Secure image encryption and authentication using the photon counting technique in the Gyrator domain

“© 2015 IEEE. Pe sonal use o his ma e ial is pe mi ed. Pe mission om IEEE mus be ob ained o all o he uses, in any cu en o u u e media, including ep in ing/ epublishing his ma e ial o ad e ising o p omo ional pu poses, c ea ing new collec i e wo ks, o esale o edis ibu ion o se e s o lis s, o euse o any copy igh ed componen o his wo k in o he wo ks.” 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). Re e ences [1] J. W. Goodman. S a is ical Op ics. Wiley, New Yo k, 2000. [2] B. Ja idi. Nonlinea join powe spec um based op ical co ela ion. Appl. Op ., 28(12):2358–2367, 1989. [3] B. Ja idi and J. L. Ho ne . Op ical pa e n ecogni ion o alida ion and secu i y e i ica ion. Op . Eng., 33(6):1752– 1756, 1994. [4] Z. Liu, D. Chen, J. Ma, S. Wei, Y. Zhang, J. Dai, and S. Liu. 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