit errror
pattterns associatted with each non-zero syn
ndrome. Witho
out
inttroducing addittional parameter which may be favorable for
f
possible eavesdropping, the sy
yndrome measurement is do
one
d
man
nner for an atteempt to reconcile
twiice in slightly distinctive
possible errors in
i the codewo
ord such that two
t
set of errror
pattterns in respecct to two set off syndromes are made availab
ble
forr matching an
nalysis. Thus, a simple con
ncept of logiccal
reaasoning is feattured by analy
yzing the codew
word in forwaard
andd reverse ordeers. It is utilizin
ng an idea thaat the exact errror
patttern should reemain the samee regardless off the direction in
whhich analysis is performed, i.e., whether from the mo
ost
siggnificant bit (M
MSB) toward th
he least significcant bit (LSB) or
vicce versa as sho
own in Fig. 3. In
I QKD appliccation, syndrom
me
in forward order is the syndrom
me calculated when a block of
fted key is anallyzed in forwaard order (MSB
B  LSB) while
sift
synndrome in reveerse order is th
he syndrome caalculated when
na
bloock of sifted keey is analyzed in
n reverse orderr (LSB  MSB
B).
Fig. 3. Th
he order of analysiss with respective syndrome.
s
Indeed syndrom
me in forward order is the sy
yndrome that has
h
beeen used in Winnow
W
protocol. The differrence between
n a
block ssyndrome of ssifted key deduuced by Alice and the one
deduceed by Bob in foorward order aas well as reverrse order, are
corresppondingly com
mputed by Bob tto determine thhe associated
error p atterns in bothh orders as show
wn in Fig. 4. Itt can be seen
that errror patterns associated witth non-zero ssyndrome in
forwardd order are a coollection of coddewords with w
weight of one
or two in every cosett of the standarrd array duringg preparatory
while error pattterns associateed with non-zeero syndrome
stage, w
in reveerse order are those of forw
ward order but experienced
straighht left right flippping. Such adjjustment is maade such that
posteriior matching analysis and error correcttion can be
perform
med by Bob inn reference to cconventional foorward order.
Wheneever syndrome measurement does not resuult in all-zero
syndro mes in forwaard order and that of in reeverse order,
mum occurrencee of two bits oof error in a block of sifted
maxim
key iss detected. O
Otherwise, the differences are all-zero
syndro mes, intimatinng that the blocck of sifted keyy is errorless.
Matchiing analysis is tthen carried ouut to determine the identical
error p attern associatted with differeence between ssyndromes in
respecttive order, ruuling out irrellevant error ppatterns and
pinpoinnting the exactt one for successsful error corrrection.
The algorithm off proposed reeconciliation pprotocol that
B84 protocol in single pass wiith maximum
rectifiees errors of BB
of douuble-bit error ccorrecting capaability is show
wn in Fig. 5.
First off all, the positiion of bits in A
Alice’s and Boob’s string of
sifted kkey is randomlly permuted viia folio interlaacement such
that poossible sequennt errors are ddispersed at rrandom. The
shuffleed strings of siifted key are ppartitioned by both parties
into bloocks that compprise seven bitts out of the total bits each.
Alice hhas syndrome oof the first blockk of sifted key calculated in
both foorward and reveerse orders usinng her portion oof sifted key,
and theen sent to Bobb via (noiseless) public yet aauthenticated
classicaal channel. Meeanwhile, Bobb also has synddrome of the
first bllock of sifted key calculatedd in both ordeers using his
portionn of sifted key. Syndrome meeasurement is ccarried out by
Bob ussing received ssyndromes in ttandem with hhis calculated
syndro mes to compuute difference between theirr syndromes
and dettermine associated error patteerns in both orders.
The matching analyysis is carried oout by Bob to ddetermine the
identic al error patteern associated with differennce between
syndro mes in respecttive order. Thee conditional decision to be
made bby Bob will bbe if there is a match of iddentical error
patternn after performiing matching aanalysis, error correction is
perform
med by addingg his block of sifted key undder test with
pinpoinnted error patteern bitwise usiing binary XO
OR operation.
Otherw
wise, there is nnot a match of iidentical error pattern after
perform
ming matchingg analysis. Erroor correction is skipped and
his bloock of siftedd key under test is discarrded with a
notificaation sent to A
Alice via the cclassical channnel such that
correspponding blockk of her sifteed key is disscarded too.
Proceddures are kept rrepeated for ennsuing blocks of sifted key
before the last blockk is analyzed. F
For all the blocks of sifted
key thaat are successsfully reconciled, the fourthh bit in each
block i s reserved while the rest are ddiscarded by booth parties on
maintenance.
accounnt of privacy m
Fig. 4. The error patterrns associated with
h difference betweeen syndromes in rrespective orders.
Fig. 5. Flow of the pro
oposed algorithm fo
for reconciliation.
Fig. 6. Reconciliiation and privacy maintenance utilizzing the proposedd algorithm.
The reconciliaation and priv
vacy maintenan
nce utilizing the
t
prooposed protoccol is shown
n through a self-explanato
ory
exaample in Fig. 6.
6 Remarkably,, a 7-bit block of
o sifted key may
m
be interspersed with
w three or fou
ur bits of error,, but such a block
onciliation in accordance
a
wiith
willl be discardeed during reco
fouurth step of thee proposed prottocol.
III. PERFORMANC
CE EVALUATIO
ONS
a
for reeconciliation iss simulated usin
ng
The proposed algorithm
MA
ATLAB® softtware which su
upports matrix
x operations th
hat
aree fundamental to
t error correcttion. The simulation is initiatted
by generating tw
wo strings of sifted key; on
ne is errorless in
nterjected with
h sequent errors.
refference while the other is in
Booth strings un
nderwent segm
mentation, ran
ndom shufflin
ng,
synndrome comp
putation, mattching analyssis, appropriaate
recconciliation, privacy
p
mainteenance and co
ombination. The
T
sim
mulation is repeeated using diffferent initial QB
BER, i.e., QBE
ER
priior to reconciliation, and evalluated against final
f
QBER, i.e.,
QB
BER right aftter reconciliattion, which iss the output of
sim
mulation.
F
Figure 7 show
ws the simulation result in comparison
c
wiith
Wiinnow protoco
ol applying parrity check and Hamming cod
de.
Thhe line that corrresponds to Winnow
W
protoccol is plotted by
b
dirrectly applying
g the data readiily available in
n [18]. Length of
sift
fted key of abou
ut 3000 bits and optimized block size are used
in this simulation
n. The final QB
BER posts a rise in response to
ols,
inccrement of inittial QBER forr both reconcilliation protoco
butt the percentag
ges recorded fo
or proposed alg
gorithm are low
wer
thaan those of Wiinnow protoco
ol. The differen
nce is noticeab
bly
cleear for initial QBER
Q
ranging from
f
4% to 11%
%. It is due to the
t
cappability of the proposed algo
orithm in correecting up to tw
wo
bitts of error in an erroneous 7-biit block of sifteed key, which iss a
feaature not posssessed by Wiinnow protoco
ol. Furthermorre,
unlike the propossed algorithm, Winnow proto
ocol is incapab
ble
d discarding thee erroneous blo
ocks of sifted key
k
of identifying and
ber of remainiing errors at the
t
whhich constitutes toward numb
endd of reconciliaation in singlee pass. Hinging
g on the limitted
sinngle-bit error correcting
c
capaability, several iterations are in
neeed for compleete reconciliation using Win
nnow protocol in
genneral.
F
Figure 8 show
ws the simulation result in comparison
c
wiith
im
mproved Winn
now protocol applying parrity check an
nd
connvolutional co
ode. The line that correspo
onds to Winno
ow
prootocol applyin
ng convolution
nal code is plo
otted by directtly
appplying the dataa readily availab
ble in [12]. Len
ngth of sifted key
k
of 100000 bits is used in theirr simulation in
n which the daata
preesented are av
veraged valuess of 100 trials. The trend th
hat
corrresponds to proposed alg
gorithm outpeerforms that of
Wiinnow protoco
ol applying con
nvolutional, allthough any odd
num
mber of erron
neous bit(s) in a block of siifted key can be
corrrected via the improved Win
nnow protocol.
55
Fig. 7.. Graph of final QB
BER versus initiall QBER in referencce to Winnow
protocol appllying parity check and Hamming codde.
Fig. 8.. Graph of final QB
BER versus initiall QBER in referencce to Winnow
protocol applyiing parity check annd convolutional ccode.
IV. CONCLUSSIONS
The quuantum error ccorrecting codde such as Ham
mming code
which uused in Winnoow protocol is ffound to be moore attractive.
Howevver, the Winnow
w protocol cann only correct oone error out
of seveen bits. In this paper, a new reconciliation protocol has
been ddeveloped to ennhance the errror correcting capability in
BB84 protocol. A ssingle pass reeconciliation pprotocol that
capablee corrects up tto two bits of error in an errroneous 7-bit
block oof sifted key has been pressented by appllying simple
Hamm
ming (7, 4, 3) coode. The syndrrome measurem
ment is done
twice iin slightly distiinctive mannerr such that twoo set of error
patternns in respect to two set of synndromes are maade available
for maatching analysiis. Thus, it is featured by aanalyzing the
codewoord in forwardd and reverse orders where thhe exact error
patternn should remaiin the same rregardless of tthe direction
whetheer from the MS
SB toward the L
LSB or vice verrsa. With this
new innterpretation oof Hamming code’s syndroome and an
unpreccedented matchhing analysis, ooccurrence of tthree or four
bits off error in the errroneous blockk of sifted keyy can also be
fied by the propposed reconciliiation protocol.
identifi
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Siao Ping Lee received
d his Bachelor of
Engineeriing and Masster of Scien
nce
degrees majoring in Computer &
Sy
ystems
fro
om
Communiication
Universitii Putra Malaysia, Serdan
ng,
Selangor, Malaysia, in 2009 and 2013
h
respectiveely. He was undertaking his
research on optical communicatio
ons
specializeed in error corrrection coding.
C
Chee Kyun Ng received his Bachelor of
E
Engineering aand Master of Science
ddegrees majooring in Coomputer &
C
Communicationn Systems from
m Universiti
P
Putra Malayssia, Serdang,, Selangor,
M
Malaysia, in 19999 and 2002 respectively.
H
He has alsoo completed his PhD
pprogramme iin 2007 m
majoring in
C
Communicationns
and
Network
Engineeering at thee same univversity. He iis currently
undertaaking his ressearch on infformation com
mmunication
technollogy (ICT) tow
wards ageing peeople. Since frrom his study
program
mmes, he has ppublished overr 100 papers in journals and
in confferences.
M
Makhfudzah Mokhtar reeceived her
B
B.Eng. degree from thee Universiti
K
Kebangsaan Maalaysia in 20000. She started
hher career inn education ffield at the
D
Department
of
Compputer
and
C
Communicationn Systems E
Engineering,
F
Faculty of Enggineering, Uniiversiti Putra
M
Malaysia as a tutor. In 2007, she
compleeted her Ph.D. degree from
m University oof Essex and
served as a lecturer inn the same instiitution. Since 22001, she has
been innvolved in optiical communiccation systems research and
her ressearch interestss focus on chaannel coding, O
Optical Code
Divisioon Multiple Acccess and quanttum key distribbution. She is
a mem
mber of the Institute of E
Electrical and Electronics
Engineeers (IEEE) andd the Photonics Society undeer the IEEE.
© 20015 by the authhors; licensee A
Asia-Pacific Addvanced
Netw
work. This articcle is an open-aaccess article ddistributed undeer
the teerms and condiitions of the Crreative Commoons Attribution
n
licensse (http://creattivecommons.oorg/licenses/byy/3.0/).