Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
6, N
o
. 3
,
Ju
n
e
201
6, p
p
. 1
002
~ 10
10
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
3.9
420
1
002
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Secure Digital Signature Scheme
Based on Elliptic Curves for
Intern
et of Things
Sumanth K
o
p
pula
,
Jay
abhaska
r
M
u
t
h
ukuru
Department o
f
C
o
mputer Scien
c
e
and Engin
eerin
g,
K L
Univ
ersity
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Nov 15, 2015
R
e
vi
sed M
a
r
3,
2
0
1
6
Accepted
Mar 15, 2016
Advances in
th
e info
and
co
mmuni
cation k
nowledge h
a
ve led
to
the
emergence of In
ternet of things
(IoT). In
ternet o
f
things (loT)
is worthwhile
to m
e
m
b
ers
,
trade, and s
o
cie
t
y s
eei
ng that it
generates a broad range of
services b
y
in
terconnecting
nu
merous
devices and
information objects.
Throughout the interactions among the
many
ubiquitous things, security
problems emerge as notew
orth
y
,
and it is signi
fi
cant
to set
up m
o
re suitab
l
e
solution for security
protection
.
Noneth
eless, as loT devices have limited
resource constr
aints
to appoin
t
str
ong protection mechanisms, they
are
vulnerab
l
e
to s
ophisticated
security
at
tacks. For this r
eason, a sensible
authen
tic
ation m
echan
is
m
that cons
iders
each us
e
f
ul res
ource con
s
traints
and
safet
y
is r
e
quired. Our proposed
schem
e
uses th
e standards of
E
llipt
i
c Curv
e
digital signatu
re scheme and ev
aluates
s
y
s
t
em
at
ica
l
l
y
th
e effi
ci
e
n
c
y
of ou
r
scheme and obs
erves that
our s
c
heme with
a smaller key
size and lesser
infrastructure p
e
rforms on pa
r with the prevailing schemes without
c
o
mpromi
si
ng t
h
e
se
c
u
ri
ty
l
e
ve
l.
Keyword:
In
tern
et o
f
th
ing
s
Ellip
tic cu
rv
e
d
i
g
ital sign
ature
Dig
ital sign
ature
Ellip
tic cu
rv
es
Copyright ©
201
6 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Sum
a
nt
h K
o
pp
ul
a,
Depa
rt
m
e
nt
of
C
o
m
put
er Sci
e
nce a
n
d
E
ngi
n
eeri
n
g,
K L Un
iv
ersity,
V
a
dd
esw
a
r
a
m
5
225
02
, Gun
t
ur
D
i
st
r
i
ct, Andh
r
a
Pr
ad
esh
,
Ind
i
a.
Em
a
il: k
o
p
p
u
l
asu
m
an
th
@liv
e.co
m
1.
INTRODUCTION
Ad
va
nces i
n
t
h
e wi
re
d, wi
re
l
e
ss, cel
l
u
l
a
r and se
ns
or net
w
o
r
k
s
ha
ve l
e
ft
a pret
t
y
good
base fo
r t
h
e
i
n
t
e
rnet
of t
h
i
ngs
(I
oT
). It
'
s
a no
vel
pa
ra
di
gm
whi
c
h t
a
kes acco
u
n
t
o
f
eve
r
y
day
p
h
y
s
i
cal
wo
rl
d
ob
ject
s
t
h
r
o
u
g
h
e
n
abl
i
ng
i
n
t
e
r
p
l
a
y
a
m
ong t
h
em
vi
a t
a
rget
e
d
a
d
d
r
e
ssi
ng
schem
e
s. Int
e
rnet
o
f
t
h
i
ngs
(
I
o
T
)
refe
r
s
bac
k
t
o
t
h
e net
w
or
k
i
n
t
e
rco
nnect
i
on
of eve
r
y
d
a
y
devi
ces. A
n
Io
T i
s
a gl
obal
-
vast
com
m
uni
t
y
of i
n
t
e
r
-
l
i
nke
d
devi
ces
uni
qu
el
y
addressa
bl
e, head
q
u
art
e
r
e
d o
n
a us
ual
com
m
uni
cat
ion
pr
ot
oc
ol
. I
n
t
h
e I
o
T,
per
s
on
s are
b
oun
d
e
d
b
y
u
tilizin
g
on
e-of-a-k
i
n
d fo
rm
s of co
m
p
u
tin
g
it
e
m
s wh
ich
m
i
g
h
t
b
e
b
illio
n in
n
u
m
b
e
r, v
a
ri
o
u
s
in
size, and
cap
a
b
ilities to
remain
a co
mm
u
n
i
catio
n
wit
h
e
ach
o
t
h
e
r
d
e
v
i
ce. It is
pred
ictab
l
e th
at aroun
d 50
b
illio
n
su
ch
ob
j
ects
w
ill b
e
in
tercon
n
e
cted
to
th
e in
tern
et b
y
m
ean
s o
f
2020
.Th
e
se D
e
v
i
ces are h
a
v
i
n
g
co
nstrain
e
d
cap
a
b
ilities, and
calcu
latin
g
reso
urces rang
es
fro
m
Rad
i
o
Freq
u
e
n
c
y Id
en
tif
i
catio
n
(RFID)
tag
s
to
em
bedde
d i
n
st
rum
e
nt
s, PD
A,
and se
ns
or
n
o
d
es.
IoT
j
o
i
n
s t
h
e p
h
y
s
i
cal
wo
rl
d wi
t
h
t
h
e i
n
f
o
rm
at
i
on w
o
rl
d, a
n
d
prese
n
ts am
bient offe
rings,
and ap
pl
i
cat
i
o
ns.
The
I
o
T
n
e
t
w
o
r
ks
pe
rm
it
s user
s,
de
vi
ces, an
d
p
u
r
p
o
s
es i
n
uni
que
physical places to
ke
ep up a c
o
rres
p
ondence
se
a
m
lessly with one anot
her. B
r
iefly, the IoT
allows
except
i
o
nal
ve
rbal
e
x
cha
n
ge
pat
t
e
rns l
i
ke:
p
e
rso
n
-
t
o
-
p
e
r
so
n,
pers
o
n
-t
o-
ob
ject
,
ob
ject
-t
o
-
ob
ject
, a
n
d o
b
j
ect
-t
o-
person. Still, the
decentralized a
nd
dispensed nature
of the
IoT face ch
allenges in
authe
n
tication, entry
cont
rol
,
an
d i
d
ent
i
f
i
cat
i
on m
a
nagem
e
nt
. T
h
e
r
e are
m
o
re t
h
an a
fe
w c
h
al
l
e
nge
s t
o
desi
g
n
pr
ot
ect
i
o
n
o
p
t
i
o
n
s
wi
t
h
i
n
t
h
e I
o
T
l
i
k
e co
nst
r
ai
nt
s, an
d
het
e
r
o
g
e
neo
u
s c
o
nve
r
s
at
i
on,
res
o
u
r
c
e
con
s
t
r
ai
nt
s,
a
nd
di
s
p
ense
d
n
a
t
u
re.
Id
en
tity
m
a
n
a
g
e
m
e
n
t
o
f
d
e
v
i
ces in
th
e Io
T is lik
ely o
n
e
o
f
th
e p
r
im
ary
ta
sk
, and
can
b
e
co
m
p
leted
b
y
u
s
ing
effective a
u
thentication sc
he
m
e
s that are
easy, secu
re,
an
d l
i
ght
wei
ght
.
I
n
t
h
e
IoT, the
r
e are
a
b
undant
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
S
ecure
Dig
ita
l
S
i
gn
a
t
u
r
e
S
c
h
e
me Ba
sed
o
n
Ellip
tic Cu
rves fo
r
In
tern
et
o
f
Th
ing
s
(Suman
t
h
K
o
ppu
la
)
1
003
n
u
m
b
e
rs
of
heterog
e
n
e
ou
s t
h
ing
s
ch
attin
g to
each
o
t
h
e
r. Ev
ery sing
l
e
d
e
v
i
ce
will h
a
v
e
t
o
b
e
ab
le to
au
th
en
ticate for th
e
p
e
ri
o
d
of
th
e sho
r
t tim
e. Du
e t
o
th
e
size o
f
econ
o
m
ics,
m
o
re th
an
en
orm
o
u
s
qu
an
titi
es of
th
in
gs m
a
y j
u
st req
u
e
st au
t
h
en
ticatio
n
app
r
ov
al at th
e sa
m
e
ti
me. To
th
is i
n
ten
tion
,
ligh
t
weigh
t
, scalab
l
e
, and
secu
re au
th
en
ticatio
n
sch
e
m
e
is essen
tial wit
h
th
e in
ten
tion
to
au
th
en
ticate o
r
g
a
n
i
zation
s
o
f
d
e
v
i
ces, an
d now
not
t
h
e i
n
di
vi
d
u
al
de
vi
ces t
o
obt
ai
n
com
f
o
r
t
a
bl
e g
r
ou
p c
o
m
m
uni
cat
i
on.
2.
ELLIPTIC CURVE
ARITHMETIC
Ellip
tic cu
rv
e
cryp
tog
r
aph
y
is b
a
sed
on
th
e
arith
m
e
t
i
c o
f
po
in
ts on
an
ellip
tic cu
rv
e [1
],[2
]. Ellip
tic
curves a
r
e c
h
aracterized
by cubic equa
tion
s
alik
e to
tho
s
e u
s
ed
for co
m
p
u
tin
g th
e ci
rcu
m
feren
c
e of an
el
lip
se.
An ellip
tic cu
rv
e E ov
er a
field
K is
d
e
fi
n
e
d
b
y
a eq
u
i
v
a
lence [3
]:
y
2
+ a
1
xy
+ a
3
y =
x
3
+ a
2
x
2
+ a
4
x + a
6
(1
)
Whe
r
e a
1
, a
2
, a
3
, a
4
, a
6
∈
k
and
∆
≠
0,
w
h
er
e
∆
is d
e
fin
e
d
as
fo
llows:
∆
= -d
2
2
d
8
—
8d
3
4
— 2
7d2
6
+
9d
2
d
4
d
6
;
Whe
r
e d
2
= a
1
2
+ 4a
2
,
d
4
=
2a
4
+ a
1
a
3
,
d
6
= a
2
3
+ 4a
6
an
d
d
8
= a
1
2
a
6
+
4a
2
a
6
a
l
a
3
a
4
+ a
2
a
2
3
a
2
4
Set o
f
all p
o
i
nts (x
, y), wh
ich
fu
l
f
ils th
e ab
ov
e equ
a
tion
,
are th
e p
o
i
n
t
s o
n
th
e ellip
tic cu
rv
e. The
q
u
a
n
tity o
f
po
in
ts o
n
an
ellip
tic cu
rv
e, n
,
is th
e o
r
d
e
r of elli
p
tic cu
rv
e, (#
(E(F
p
)). Th
e set o
f
po
in
ts of E (F
p
)
co
m
p
o
s
ed
wit
h
add
itio
n
op
eratio
n
fo
rm
s a
n
ab
elian
group
with
po
in
t at in
fin
ity,
∞
as
th
e id
en
tity ele
m
en
t.
Th
e Equ
a
lity {1
} is called as weierstrass
eq
u
a
tion
.
Th
e co
nd
itio
n
∆
≠
0
en
su
res th
at
th
e ellip
tic curv
e is
pl
ane,
i
.
e, t
h
ere
are
n
o
poi
nt
s a
t
whi
c
h t
h
e
c
u
r
v
e
has t
w
o
o
r
m
o
re di
ver
g
e
n
t
t
a
nge
nt
l
i
n
es
.
If t
h
e
fi
el
d r
e
p
r
esent
a
t
i
v
e P
i
s
not
e
q
ual
t
o
2
or
3 i
.
e., prim
e
field, and t
h
en
t
h
e
perm
i
ssi
bl
e cha
nge
o
f
Varia
b
les (
x
,
y
)
→
((
x-
3a
1
2
-12a
2
)/36
, (y
-3a
1
x
)
/2
16
- (a
1
3
+4a
1
a
2
-1
2a
3
)/36) tran
sfo
r
m
E to
t
h
e curv
e,
Y
2
=
x
3
+ a
x
+
b; whe
r
e a, b
∈
k
(2
)
The
∆
is 16(
4
a
3
+ 27b
2
)
2.
1.
Point Additi
on
To
talin
g
o
f
po
i
n
ts o
n
an
ellip
tic cu
rv
e is d
e
fi
n
e
d
b
y
Ch
o
r
d
an
d
Tan
g
en
t rule. Let P = (x
1
, y
1
) and Q =
(x
2
, y
2
)
b
e
th
e t
w
o
d
i
ssim
i
lar p
o
i
n
t
s on
an
ellip
tic cu
rv
e
E.
Th
en
th
e su
m
R, o
f
P and
Q,
is d
e
fin
e
d
as
follo
ws:
Draw a lin
e att
ach
ing
P an
d
Q sp
read it to
i
n
tersect t
h
e ellip
tic curv
e at a
th
ird po
in
t.
At
th
at po
in
t, th
e
su
m
R,
i
s
t
h
e ne
gat
i
v
e
of t
h
e t
h
i
r
d p
o
i
n
t
.
Ne
gat
i
v
e
of
a poi
nt
i
s
defi
ned
by
re
fl
ect
i
on
of t
h
e p
o
i
n
t
near t
h
e x
-
axi
s
. The
d
oub
le R, of P, is d
e
fi
n
e
d
as fo
llo
ws: Draw t
h
e tang
en
t lin
e to
th
e ellip
tic cu
rv
e at P. Let
it in
tercon
n
ect
s th
e
ellip
tic cu
rv
e at an
o
t
h
e
r po
in
t. Th
en
t
h
e
d
ouble R is
th
e
reflectio
n
o
f
th
is
p
o
in
t n
e
ar th
e x-ax
is.
2.
2.
Po
int Multiplica
t
i
o
n
It is also
k
n
o
wn
as Scalar m
u
ltip
licatio
n
.
It t
h
e arith
m
e
tic
o
p
e
ration
wh
ich calcu
lates k
p
wh
ere k
i
s
an
i
n
teg
e
r an
d
p
is
a
p
o
i
n
t
o
n
ellip
tic cu
rv
e.
It is co
m
p
leted
b
y
rep
e
titiv
e ad
d
ition
.
Fo
r instan
ce
Q = kp
mean
s
Q i
s
achi
e
ved
by
addi
ng
p*
k t
i
m
e
s t
o
i
t
s
el
f (
p
+ p + p...
.k t
i
m
es). C
r
y
p
t
a
na
l
y
si
s i
nvol
ves
det
e
rm
i
n
i
ng k gi
v
e
n
P and
Q. Th
is
p
r
o
c
ed
ure
d
o
m
in
ates th
e im
p
l
e
m
en
ta
tio
n
time of ellip
tic curv
e cryp
tog
r
aphic sch
e
m
e
s.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
10
0
2
– 10
10
1
004
Fig
u
re
1
.
Th
e im
p
l
e
m
en
tatio
n
ti
m
e
o
f
ellip
tic curv
e cryp
t
o
grap
h
i
c
sch
e
m
e
s
2.
3.
Operations de
fined
for
E(F
p
): y
2
= x
3
+a
x +b
1.
Identity:
P +
∞
=
∞
+ P
= P
f
o
r
a
ll >
∈
E(
F
p
)
2.
N
e
gat
iv
es:
If
P = (x,y
)
∈
E(
F
P
), then (
x
,y
) + (x, -y
) =
∞
. The p
o
i
n
t
(
x
, -y
) i
s
den
o
t
e
d by
–P an
d i
s
cal
le
d
n
e
g
a
tiv
e of
P.
No
te t
h
at P i
n
deed
is a po
in
t i
n
E(F
P
).
3.
Point Addi
tio
n
:
Let P=(x
1
,y
1
)
∈
E(
K) a
nd
Q
=
(x
2
,y
2
)
∈
E(
K)
; where P
≠
+
Q,
th
en
P+Q
= (x
3
,y
3
) w
h
ere
,
x
3 =
(
y
2
- y
1
/ x
2
– x
1
)
2
- x
1
– x
2
and
y
3 = (
y
2
- y
1
/ x
2
– x
1
)
2
(x
1
– x
3
) -
y
1
4.
Po
int Do
ubling
:
Let P=
(x
1
,y
1
)
∈
E(
K
)
, the
n
2P =
(x
3
,y
3
)
w
h
ere
,
x
3 =
(3
x
1
2
+ a / 2 y
1
)
2
- 2x
1
and y
3 =
(3x
1
2
+
a / 2 y
1
)
2
(
x
1
– x
3
)
- y
1
2.
4.
Elliptic Curve
Discrete
logarithm pr
oble
m
Assu
m
e
d
ellip
t
i
c cu
rv
e
p
a
ram
e
ters and
a p
o
i
n
t
P
∈
E(F
p
), fi
nd t
h
e u
n
i
q
ue i
n
t
e
ge
r k,
0
≤
k
≤
n
1
,
s
u
c
h
th
at P=k
G
, where n1
is o
r
d
e
r o
f
E. ECDLP is alik
e to
th
e
Discrete Log
a
rith
m
Pro
b
l
em an
d
is th
e elli
p
tic
cur
v
e
refe
rent
of
DLP
.
I
n
t
h
e
ECDLP
,
the s
u
b
g
r
o
u
p
Z
p
*
is altered
b
y
th
e
g
r
ou
p
o
f
po
in
t
s
on
an ellip
tic cu
rv
e
o
v
e
r a fin
ite
field
.
In add
ition
,
un
lik
e t
h
e
Discrete Log
a
rith
m
Prob
lem
a
n
d in
teg
e
r
fact
o
r
ization
p
r
ob
le
m
,
n
o
su
b expo
n
e
n
tial-ti
m
e
alg
o
r
ithm is k
n
o
wn
for th
e EC
DLP.
ECDLP is consid
ered
to
b
e
sig
n
i
fican
tly stro
ng
er
th
an
DLP, th
erefo
r
e ellip
tic
cu
rv
e si
g
n
a
t
u
re sch
e
m
e
g
i
v
e
s a greater st
ren
g
t
h
-
p
e
r-k
ey-b
it th
an th
eir
d
i
screte
l
oga
ri
t
h
m
i
c count
e
r
pa
rt
s.
3.
ELLIPTIC CURVE
CRYPTOGRAPHY
Th
e u
s
ag
e of Ellip
tic Cu
rv
e Cryp
tog
r
aph
y
was p
r
im
ar
ily
ad
v
i
sed
b
y
Neal Ko
b
litz [4
] an
d
Victo
r
S.
Miller [5
]. Ell
i
p
tic cu
rv
e cryp
to
system
s
o
v
e
r fi
n
ite field
h
a
v
e
so
m
e
b
e
n
e
fits lik
e t
h
e k
e
y size can
b
e
co
nsid
er
ab
ly smaller
co
m
p
ar
ed
to add
itio
n
a
l cr
yp
to
systems lik
e RSA,
D
i
f
f
i
e-H
e
llm
an
si
n
ce
o
n
l
y ex
ponen
tial-
ti
m
e
at
tack
is
k
nown so far i
f
th
e curv
e is
carefu
lly
ch
osen
[4
],[6
] an
d
Ellip
tic Cu
rv
e
Cryp
tog
r
aph
y
d
e
p
e
n
d
o
n
t
h
e d
i
fficu
l
ty o
f
exp
l
ain
i
ng
th
e Ellip
tic Cu
rv
e Di
screte Lo
g
a
rith
m
P
r
ob
lem
ECDLP, wh
ich
states th
at,
“Giv
en
an
ellip
tic cu
rv
e E
well-d
e
fined
on
a fin
ite field
F
P
, a po
in
t P
∈
E (
F
P
)
of a
n
or
d
e
r n
,
an
d a
poi
nt
Q
∈
E
(F
P
) , fi
nd t
h
e i
n
t
e
ge
r k
∈
[0
,n
−
1]
such t
h
at
Q = k P. The i
n
t
e
ge
r k i
s
na
m
e
d as t
h
e di
scret
e
l
ogari
t
h
m
of Q t
o
t
h
e ba
se P
,
den
o
t
e
d
k =
l
o
g
P
Q”.
3.
1.
Elliptic Curve
Encry
ption/Decryption
Co
n
s
i
d
er a m
e
ssag
e
‘Pm
’
d
i
rected
fro
m
A t
o
B. ‘A’ p
i
ck
s a ran
d
o
m
p
o
s
itiv
e in
teg
e
r ‘k’, a p
r
i
v
ate
key
‘
n
A
’
a
n
d
pr
o
duces
t
h
e
p
ubl
i
c
key
P
A
= n
A
×
G a
n
d
p
r
od
uces t
h
e ci
p
h
er t
e
xt
‘C
m
’
be m
a
de u
p
of
pai
r
o
f
poi
nt
s C
m
= {kG
,
P
m
+ kP
B
} wh
ere
G is the b
a
se
po
in
t selected
on
th
e
Ellip
tic Cu
rv
e, P
B
=
n
B
×
G i
s
the
pu
bl
i
c
key
of
B
wi
t
h
pri
v
at
e
key
‘
n
B
’
. T
o
decry
p
t
t
h
e
ciph
er tex
t
, B reprodu
ces th
e
1
s
t p
o
i
n
t
in
th
e pair by
B’s secret &
ded
u
c
ts th
e
result fro
m
th
e
2
nd po
in
t P
m
+ kP
B
–
n
B
(k
G)
=
P
m
+ k(n
B
G) – n
B
(k
G) =
P
m
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
S
ecure
Dig
ita
l
S
i
gn
a
t
u
r
e
S
c
h
e
me Ba
sed
o
n
Ellip
tic Cu
rves fo
r
In
tern
et
o
f
Th
ing
s
(Suman
t
h
K
o
ppu
la
)
1
005
4.
DIGIT
A
L SI
GNATURE S
C
HE
MES
Dig
ital sign
atures are b
e
i
n
g used
to attain
integ
r
ity, non
-rep
ud
iatio
n and
au
th
en
ticatio
n
o
f
t
h
e
d
i
g
ital
dat
a
i
n
t
r
ansm
i
ssi
on am
ong
di
ssi
m
i
l
a
r end
users
.
Di
gi
t
a
l
si
gnat
u
re o
ffe
rs co
rrect
arc
h
i
t
ect
ure fo
r se
ndi
ng
secu
re m
e
ssag
e
s b
y
way of
u
tilizin
g
ex
ceptio
n
a
l algo
rithm
s
. Th
e d
i
g
ital sig
n
a
t
u
re al
go
rith
m
s
co
mmo
n
l
y
co
nsistin
g
of
t
h
r
e
e sub
ph
ases:
1)
Key
ge
nerat
i
on
sy
m
m
e
t
r
i
c
or
asy
m
m
e
t
r
i
c
al
go
ri
t
h
m
.
2
)
Sign
ing
al
go
rith
m
.
3) Signature
ve
rification algorith
m
The symmetric
key algorithm gene
rates single key th
at is share
d
by se
nder and recei
ver. On ot
her
han
d
, t
h
e asy
m
m
e
t
r
i
c
key
al
gori
t
h
m
generat
e
s t
w
o
key
s
:
p
ubl
i
c
an
d
pri
v
a
t
e key
s
. The
p
ubl
i
c
key
s
a
r
e share
d
bet
w
ee
n t
w
o p
a
rt
i
e
s;
i
n
cont
r
a
st
t
h
e pri
v
at
e
key
s
are kee
p
i
ng sec
r
et
. D
u
ri
ng sec
o
nd
pha
se si
gni
n
g
al
g
o
r
i
t
h
m
th
e d
i
g
ital sig
n
a
ture is gen
e
rated
b
y
tak
e
n
p
l
ain
tex
t
i.e.
p
r
i
v
ate k
e
y,
sen
s
itiv
e d
a
ta, an
d m
e
ssag
e
as in
pu
t.
Aft
e
r
t
h
at
, t
h
e
sen
d
er se
n
d
s t
h
e m
e
ssage al
on
g
wi
t
h
gene
rat
e
d si
gnat
u
re
t
o
t
h
e i
n
t
e
n
d
e
d
reci
pi
ent
.
Si
gnat
u
r
e
verification al
gorithm
is executed at
recipi
ent end t
o
e
n
s
u
re t
h
e
receive
d
data [7].
A
valid
digital signat
u
re
gives a
receiver the
reason t
o
adm
it
m
e
ssa
ge and e
n
s
u
re
the m
e
ssage was created a
nd comm
unicate
d
by a
k
nown sen
d
e
r, no
t altered in
tran
sit.
Dig
i
tal sig
n
a
tu
re
h
a
s n
u
m
e
rous
s
c
hem
e
s, such
as R
S
A
,
DS
A
an
d
ECDSA,
wh
ich
are
u
s
ed to im
p
o
s
e th
e secu
rity of
d
i
ff
eren
t tran
saction
[7
].
Dig
ital sign
ature sch
e
m
e
s were
en
h
a
n
c
ed
in
ord
e
r to
o
v
e
rcome so
m
e
o
f
vu
ln
erab
ilities. So
m
e
i
m
p
r
o
v
e
men
t
tech
n
i
ques o
f
d
i
g
ital si
g
n
a
t
u
re
sch
e
m
e
s are attain
ed
with
re
s
p
ect to va
rious
perceptions.
Fi
gu
re
2.
The
i
m
provem
e
nt
t
echni
que
s
of
di
gi
t
a
l
si
gnat
u
re
schem
e
s t
h
at
are at
t
a
i
n
ed
wi
t
h
respect
t
o
var
i
ous
perce
p
t
i
o
ns
In
R
S
A, it is fau
lt to
leran
ce
persp
ectiv
e,
whereas
in DSA,
they are s
p
ee
d
of
operation c
o
m
putational
pers
pect
i
v
e an
d l
o
n
g
t
i
m
e of com
put
at
i
ons
pers
pect
i
v
e.
And in EC
DSA, they are e
fficiency pers
pective and
spee
d of o
p
erat
i
on
c
o
m
put
at
i
o
nal
pe
rs
pect
i
v
e
5.
PERFO
R
MA
NCE CO
MP
A
R
ISO
N
The pe
rform
a
nce m
easurements ha
ve bee
n
categ
orized according to the depe
nde
n
t varia
b
les.
Refere
nces [8] regarding the
chosen
algo
rit
h
m
s
with
resp
ect to
th
eir p
e
rform
a
nce and com
p
ared to the level
o
f
security p
r
ov
id
ed
.
Tabl
e
1. T
h
e
p
e
rf
orm
a
nce m
e
asurem
ent
s
acc
or
di
n
g
t
o
t
h
e
d
e
pen
d
e
n
t
va
ri
a
b
l
e
s
Algorit
hm
F
a
m
i
ly
Security
lev
e
l
(in
bits)
80
128
192
256
RSA I
n
teger
factor
izatio
n
1024
3072
7680
1536
0
DSA Discr
e
te
logar
ith
m
1024
3072
7680
1536
0
ECDSA
Elliptic
Curves
160
256
384 512
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
10
0
2
– 10
10
1
006
R
S
A a
nd
DS
A
al
go
ri
t
h
m
s
are sus
p
en
de
d
fr
o
m
im
provem
e
nt
of t
h
ei
r
per
f
o
r
m
a
nce for
t
h
e
reas
on t
h
at
in
stallin
g
su
ch alg
o
rith
m
o
n
lig
h
t
-weigh
t
d
e
v
i
ces
will
ad
v
e
rsely affect th
eir p
e
rforman
ces an
d
d
e
l
a
y th
e
d
ecry
p
tio
n
p
r
ocess. ECDSA in
equ
i
v
a
len
t
co
u
l
d
b
e
a
aux
iliary fo
r RSA & DSA syst
e
m
, th
eir co
m
p
tab
ilit
y to
b
e
in
stalled
in
an
y syste
m
with
d
i
fferen
t
m
e
m
o
ry si
zes and
C
P
U
descri
pt
i
on a
n
d pa
ram
e
t
e
rs, EC
D
S
A
p
r
o
v
i
d
e
th
e sam
e
lev
e
l
o
f
secu
rity as RSA and
DSA b
u
t
with
sm
all
e
r k
e
ys: Th
e lesser k
e
y sizes
o
f
EC
DSA po
ssib
ly
al
l
o
w f
o
r l
e
ss
com
put
at
i
onal
l
y
abl
e
l
i
ght
-
w
e
i
ght
de
vi
ces a
n
d wi
rel
e
s
s
sy
st
em
s t
o
use cry
p
t
o
gra
p
hy
fo
r
secure
dat
a
t
r
ansm
i
ssions
, dat
a
ve
ri
fi
cat
i
on an
d o
ffe
rs l
e
ss heat
ge
nerat
i
o
n an
d l
e
ss po
wer c
o
n
s
um
pt
i
on, l
e
ss st
ora
g
e
space a
n
d
offers an optimized me
m
o
ry and
bandwidt
h a
n
d faster signature
gene
ration.
6.
E
X
ISTING SYSTEM
Th
e sch
e
m
e
is
ap
t fo
r a signer who
h
a
s limited
co
m
p
u
t
i
n
g
cap
a
b
ility l
i
k
e
, a si
g
n
e
r usin
g
h
i
s sm
art
C
a
rd
whi
c
h st
o
c
ks
hi
s secret
k
e
y
and
di
spl
a
y
s
a m
e
ssage on
a ext
e
r
n
al
Key
pai
r
phase
o
f
t
h
i
s
schem
e
i
s
sam
e
as the EC
DSA
schem
e
.
6.
1.
Si
gn
atu
re Ge
neration
Usi
n
g se
n
d
er
’
s
p
r
i
v
at
e
key
,
sen
d
e
r
gene
r
a
t
e
s t
h
e si
gnat
u
re
f
o
r
m
e
ssage M
usi
n
g
t
h
e su
bse
q
uent
steps:
(1) Select a
un
i
q
u
e
and
un
pred
ictab
l
e in
teg
e
r
k
in th
e i
n
terv
al [1
,n
-1
]
(2
) C
o
m
pute k
g
=
(
x
1
,y
1
), where x
1
is an
in
t
e
g
e
r
(3
) C
o
m
pute r
= x
1
m
od
n;
I
f
r =
0,
t
h
e
n
go
t
o
st
e
p
1
(4
) C
o
m
put
e h
=H(M
)
,
whe
r
e
H i
s
t
h
e S
H
A
-
51
2
[
9]
(5
) C
o
m
pute s
= k
-1
(h
+ dr
) mo
d n
;
I
f
s = 0, t
h
en go
t
o
step
1
(6) Th
e si
g
n
a
t
u
re
o
f
send
er fo
r m
e
ssag
e
M is th
e in
teger
p
a
ir (r, s)
6.
2.
Signature Ver
i
fication
The recei
ver c
a
n aut
h
orize the authe
n
ticity
of se
nder’s signature (r, s)
for
m
e
ssage M with the aid of
execution
t
h
e followi
ng:
(
1
)
O
b
tain
signato
r
y
A
’
s
pub
lic
k
e
y (
E
, q
,
n, Q
)
(2
)
Verify
t
h
at
values
r a
n
d
s
are in the i
n
terv
al [1
,n
-1
]
(3
) C
o
m
pute w
= s
-1
m
od
n.
(4) C
o
m
pute h
= H(M
)
,
where
H is t
h
e sam
e
secure
ha
sh algorithm
used by
A.
(5
) C
o
m
put
e u
1
=
h
w
m
od
n
(6
) C
o
m
pute u
2
=
rw
m
od
n
(7
) C
o
m
put
e u
1
G
+
u2
Q =
(
x
0,y
0
)
(8
) C
o
m
put
e v
= x
0
m
od
n
(9
) T
h
e si
gnat
u
re f
o
r m
e
ssage
M
is ve
rified
o
n
ly
if
v =
r
6.
3.
A
P
o
ssible attack
Th
e secret k
e
y
k
u
s
ed
for sig
n
i
n
g
t
w
o
o
r
m
o
re
m
e
ssag
e
s will h
a
v
e
to
b
e
produ
ced
sep
a
rately. In
part
i
c
ul
a
r
, a
ddi
t
i
onal
secret
k
sho
u
l
d
be
use
d
fo
r si
g
n
i
n
g sel
ect
m
e
ssages, i
n
any
ot
he
r cas
e t
h
e p
r
i
v
at
e k
e
y
d
can al
so
be
rec
ove
re
d.
Ho
we
ver i
f
a ra
n
d
o
m
or p
s
eu
d
o
-
r
a
nd
om
num
ber gene
rat
o
r i
s
u
s
ed, t
h
en t
h
e t
h
reat
of
m
a
ki
ng a
repe
at
ed k
val
u
e i
s
ne
gl
i
g
i
b
l
e
.
I
f
sam
e
secr
et k
is u
s
ed
t
o
pr
odu
ce si
gnature of two
different
m
e
ssages m
1
and
m
2
th
en
and th
ere it will effect in
t
w
o sign
atures
(r,s
1
) a
n
d
(
r
, s
2
).
s
1
= k
-1
(h
1
+
dr)
s
2
= k
-1
(h
2
+ dr)
;
wh
er
e h1
=
SHA
512
(
m
1
)
an
d h2
= SH
A5
12
(
m
2
).
ks
1
- ks
2
= h1
+dr
-
h
2
-d
r
k
=
(h
1-
h2)
/ (
s
1
-
s2
)
d =
(
k
s-
h)/
r
7.
PROP
OSE
D
SYSTE
M
In this
va
riant
there
is
no
need t
o
find invers
e
in each
key
gene
ration and si
gning s
ection. This
schem
e
i
s
deve
l
ope
d
wi
t
h
o
u
t
m
odul
ar i
n
ve
rs
i
on
p
r
oces
s i
n
Si
gnat
u
re
ge
ne
rat
i
o
n
an
d
Veri
fi
cat
i
on al
go
ri
t
h
m
s
.
7.
1.
No
ta
tio
n
s
To
be a
p
pr
op
ri
at
e i
n
ex
pl
a
n
at
i
o
n
o
f
our
work the elem
ents are
define
d a
s
d:
pri
v
ate key
Q:
Pub
lic k
e
y
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
S
ecure
Dig
ita
l
S
i
gn
a
t
u
r
e
S
c
h
e
me Ba
sed
o
n
Ellip
tic Cu
rves fo
r
In
tern
et
o
f
Th
ing
s
(Suman
t
h
K
o
ppu
la
)
1
007
m:
m
e
ssage
H(
) :
a
sec
u
re
one
-
w
ay
has
h
f
unct
i
o
n
r, s
1
, s
2
:
Si
gnature
elem
ents
q:
fi
el
d or
der
FR:
field rep
r
esen
tatio
n
a, b:
coe
fficients
G:
b
a
se po
in
t
n:
Or
der
o
f
G
h:
c
o
-factor
7.
2.
Key pair Gen
erati
on
Key
pai
r
d a
n
d
Q m
a
de by
t
h
e
Si
g
n
er
as
fol
l
o
ws
INP
U
T:
D= (q,
FR, a, b,
G
,
n,
h)
(
1
)
Ch
oo
se a d
i
sti
n
ctiv
e an
d unpr
ed
ictab
l
e i
n
teg
e
r,
d
,
w
ith
in th
e in
ter
v
al
[1
,
n
-
1
]
(2
)
C
o
m
put
e
Q
←
(d
g)
(3
)
Return
(
Q
,
d
)
OUTP
UT:
Q,
d
7.
3.
Signature Ge
neration
The si
gne
r ca
n
si
gn
m
e
ssage m
as fol
l
o
ws
INP
U
T:
D= (q,
FR, a, b,
G
,
n,
h), d,
m
Beg
i
n
repeat
k = R
a
n
dom
[1
, 2
,
…,
n
-
1]
P =
kG
c=X
-
Co
-o
rd
in
ate (
P
)
e = H
(m
) m
od n
s
1
= eck m
od n
s
2
= (d
c + 1)
k
m
o
d
n
R = eP
r =
X-C
o
-
o
rdi
n
ate(R)
u
n
til r
≠
0 an
d s
1
≠
0 a
n
d s
2
≠
0 retur
n
(r
,
s
1
, s
2
)
End
OUTP
UT:
Sign
atur
e (r
, s
1
, s
2
)
7.
4.
Signature Ver
i
fication
To
veri
fy
the si
gnat
u
re
(
r
, s
1
, s
2
) on m
e
ssage
m
,
receiver
does the
following:
INP
U
T:
D= (q,
FR, a, b,
G
,
n,
h), Q,
m
,
Signature (r
,
s
1
, s
2
)
Beg
i
n
ifr, s
1
, s
2
do
esn’
t
b
e
l
o
ng
s
t
o
[
1
,…, n-
1
]
th
en
Return (“Re
jec
t
the signature”
)
end if
e= H(m
)
t = es
2
U1
= tG
U2
= s
1
Q
W=
U1
–
U
2
v
=
X
-
Co
-o
rd
inate
(W
)
if v = r th
en
Return (“
Acce
pt the
signature”)
else
Return
(“Re
je
ct the si
gnature”)
end if
end
OUTP
UT:
Ac
cept
a
nce
o
r
rej
ect
i
on of
t
h
e si
gnat
u
re
.
7.
5.
Proo
f
Of
Signa
t
u
re Verification
We begi
n wi
t
h
W=U
1
-U
2
By su
b
s
titu
ting U1
with
tG and
U2 with s
1
Q
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
10
0
2
– 10
10
1
008
W = t
G
- s
1
Q
By su
b
s
titu
ting t with
es
2
an
d Q wi
t
h
dG
W = e
s
2
G -
s
1
dG
By su
b
s
titu
ting s
1
with ec an
d
s
2
w
ith
(d
c +
1)
k
W
= e(
dc +
1
)
k
G
– ec
dG
= edc
k
G + e
k
G
– ec
kdG
= ekG
= eP
= R
v =
X
-C
o-
or
di
n
a
t
e
(
W) a
n
d
r =
X-C
o
-
o
rdi
n
ate(R)
There
f
ore
v =
r.
K can
not
be r
e
sol
u
t
e
al
t
h
o
u
gh si
m
i
l
a
r secret
key
i
s
used t
o
si
gn t
w
o
di
ffe
re
nt
m
e
ssages. S
o
t
h
i
s
Syste
m
is not
vulne
r
able t
o
attack
on sam
e
secret.
8.
RESULTS
A
N
D
DI
SC
US
S
I
ON
EC
C
can be i
m
pl
em
ent
e
d i
n
soft
wa
re an
d har
d
ware [
10]
.
Soft
wa
re EC
C
im
pl
em
ent
a
ti
on p
r
o
v
i
d
e
m
oderat
e
spee
d,
hi
g
h
e
r
po
w
e
r c
ons
um
pt
i
on a
n
d
al
so
ha
v
e
ve
ry
l
i
m
i
t
e
d phy
si
cal
sec
u
ri
t
y
w.r
.
t
key
st
ora
g
e
.
Wh
ere as h
a
rdware im
p
l
e
m
en
tatio
n
imp
r
ov
es
p
e
rfor
man
ce in
term
s o
f
flex
ib
i
lity. Also
h
a
rdware
i
m
p
l
e
m
en
tatio
n
p
r
ov
id
es greato
r security
since they cannot be easily
m
o
dified
or rea
d
by an outsi
de attacker.
Th
is section
rep
r
esen
ts im
p
l
emen
ta
t
i
on re
su
l
t
s
of
o
u
r
Pr
o
p
o
se
d Sc
hem
e
Basep
o
i
n
t
=
(
4
2
582
623
172
388
835
044
654
159
270
140
906
591
363
556
877
0,
2
035
201
141
629
041
078
739
914
579
573
468
920
279
826
419
70)
Jay
a
B
h
as
kar i
n
gen
k
ey
Basep
o
i
n
t::g
enK
e
y =
(425
8262
317
238
883
504
465
415
927
014
090
659
136
355
687
70
,
2
035
201
141
629
041
078
739
914
579
573
468
920
279
826
419
70)
p
r
i
v
ate_A
=
340
282
366
920
938
463
463
374
607
431
768
211
455
Ellip
ticCu
rv
e:
y^2
= x
^
3
+ 146
150
163
733
090
291
820
368
483
271
628
301
965
378
505
932
4x +
1
632
357
913
061
681
105
466
049
194
032
715
795
305
483
454
13 (
m
o
d
1
461
501
637
330
902
918
203
684
832
716
283
019
653
785
059
327
)
created s
u
ccess
f
ully!
p
u
b
lic_A
=
(1
93
596
275
460
689
438
633
057
135
026
141
223
361
451
460
712
,
8
525
856
310
300
448
737
103
525
015
533
331
483
771
456
661
26)
Pub
lic_
A
on
t
h
e curv
e is tru
e
8.
1.
Si
gn
atu
re Ge
neration
Select r
a
ndo
m
n
u
m
b
e
r
= 146
15
016
373
309
029
182
036
871
976
068
267
798
841
648
049
61
C
o
m
put
e base
poi
nt
*
ra
n
dom
n
u
m
b
er=P
P = (531
158
657
844
619
155
995
167
414
799
432
702
697
095
257
705
,
1
803
449
186
458
949
746
512
182
733
289
892
184
316
805
095
76)
c = 531
158
6578
446
191
559
951
674
147
994
327
026
970
952
577
05
h
e
x
:
-
5d5
2
e
9
c
b5
d88
9f6
dd7
ab9
f
28
415
d2
c7
bfd
865
9f3
d
ec:-5
327
851
69
157
166
761
525
766
418
400
736
964
836
206
205
427
has
h
:
[
B
@
1c5
f
de0
Ori
g
i
n
al
M
e
ssage:
Pa
ul
hat
e
d sch
o
o
l
.
He di
d n
o
t
d
o
hi
s hom
e
e = 928
716
4681
737
361
566
779
207
792
060
898
150
484
372
870
12
s
1
=eck
=738
697
728
917
748
481
767
219
149
975
279
592
392
030
137
000
964
288
383
764
618
857
587
084
944
0
2
226
606
005
217
654
813
6
s
2
=(
d
c
+1
)k-
-
>
1
704
271
359
825
084
431
055
318
977
373
884
102
275
096
852
04
R=eP = (98
9057
722
868
231
206
769
763
389
899
805
110
651
529
187
912
,
2
033
914
330
947
679
125
956
003
962
783
622
185
995
185
289
12
r
=
x-
co-o
rd
(
R
)
=
98
905
772
286
823
120
676
976
338
989
980
511
065
152
918
791
2
8.
2.
Si
gn
atu
re Ve
r
i
ficatio
n
h
e
x
:
-
5d5
2
e
9
c
b5
d88
9f6
dd7
ab9
f
28
415
d2
c7
bfd
865
9f3
d
ec:-5
327
851
69
157
166
761
525
766
418
400
736
964
836
206
205
427
has
h
:
[
B
@
1b
5
3
40c
Ori
g
i
n
al
M
e
ssa
ge:
Pa
ul
hat
e
d
sch
ool
.
He
di
d
not
d
o
hi
s
hom
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
S
ecure
Dig
ita
l
S
i
gn
a
t
u
r
e
S
c
h
e
me Ba
sed
o
n
Ellip
tic Cu
rves fo
r
In
tern
et
o
f
Th
ing
s
(Suman
t
h
K
o
ppu
la
)
1
009
E =
9
287
164
68
173
736
156
677
920
779
206
089
815
048
437
287
012
t=es
2
G
=
6
6054
246
628
614
516
487
199
143
288
001
409
883
455
206
929
4
Co
m
p
u
t
e U1
=tG
=
(
8676
564
30
810
165
309
875
458
600
806
608
047
488
460
704
882
,
1
293
539
316
315
317
053
662
292
883
889
929
244
439
313
430
206
)
C
o
m
put
eU2=s
1
Q
=
(
717
328
24
241
819
992
935
376
226
987
838
881
939
840
564
741
8,
1
032
047
338
562
966
998
576
808
247
653
816
832
960
909
777
35)
Co
m
p
u
t
eW
=U1
-
U
2
=
(
989
057
722
868
231
206
769
763
389
899
805
110
651
529
187
912
,
2
033
914
330
947
679
125
956
003
962
783
622
185
995
185
289
12)
v
=
x-
co
or
d(W
)
=
98
905
772
286
823
120
676
976
338
989
980
511
065
152
918
791
2
We
obtain v=r, he
nce Si
gnat
u
re is acce
pted
We com
p
are t
h
e res
u
l
t
s
o
f
E
C
DSA a
n
d o
u
r
pr
op
ose
d
sy
st
em
t
h
at
presen
t
s
t
h
e no
o
f
P
o
i
n
t
Ad
di
t
i
on
an
d
Scalar Multip
licatio
n
o
p
e
ratio
n
s
for Sign
ing
and
Si
gn
atu
r
e Verificatio
n
p
r
o
cess. ECDSA
u
s
es i
n
v
e
rsi
on
ope
rat
i
o
n i
n
b
o
t
h
si
g
n
i
n
g an
d Si
g
n
at
ure
V
e
ri
fi
cat
i
on b
u
t
our P
r
o
p
o
se
d
Sy
st
em
doesn
’t
use any
i
n
v
e
rsi
o
n
ope
rat
i
o
ns i
n
Si
gni
ng
a
n
d
S
i
gnat
u
re
Ve
ri
f
i
cat
i
on.
We i
m
pl
em
ent
e
d o
r
i
g
i
n
al
EC
DS
A a
n
d
ou
r
pr
op
os
e
d
sch
e
m
e
an
d
com
p
ared
th
eir perfo
r
m
a
n
ce over Ellip
tic Curv
e an
d presen
ted
th
e resu
lts
belo
w.
Tabl
e
2.
Im
pl
em
ent
e
d o
r
i
g
i
n
a
l
EC
DS
A a
n
d
ou
r
pr
o
pos
ed
s
c
hem
e
and c
o
m
p
ared t
h
ei
r
p
e
rf
orm
a
nce o
v
e
r
Ellip
tic Cu
rv
e
Algor
ith
m
E
C
DSA
Pr
oposed
Algor
ith
m
No.
of Secr
et key
s
1
1
I
nver
s
e in Signing
Yes
No
No.
of scalar M
u
ltiplication oper
a
tions in signing
1
2
Inverse in Signature Verification
Yes
No
No.
of Point Addition oper
a
tions in
Ver
i
fication
1
1
No.
of scalar
M
u
ltiplication oper
a
tion
s
in Ver
i
fication
2
2
From
Fi
gu
re 3 Pr
o
p
o
s
ed S
i
gnat
u
re sc
he
m
e
im
pl
em
ented p
o
o
rl
y
i
n
si
gnat
u
re
gene
rat
i
on si
nc
e
secu
rity is inv
e
rsely propo
rtion
a
l to
p
e
rfo
r
m
a
n
ce
o
f
th
e syste
m
.
0
20
40
60
80
10
0
Pro
p
os
e
d
Sy
st
e
m
EC
D
S
A
Fi
gu
re
3.
Pr
o
p
o
se
d Si
gnat
u
re
schem
e
From
Fi
gu
re
4
,
p
r
o
p
o
sed
sch
e
m
e
si
gnat
u
re
veri
fi
cat
i
o
n al
go
ri
t
h
m
perf
or
m
e
d bet
t
e
r
wh
en c
o
m
p
ared
t
o
t
h
e exi
s
t
i
n
g
veri
fi
cat
i
o
n s
c
hem
e
. Thi
s
i
s
desi
ra
bl
e be
cause t
o
t
h
e a
ppl
i
cat
i
o
n-
ori
e
nt
ed
poi
nt
of
vi
ew,
m
e
ssage i
s
a
u
t
h
o
r
i
zed
by
t
h
e
i
ndi
vi
dual
o
n
l
y
o
n
ce,
b
u
t
ve
ri
f
i
cat
i
on m
a
y
be req
u
i
r
e
d
m
a
ny
t
i
m
e
s.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
10
0
2
– 10
10
1
010
Fi
gu
re
4.
Pr
o
p
o
se
d sc
hem
e
signat
u
re
ve
ri
fi
c
a
t
i
on al
g
o
r
i
t
h
m
Ou
r a
p
pl
i
cat
i
ons
req
u
i
r
i
n
g
Si
gnat
u
re
ve
ri
fi
c
a
t
i
on
m
o
re fre
que
nt
l
y
t
h
a
n
S
i
gnat
u
re Ge
ner
a
t
i
on, he
nce
pr
o
pose
d
s
c
he
m
e
i
s
best
s
u
i
t
a
bl
e f
o
r
I
n
t
e
r
n
et
o
f
T
h
i
n
gs.
9.
CO
NCL
USI
O
N
In t
h
e E
x
i
s
t
e
nc
e sy
st
em
,
i
f
t
h
e sam
e
random
num
ber i
s
gen
e
rat
e
d w
h
i
c
h i
s
used t
o
si
g
n
t
h
e m
e
ssage,
then t
h
ere
is a
chance
of
dec
r
ypting
t
h
e
pri
v
ate key
by the
attacker. B
u
t i
n
ou
r pr
o
p
o
s
ed
schem
e
,
eve
n
i
f
t
h
e
sam
e
rand
om
n
u
m
b
er i
s
used at
t
acker can
’t
decry
p
t
t
h
e pri
v
at
e key
.
M
o
d
u
l
a
r i
n
versi
on
ope
rat
i
o
n i
s
addi
t
i
onal
t
i
m
e
cons
um
i
n
g o
p
erat
i
o
n [
1
1]
for c
onst
r
ai
ned
devi
ces
.
O
u
r p
r
op
ose
d
Di
gi
t
a
l
Si
gnat
u
re
schem
e
i
s
devel
ope
d
wi
t
h
o
u
t
m
odu
l
a
r i
nve
rsi
o
n pr
ocess i
n
Si
gnat
u
re
gene
r
a
t
i
on an
d Ve
r
i
fi
cat
i
on al
go
r
i
t
h
m
s
. B
u
t
m
o
d
u
l
a
r
in
v
e
r
s
ion
o
p
e
ratio
n
is
u
s
ed
in
ex
isten
ce syste
m
. Co
n
s
ider
ing
th
e abov
e,
ou
r pr
oposed
d
i
g
ital sign
atur
e
schem
e
is
m
o
re secure a
n
d efficient whe
n
c
o
m
p
ared t
o
the
existing sc
he
me.
REFERE
NC
ES
[1]
V. S. Mi
ller
,
“
U
se of E
llip
ti
c Cu
rves in Cr
yp
togr
aph
y
,”
Springer-
Verlag
Berlin H
e
idelberg
, 1986.
[2]
N. Koblit
z,
“
E
ll
i
p
tic
curve
cr
yp
to
s
y
s
t
em
s
,
”
Ma
the
m
atics of
Computation
, vol. 48
, p
p
. 203-209
, 198
7.
[3]
A. Khalique
an
d K. S. S. Sood
v, “
I
m
p
lem
e
ntat
ion of Ellip
ti
c Curve Digita
l Signature Algori
t
h
m
,”
Internation
a
l
Journal of Computer App
lica
tion
s
, 2010.
[4]
N. Koblit
z,
“
E
ll
i
p
tic
Curve
Cr
ypt
o
s
y
stem
s,”
Ma
th
ematics o
f
Comp
utation
, vol. 48
,
pp. 203-209
, 19
87.
[5]
V. Miller
,
“
U
ses of Ellip
tic C
u
rve in Cr
ypto
graph
y
,”
Advan
ces in Cryptogr
aphy, Pr
oceed
in
gs of Crypto’85
,
L
ectur
es
not
es
o
n
Computer
Sciences, 218
, S
p
ringer-Verlag
, pp.
417-426, 1986
.
[6]
D. Hankerson,
et.
a
l
.
,
“
G
uide
to
E
llipt
i
c
Curve
Cr
yptograph
y
”
.
[7]
A. Ro
y
and S. Karforma, “A
Survey
on Di
gital Signatures and
Its Applications,”
Journal o
f
Computer and
Information Technology
, vol. 3
,
pp. 45-69
, 2012
.
[8]
T. Long, Xi
aox
i
a L. I
.
U, “Two Im
provem
e
nts to
Digital
Signature Schem
e
Based on the Ellip
ti
c Curve
Cr
y
p
tos
y
stem,”
Proceed
ings of the 2009 International Works
hop on Information Security and A
pplicat
ion
, Nov-
2009.
[9]
V. Pallipam
u, et
al., “
A
Surve
y
on Digital Signa
tures,”
International Journal of Adv
anced Resea
r
ch in Computer
and Communica
tion
Engineering
(
IJARCCE
)
,
vol. 3
,
pp
. 7243-72
4s 6, 2014
.
[10]
Ma
risa
W.
Po,
et al.
,
“
I
ssues in E
llipt
i
c Curv
e Cr
yp
y
ograph
y
im
ple
m
entation
,
”
In
ternetworking Ind
onesial Journa
l
,
vol/issue: 1(1),
2
009.
Evaluation Warning : The document was created with Spire.PDF for Python.