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DOI: 10.1109/STSIVA.2015.7330460
Secu e Image Enc yp ion and Au hen ica ion using he Pho on Coun ing
Technique in he Gy a o Domain
Juan M. Vila dy O.
G upo de ´
Op ica e In o m´
a ica
Uni e sidad Popula del Cesa
Valledupa (Cesa )–Colombia
[email p o ec ed]
Ma ´
ıa S. Mill´
an and Elisabe P´
e ez–Cab ´
e
G upo de ´
Op ica Aplicada y P ocesado de Imagen
Uni e si a Poli `
ecnica de Ca alunya
Te assa (Ba celona)–Spain
[email p o ec ed], [email p o ec ed]
Abs ac
In his wo k, we p esen he in eg a ion o he pho on
coun ing echnique (PhCT) wi h an enc yp ion sys em in he
Gy a o domain (GD) o secu e image au hen ica ion. The
enc yp ion sys em uses wo andom phase masks (RPMs),
one RPM is de ined a he spa ial domain and he o he RPM
is de ined a he GD, in o de o encode he image o enc yp
(o iginal image) in o andom noise. The o a ion angle o
he Gy a o ans o m adds a new key ha inc eases he
secu i y o he enc yp ion sys em. The dec yp ion sys em
is an in e se sys em wi h espec o he enc yp ion sys em.
The PhCT limi s he in o ma ion con en o an image in a
nonlinea , andom and con olled way; he pho on-limi ed
image only has a ew pixels o in o ma ion, his ype o image
is usually known as spa se image. We apply he PhCT o e
he enc yp ed image. The esul ing image in he dec yp ion
sys em is no a copy o he o iginal image, his dec yp ed
image is a andom code ha should con ain he su icien
in o ma ion o he au hen ica ion o he o iginal image
using a nonlinea co ela ion echnique. Finally, we e alua e
he peak- o-co ela ion ene gy me ic o di e en alues o
he pa ame e s in ol ed in he enc yp ion and au hen ica ion
sys ems, in o de o es he e i ica ion capabili y o he
au hen ica ion sys em.
1. In oduc ion
The pho on coun ing echnique (PhCT) allows o con-
ol he in o ma ion con en o an image in a nonlinea and
andom o m [
12
,
1
]. Recen ly, he PhCT has been in e-
g a ed wi h he double andom phase encoding (DRPE) in
he Fou ie domain [
9
,
6
], o secu e image enc yp ion and
au hen ica ion [
7
,
5
,
8
]. The PhCT in oduces a nonlinea i y
in o he DRPE sys em, his nonlinea i y inc eases he secu-
i y o he enc yp ion and au hen ica ion sys ems agains o
some a acks [
8
]. The PhCT can be applied o he o iginal
image o enc yp ed image. On he one hand, i he PhCT is
applied o he o iginal image, a pho on-limi ed image wi h
only ew pixels is ob ained. This image pe mi s o hide he
sec e in o ma ion o he o iginal image om isual inspec-
ion (in o ma ion hiding) [
7
]. On he o he hand, i he PhCT
is applied o he enc yp ed image, a spa se enc yp ed dis i-
bu ion is p oduced, and a educ ion o he ansmi ed/s o ed
in o ma ion is achie ed (in o ma ion comp ession) [7].
In his wo k, he in eg a ion o he PhTC wi h he DRPE
in he Gy a o domain (GD) o secu e image au hen ica-
ion is p esen ed. We apply he PhCT o e he enc yp ed
image wi h he pu pose o comp essing his image. The
au hen ica ion sys em is based on a nonlinea co ela ion
echnique [
2
,
3
]; in his sys em he dec yp ed image is com-
pa ed wi h he o iginal image o e i y i s au hen ici y. We
show ha he peak- o-co ela ion ene gy (PCE) [
11
] is im-
p o ed o ce ain alues o he o a ion angle o he Gy a o
ans o m (GT) and he nonlinea i y applied in he co ela-
ion echnique, in compa ison wi h he p e ious esul s o he
PCE o he in eg a ion o he PhTC wi h he DRPE in he
Fou ie domain (FD). This imp o emen o e he PCE me ic
allows a be e e i ica ion capabili y o he au hen ica ion
sys em.
The pape is o ganized as ollows: Sec ion 2 and 3 in o-
duce he GT and PhCT, espec i ely. The in eg a ion o he
PhCT wi h he DRPE in he GD o secu e image au hen ica-
ion is desc ibed and illus a ed wi h an example in Sec ion
4. In Sec ion 5, we e alua e and compa e he PCE me ics
o he in eg a ion o he PhCT wi h he DRPE in he FD and
GD. Conclusions a e ou lined in Sec ion 6.
2. Gy a o ans o m (GT)
The GT is ma hema ically de ined as a linea canonical
in eg al ans o m which p oduces he wis ed o a ion in
posi ion–spa ial equency planes o phase space [
10
]. The
GT a pa ame e
α
, which is he o a ion angle, o a wo-
dimensional unc ion
(x, y)
can be w i en in he ollowing
o m
α(u, ) = Gα{ (x, y)}
=
+∞
Z
−∞
+∞
Z
−∞
(x, y)Kα(u, , x, y)dxdy, (1)
Kα(u, , x, y) = ei2π[(u +xy) co α−( x+uy) csc α]
|sin α|,(2)
α=pπ
2,whe e: 0≤α < 2π, and 0≤p < 4,(3)
whe e
x
and
y
deno e he coo dina es a he spa ial domain,
u
and
indica e he ou pu coo dina es in he GD and
Kα
is
he gy a o ke nel. Fo
p= 0 (α= 0)
, i co esponds o he
iden i y ans o m. Fo
p= 1 (α=π/2)
, i educes o he
di ec Fou ie ans o m wi h o a ion o he coo dina e a
π/2
. Fo
p= 2 (α=π)
, he e e se ans o m is ob ained.
Fo
p= 3 (α= 3π/2)
, i co esponds o he in e se Fou ie
ans o m wi h o a ion o he coo dina e a
π/2
. The GT
has a pe iod o 4 wi h espec o
p
and
2π
o
α
. The in e se
GT co esponds o he GT a o a ion angle
−α
. The GT is
addi i e wi h espec o he o a ion angle, GαGβ=Gα+β.
3. Pho on Coun ing Technique (PhCT)
We can con ol he expec ed numbe o inciden pho-
ons (coun s,
Np
) o e a cap u ed image by using he PhCT.
The e o e, in gene al, a pho on-limi ed image has less in o -
ma ion han he o iginal coun e pa [
7
]. The p obabili y o
coun ing
lj
pho ons a pixel
j
can be shown o be Poisson
dis ibu ed [12, 1]
Pd(lj;λj) = [λj]lje−λj
(lj)! , lj= 0,1,2, ..., (4)
whe e he Poisson pa ame e
λj
is gi en by
λj=Npg(xj)
,
wi h
g(xj)
being he no malized i adiance a pixel
xj
such
ha
PM
j=1 g(xj)=1
and
M
equals he o al numbe o
pixels in he image.
The PhCT can be applied o a eal- alued image
(x)
o he complex- alued image
d(x)
[
7
]. Fo he case o he
eal- alued image
(x)
, we ob ain he pho on-limi ed image
ph(x)
when he equa ion
(4)
is applied o he no malized
dis ibu ion g(x) = (x)/PM
j=1 (xj).
When he PhCT is applied o he complex- alued im-
age d(x), a pho on-limi ed ampli ude in o ma ion |d(x)|, is
gene a ed om he no malized ampli ude image
g(x) =
|d(x)|/PM
j=1|d(xj)|
using he equa ion
(4)
. The pixels
ha ecei e a leas one pho on coun a e conside ed in he
pho on-limi ed enc yp ed unc ion
dph(x)
. Only hese pixels
con ain in o ma ion o he phase o d(x).
4. In eg a ion o he PhCT wi h he DRPE in
he GD o Secu e Image Au hen ica ion
We desc ibe he enc yp ion sys em based on a DRPE
in he GD. Le
(x, y)
be he eal image o be enc yp ed
(o iginal image) wi h alues in he in e al
[0,1]
, and
(x, y)
and
hα(u, )
be wo andom phase masks (RPMs) gi en by
(x, y) = exp{i2πm(x, y)},
h(u, ) = exp{i2πn(u, )},(5)
whe e
m(x, y)
and
n(u, )
a e no malized posi i e unc ions
andomly gene a ed, s a is ically independen , uni o mly
dis ibu ed in he in e al
[0,1]
and de ined a he spa ial
domain and he GD, espec i ely. In he i s s ep o he
enc yp ion sys em, he o iginal image
(x, y)
is mul iplied
by he RPM (x, y)and his p oduc is gy a o ans o med
wi h he o a ion angle
α
. The esul o he p e ious GT is
mul iplied by he RPM h(u, )and inally, his las p oduc
is gy a o ans o med wi h he o a ion angle
−α
. The inal
enc yp ed image
e(x, y)
is a complex- alued image de ined
by
e(x, y) = G−α{h(u, )Gα{ (x, y) (x, y)}} .(6)
The complex- alued enc yp ed image
e(x, y)
has ampli ude
|e(x, y)|
and phase
φe(x, y)
in o ma ion, so his image can
be w i en as
e(x, y) = |e(x, y)|exp{iφe(x, y)}
. The secu-
i y keys o he enc yp ion sys em a e gi en by he o a ion
angle
α
o he GT and he RPM
h(u, )
. These keys will be
equi ed o dec yp ion.
In o de o ob ain a pho on-limi ed enc yp ed image
eph(x, y)
, we applied he PhCT wi h a numbe o pho on
coun s
Np
o e he complex- alued enc yp ed image
e(x, y)
using he p ocedu e desc ibed in he sec ion 3.
The dec yp ion sys em uses he e e se p ocess o he
enc yp ion sys em. The inpu s o he dec yp ion sys em a e:
he pho on-limi ed enc yp ed image
eph(x, y)
, he o a ion
angle
α
o he GT and he RPM
h(u, )
. In he i s s ep o
he dec yp ion sys em, we apply he GT wi h he o a ion
angle
α
o
eph(x, y)
and his esul is mul iplied by he com-
plex conjuga e o he RPM
h(u, )
. The esul ing p oduc is
gy a o ans o ming wi h he o a ion angle
−α
and inally,
we ob ain he dec yp ed image
ph(x, y)
when he absolu e
alue unc ion is applied o he las esul o he GT. The
eal- alued dec yp ed image ph(x, y)is gi en by
ph(x, y) = G−α{h∗(u, )Gα{eph(x, y)}},(7)
whe e he supe sc ip
∗
deno es he complex conjuga ion
ope a ion.
The digi al esul s o he enc yp ion sys em, he PhCT and
he dec yp ion sys em ollowing he s eps desc ibed abo e
a e illus a ed wi h an example in igu e 1. The o iginal
image
(x, y)
has
512 ×512
pixels and i is p esen ed in
igu e 1(a). The andom dis ibu ion code
m(x, y)
o RPM
(x, y)
is shown in igu e 1(b). The andom code image
n(u, )
o RPM
h(u, )
has di e en alues bu he same
appea ance o he image p esen ed in igu e 1(b). The ampli-
ude and phase in o ma ion o he enc yp ed image
e(x, y)
a e depic ed in igu es 1(c) and 1(d), espec i ely, o he
o a ion angle
p= 3/4 (α= 3π/8)
. Figu e 1(e) shows he
ampli ude o he pho on-limi ed enc yp ed image
|eph(x, y)|
,
co esponding o igu e 1(c) when he o al numbe o pho on
coun s o he PhCT is se o be
Np= 103
. The dec yp ed
image
ph(x, y)
o all he co ec keys (
h(u, )
and
α
) is
shown in igu e 1( ). The digi al GT was implemen ed using
he as algo i hm o he disc e e GT based on con olu ion
ope a ion [4].
The enc yp ed image
e(x, y)
looks like andom noise ha
p o ec s he in o ma ion con en o he o iginal image, as can
be seen in igu es 1(c) and 1(d). The ampli ude o he pho on-
limi ed enc yp ed image
|eph(x, y)|
shown in igu e 1(e) is
a spa se e sion o he ampli ude enc yp ed image
|e(x, y)|
.
The o al numbe o pho ons in igu e 1(e) is
Np= 103
,
which co esponds o less han
0.4%
o he o al numbe o
pixels o he
512×512
o iginal image. The dec yp ed image
ph(x, y)
o igu e 1( ) is a andom eal- alued image ha
i is no a copy o he o iginal image. The ex con ained
in he o iginal image o igu e 1(a) canno be ecognized
om he noisy dec yp ed image
ph(x, y)
depic ed in ig-
u e 1( ). The e o e, he dec yp ed image
ph(x, y)
is no
in ended o isualiza ion, bu i has su icien in o ma ion
o au hen ica ion [7].
The au hen ica ion p ocess o he dec yp ed image
ph(x, y)
compa es his image wi h he o iginal image
(x, y)
u ilized as a e e ence, by using a nonlinea co ela-
ion echnique [
2
]. The images o be compa ed a e Fou ie
ans o med, nonlinea ly modi ied and mul iplied in he FD.
By in e se Fou ie ans o ming his p oduc , he nonlinea
co ela ion c(x, y)be ween bo h images is ob ained [2]
c(x, y) = F−1(F{ (x, y)}[F{ ph(x, y)}]∗
F{ (x, y)}
1−kF{ ph(x, y)}
1−k),
(8)
whe e he pa ame e
k
de ines he s eng h o he applied
nonlinea i y and de e mines he pe o mance ea u es o he
co ela o [
2
]. The pa ame e
k
is de ined in he in e al
[0,1].
The in ensi y ou pu o he nonlinea co ela ion
c(x, y)
wi h
k= 0
, be ween he dec yp ed image
ph(x, y)
o ig-
u e 1( ) and he e e ence o iginal image
(x, y)
o ig-
u e 1(a) is p esen ed in igu e 2. This esul allows he
au hen ica ion o he dec yp ed image
ph(x, y)
due o he
sha p peak p esen ed in he ou pu co ela ion
c(x, y)
o e
i s noisy backg ound. The maximum co ela ion alue has
been se o uni y o make he compa ison easie .
(a) (x, y)(b) m(x, y)
(c) |e(x, y)|(d) φe(x, y)
(e) |eph(x, y)|( ) ph(x, y)
Figu e 1. (a) O iginal image
(x, y)
o be enc yp ed. (b) Random
image
m(x, y)
o RPM
(x, y)
. Enc yp ed image
e(x, y)
o he
o a ion angle
p= 3/4 (α= 3π/8)
: (c) Ampli ude in o ma ion
|e(x, y)|
, and (d) Phase in o ma ion
φe(x, y)
. (e) Ampli ude in-
o ma ion o he pho on-limi ed enc yp ed image
|eph(x, y)|
wi h
Np= 103, and ( ) Real- alued dec yp ed image ph(x, y).
Figu e 2. In ensi y ou pu o he nonlinea co ela ion
c(x, y)
be-
ween he dec yp ed image
ph(x, y)
o igu e 1( ) and he o iginal
image (x, y)o igu e 1(a) wi h k= 0.
(a) ˜
(x, y)(b) ˜
ph(x, y)
(c) ˜c(x, y)
Figu e 3. (a) False image
˜
(x, y)
o be enc yp ed. (b) New de-
c yp ed image
˜
ph(x, y)
om a pho on-limi ed enc yp ed image
˜eph(x, y)
wi h
Np= 103
, and (c) In ensi y ou pu o he non-
linea co ela ion
˜c(x, y)
be ween he dec yp ed image
˜
ph(x, y)
o igu e 3(b) and he o iginal image
(x, y)
o igu e 1(a) wi h
k= 0.
In o de o es he disc imina ion capabili y o he p o-
posed sys em, we gene a e a new dec yp ed image
˜
ph(x, y)
om a alse image
˜
(x, y)
. The new enc yp ed image
˜e(x, y)
is ob ained om
˜
(x, y)
using he equa ion
(6)
o he o-
a ion angle
p= 3/4 (α= 3π/8)
. The pho on-limi ed
enc yp ed image
˜eph(x, y)
is compu ed wi h
Np= 103
, and
om his image he dec yp ed image
˜
ph(x, y)
is ob ained
by using he equa ion
(7)
and he app op ia e keys ( he RPM
h(u, )
and he o a ion angle
α
o he GT). The alse im-
age
˜
(x, y)
and he dec yp ed image
˜
ph(x, y)
a e shown
in igu es 3(a) and 3(b). The dec yp ed image
˜
ph(x, y)
o
igu e 3(b) is e y simila o he dec yp ed image
ph(x, y)
o igu e 1( ) wi h a noisy appea ance ha does no pe mi
o make ou he o iginal ex .
We compa e
˜
ph(x, y)
wi h he o iginal image
(x, y)
using he equa ion 8 (nonlinea co ela ion) wi h
k= 0
o
e i y i s au hen ici y. The ob ained in ensi y ou pu o he
nonlinea co ela ion
˜c(x, y)
is depic ed in igu e 3(c). Fo
his case, only a noisy backg ound is ob ained wi hou any
ema kable co ela ion peak. Thus, i is possible o ejec he
analyzed image and conside i as a alse image.
When an inco ec o a ion angle
α
o he GT o an inco -
ec RPM
h(u, )
o bo h a he same ime a e used in he
dec yp ion sys em, he ob ained dec yp ed images
ph(x, y)
a e s ill noisy pa e ns e y simila o igu es 1( ) and 3(b)
and he ob ained in ensi y ou pu o he nonlinea co ela-
ions
c(x, y)
be ween
ph(x, y)
and
(x, y)
a e noisy back-
g ound wi hou any sha p peak, simila ly o he ou pu plane
shown in igu e 3(c). These esul s p o e ha all he keys o
he enc yp ion sys em a e equi ed in he dec yp ion s age o
he co ec au hen ica ion o he dec yp ed image ph(x, y).
5. E alua ion o he PCE me ic
We use he PCE me ic in his sec ion wi h he pu pose o
e alua ing he e i ica ion capabili y o he p oposed sys em.
The PCE me ic, de ined as he a io be ween he maximum
in ensi y peak alue o he co ela ion and he o al ene gy o
he ou pu co ela ion plane, usually indica es he sha pness
and heigh o he ou pu co ela ion peak [
11
]. The PCE
me ic can be de ined as
PCE =ACph
R|c(x, y)|2dxdy,(9)
whe e he pa ame e
ACph
ep esen s he maximum in en-
si y peak alue o he nonlinea co ela ion be ween he
dec yp ed image
ph(x, y)
and he o iginal image
(x, y)
.
The PCE me ic is e alua ed o di e en alues o he num-
be o pho ons
Np
, he nonlinea i ies
k
and he o a ion angle
αo he GT.
We show he esul s o he PCE me ic in igu e 4(a)
when he PhCT is in eg a ed wi h he DRPE sys em in he
FD [
7
]. The equa ions
(6)
and
(7)
o he DRPE sys em o
enc yp ion and dec yp ion in he GD can be con e ed in o
a DRPE in he FD when he GTs o o a ion angles
α
and
−α
a e eplaced by he di ec and in e se Fou ie ans o m,
espec i ely. In igu e 4(a), he PCE alues apidly dec ease
wi h he numbe o pho ons, pa icula ly when
Np
is less
han
106.5
. The alues o
k
be ween
0.2
and
0.4
gi e he
bes esul s in e ms o PCE when Npis less han 105.
Figu es 4(b) and 4(c) p esen he PCE esul s o he
p oposed in eg a ion o he PhCT wi h he DRPE in he
GD o he o a ion angles
p= 0.5 (α=π/4)
and
p= 1 (α=π/2)
, espec i ely. The o a ion angles
p= 0 (α= 0)
and
p= 2 (α=π)
we e unused in he
DRPE sys em in he GD because he esul s o he GT wi h
hese o a ion angles o he o iginal image a e he same o
an in e ed o iginal images, espec i ely, and he enc yp ed
images a e no andom images. When he PCE is e alua ed
o a o a ion angle
p
di e en om 0, 0.5, 1, 2 and 3, he
PCE cu es ob ained a e e y simila o he cu es p esen ed
in igu e 4(b). The PCE cu es ob ained o he o a ion
angle
p= 3 (α= 3π/2)
a e e y simila o he cu es
p esen ed in igu e 4(c).
I we compa e he esul s o igu es 4(a) and 4(b), we can
see ha he esul s o he PCE me ic a e e y simila . On
he o he hand, he esul s o he PCE me ic om igu e 4(c)

102
103
104
105
106
107
108
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Numbe o Pho ons
PCE
0
0.2
0.4
0.6
0.8
1
k
(a)
102
103
104
105
106
107
108
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Numbe o Pho ons
PCE
0
0.2
0.4
0.6
0.8
1
k
(b)
102
103
104
105
106
107
108
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Numbe o Pho ons
PCE
0
0.2
0.4
0.6
0.8
1
k
(c)
Figu e 4. PCE alues e sus numbe o pho ons
Np
and di e en
nonlinea i ies
k
o he in eg a ion o he PhCT wi h he DRPE
sys em in: (a) he FD, (b) he GD o he o a ion angle
p=
0.5 (α=π/4)
, and (c) he GD o he o a ion angle
p= 1 (α=
π/2).
wi h he o a ion angles
p= 1
o
p= 3
a e imp o ed in com-
pa ison wi h he esul s o he PCE p esen ed in igu e 4(a),
o he nonlinea i ies
k
be ween
0
and
0.4
and he numbe o
pho ons
Np⩾104.5
. The imp o emen on he PCE alues
co esponds o a be e e i ica ion capabili y o he au hen-
ica ion sys em. We ecall ha he o a ion angles
p= 1
and
p= 3
o he GT co espond o he di ec and in e se Fou ie
ans o ms wi h o a ion o he coo dina e a
π/2
. The e o e,
he PCE alues can be imp o ed when he o a ion angles
o he GT a e equal o
p= 1
o
p= 3
o he in eg a ion
o he PhCT wi h he DRPE sys em in compa ison wi h he
PCE alues ob ained in he in eg a ion o he PhCT wi h he
DRPE in he FD.
6. Conclusions
In his pape we ha e p esen ed he in eg a ion o he
PhCT wi h he DRPE in he GD o secu e image au hen i-
ca ion. We ob ained a spa se enc yp ed image using he GT,
DRPE and he PhCT applied o he enc yp ed image. The
dec yp ed image was a noisy-like code ha i is no in ended
o isualiza ion; his image was u ilized o a success ul
e i ica ion o an o iginal image used as a e e ence pa e n
by means o a nonlinea co ela ion echnique. The o a ion
angle o he GT has imp o ed bo h he secu i y o he en-
c yp ion sys em and he PCE me ic o he au hen ica ion
sys em. The secu i y o he enc yp ion sys em was imp o ed
because a new key, co esponding o he o a ion angle o he
GT, has been in oduced in he DRPE sys em in compa ison
wi h he same enc yp ion sys em in he FD. The PCE me ic
o he au hen ica ion sys em was also imp o ed using he
o a ion angles
p= 1 (α=π/2)
and
p= 3 (α= 3π/2)
o
he GTs employed in he DRPE sys em o enc yp ion and
dec yp ion.
Acknowledgmen s
This esea ch has been unded by he Uni e sidad Popula
del Cesa om Valledupa (Cesa ), Colombia, and he Span-
ish Minis e io de Ciencia e Inno aci
´
on and Fondos FEDER
(P ojec DPI2013-43220-R).
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