Novel multi-image quantum encryption / decryption schemes

Abstract

Below, L denotes the number of bit planes, and M denotes the number of images to encrypt / decrypt. Our own convention is that bit plane 0 is the least significant bit plane.

Description

Titre de l’invention : Novel multi-image quantum encryption / decryption schemes

[0001] Due to the strong corrélations amongst adjacent pixels, bulky data volume and the handling of varions data formatting, traditional encryption schemes are not suitable for image encryption. Image encryption / decryption schemes are generally based on géométrie scrambling and diffusion using pseudo-random sequences generated from chaotic discrète dynamical Systems.

[0002] With the prospect of quantum computers, so-called quantum image encryption / decryption schemes hâve emerged as a mean to communicate images confidentially and efficiently. For faster processing, the quantum computer converts the classical image plaintext into quantum représentation and performs with quantum gates the encryption of the quantum image using some secret keys. Through measurements, the encrypted quantum image is converted into a classical image ciphertext which is transmitted to the receiver. In tum, the quantum computer of the receiver converts the classical image ciphertext into quantum représentation and performs ail the inverse quantum gates in the reverse order using the same secret keys. Through measurements, the receiver retrieves the classical image plaintext sent by the transmitter.

[0003] However, quantum encryption / decryption schemes for multiple images hâve been little addressed although they allow for a faster parallel processing of ail the images and increase the level of security greatly by shuffling ail the images together, instead of shuffling them one by one. Moreover, in the four schemes disclosed here, an attachant has no way of knowing the précisé number of images being transmitted.

[0004] We therefore disclose for the sake of quantum encryption / decryption schemes two novel quantum représentations for gray-scale multi-image and two novel quantum représentations for color multi-image quantum. In each case, one of the two quantum représentations uses blocks of images so as to decrease the number of qubits needed when encrypting large sets of images. Each of the four quantum représentations allows for accurate retrieval of the classical multi-image by measurement, contrary to some already existing ones. The conversion from classical to quantum représentation is standard process that is extensively discussed in the littérature and thus omitted here. Our four quantum encryption / decryption schemes are based on a two-stage scrambling using quantum baker maps: we first do some pixel and block independent scrambling of the images and of the bit planes, second we do some image and bit plane independent scrambling of the pixel positions.Here, "independent" means for each the control and itération parameters of the quantum baker map may be chosen differently. . Moreover, we provide a general quantum circuit to implement the quantum baker map for any values of its control parameters. For this two-stage scrambling, we may also proceed in the other order. As far as the diffusion, it is then achieved using a sine chaotification of a 2D or higher dimensional (resp 4D or higher dimensional in the case of the quantum représentation by blocks) discrète dynamical System, iterating it from a seed, and additionally applying some Chebyshev transformations on the pseudo-random numbers that are generated. The seed must be chosen to dépend on the plaintext multi-image. In the section "Details of the invention", we provide instances of such discrète dynamical Systems and seed. The Chebyshev transformation could eventually be replaced with some other chaotic map or chaotic sytem but présents some spécifie advantages clearly appearing in the section "Details of the invention".The Chebyshev transformations could also be applied in a different way than the way presented here which only serves as instance. From the applications of the three different chaotic transformations, enough pseudo-random bits are generated which get XORed with ail the bit values encoding the multi-image for diffusion of ail the images. The XOR operations are performed through the use of controlled CNOT gates. The order in which the scrambling and the diffusion are achieved could eventually be reversed. Finally, because a quantum computation consists of applying quantum gates which are nothing else than unitary matrices, the four schemes disclosed here can be programmed and used on classical machines using for instance the software MATLAB for matrix multiplication. . We gathered the novelties of our schemes in the five points below. 1. (detailed in Section 1. of the Abrégé). We disclose two novel quantum représentations for a gray-scale multi-image. For future reference, we call these two quantum représentations représentation A and représentation B respectively. Suppose there are M gray-scale images of size ?” x 2” with pixel values summed on L bit planes to be encrypted. For représentation A, when M < L, we add L-M blank images, e.g. ail black and ail white images in alternance, before converting the multi-image into quantum représentation. And in the case when M > L, we add M - L bit planes filled with bit values 0. Having the same number of bit planes and images in the quantum représentation will allow for 2D quantum scrambling on the images and the bit planes. Letting N = Max^M, L), this représentation uses a total of In + ^log^ Au+l qubits, namely 2n qubits for the pixel positions, ^1 qubits to encode the bit planes, qubits to encode the images and a single qubit to encode the bit value of a given image at a given pixel position on a given bit plane. Représentation B’s purpose is to limit the number of qubits used in the case when the number of images to transmit is large. To this aim, we divide the set of images by blocks, each of size l| Consequently, there are l\blocks , each containing l] images (counting also the blank images in the last block in the case when m])- Thus, there are the blocks and the images belonging to a given block, thus a total of B.

[0005] We now briefly comment on quantum représentations recently developed and used by other authors in relation to our quantum représentations A and B. Regarding both quantum représentations A and B, the idea to encode the bit qubits, instead of L qubits like in the well-known NEQR représentation, originates in [5]. In

[10] , the authors also deal with multi-image quantum encryption / decryptio n, but contrary to us, they adopt an NEQR based représentation instead of a bit plane représentation, to which they add qubits to encode the images. Moreover, they must eventually add blank images so as to hâve as many images as there are pixel positions because their quantum scrambling transform is the quantum 3D Arnold transform. In ternis of qubits, it means extra qubits instead of simply extra qubits like in our quantum représentation A when

[0006] M < L. Additionally, the Arnold transform has shorter scrambling period than the baker map. Finally, regarding the quantum représentation B, the idea to use blocks with quantum image encryption originates in [6], where the authors divide an image into blocks and use qubits to represent the position of the block in the image and qubits to represent the pixel position inside a block. Converting the classical images into quantum représentation uses the standard techniques widely provided in the literature. Both représentations are used to perform the quantum computation for the encryption or the decryption procès s and they both allow to retrieve the classical multi-image ciphertext in case of encryption or the classical multi-image plaintext in case of decryption accurately by projective measurements. 2. (detailed in Sections 3. and 4. of the Abrégé). Using these two novel représentations, we disclose for each a novel way to encrypt and decrypt a multi-image, using the quantum baker map for scrambling the plane of the bit planes and the images, then scrambling the pixel positions, and a combination of three different chaotic Systems or maps for diffusion. The scrambling may also be achieved in the reverse order, that is first scrambling the pixel positions, then scrambling the images and the bit planes. For convenience, we will refer to our first scheme as scheme A and to our second scheme as scheme B. Scheme B will be used for encrypting a large number of images exceeding the number of bit planes, while scheme A is more suitable for encrypting a small number of images, for instance not exceeding or being equal to the number of bit planes. In both schemes, an attachant has no way of knowing the précisé number of images getting encrypted.

[0007] Scrambling and diffusion are traditionally used in image encryption as the combination of both is known to provide some robustness for the cryptographie scheme. In [5], a quantum image encryption / decryption scheme is presented for single image encryption using a bit plane représentation model and the quantum baker map to perform some bit plane independent shuffling of the pixel positions. The novelty in our scrambling process dealing with multi-image quantum encryption / decryption is that the quantum baker map is not only used to shuffle the pixel positions for a given image (in scheme A) or a given image belonging to a given block (in scheme B) and a given bit plane, but also used to shuffle at a given pixel position (and for a given block in scheme B) the bit planes and the images. The scrambling is a two-stage scrambling.

[0008] Next, the diffusion is achieved by generating two in scheme A (resp four in scheme B) pseudo-random sequences issued fom a sine chaotification of a 2D Hénon transform [2] (resp of a 5D hyperchaotic System, e.g. the one of [9]), with initial conditions in the framework of scheme A such as the total intensity of the images and the Chebyshev polynomial of index depending again on the plaintext multi-image applied to the latter number. With such initial conditions, a slight bit value change will resuit in a completely different ciphertext multi-image. Additional initial conditions must be added in the framework of the 5D hyperchaotic System used in scheme B, see the part "details of the invention". It is important that these initial conditions again dépend on the plaintext multi-image so as to avoid CPA or CCA attacks.The sine chaotification model, like introduced in , allows for infinité key space, thus making the protocol robust against brute-force attacks and moreover increases the sensitivity to the initial conditions by increasing the Lyapunov exponents in a controllable way. The pseudo-random sequences are then used to generate secret bits which will be XORed with bit values. .

[0009] The main novelty in our diffusion process consists of cascading three different chaotic Systems or maps, thus increasing the randomness characteristics of the secret keys. More specifically, the novelty is to use the sine chaotification of a chaotic System and apply some Chebyshev transformations to the pseudo-random sequences it

[0010]

[0011]

[0012]

[0013]

[0014]

[0015] generates in order to produce secret bits for diffusion. The use of a higher dimensional chaotic System in scheme B allows for global diffusion of ail the images, instead of image independent diffusion like in scheme A. 3. (detailed in Section 2. of the Abrégé). It is known that the quantum baker map may be implemented using SWAP quantum gates and controlled SWAP quantum gates, see e.g. [5]. We disclose a general quantum circuit for the quantum baker map which can be applied with any values of its parameters. We note that since the parameters of the quantum baker maps may differ for each pixel position (and block in scheme B) when we shuffle the images and the bit planes and for each image (or image belonging to a given block in scheme B) and bit plane when we shuffle the pixel positions, it is of crucial importance to hâve a general quantum implémentation. 4. (detailed in Section 5. of the Abrégé). We extend our invention to a color multi-image to encrypt and decrypt. We namely introduce a bit plane quantum représentation for a color multi-image, as well as a bit plane quantum block représentation for a color multi-image. The qubit previously carrying the bit value for an image (eventually belonging to a block) at a pixel position on a certain bit plane is now replaced with a 3-qubit encoding the quantities of Red, Green and Blue on that bit plane. The rest of the protocol is operated like before. There are no changes in the two-stage scrambling using the quantum baker maps. As far as diffusion, more pseudo-random secret bits need to be generated which get XORed with the bit values each associated with Red, Green and Blue. For details, see Section 5. of the Abrégé. 1. Novel quantum représentations for a gray-scale multi-image Scheme A The novel quantum représentation for a gray-scale multi-image is: r__1______y2p°s-'Mlly2p^ *A > “ ^ / «=0 ^1=0 ^x=0^y=(r»^y > Pm[xv is the bit value in the z-th bit plane of the m-th image at pixel (x, j). With this représentation, it is understood that we set: -1 if L Pmlxy • ~ is odd when < m< - 1 if M < 9po^2^"l where b dénotés the Kronecker Symbol. Of course m is odd could be replaced with m is even instead. P j 1 rnlxy In our encryption / decryption scheme, up to adding some ail black and ail white images in alternance when M < L or adding some bit planes filled with bit values 0 when M>L, we imposeM-L. Scheme B

[0016] The novel quantum représentation by block for a gray-scale multi-image is:

[0017] | ^2"-! |, I t I rb > — ktmxyi > I > i nd > [xy >

[0018] with

[0019] A . .................1.....................

[0020] In this représentation, Ptmxyl is the bit value of the w-th image belonging to the f-t h block at pixel (x, y} on the Qh bit plane. With this représentation, it is understood that we set: Ptmxyl • — 0 when j < - 1 if L

[0021] and

[0022] Ptmxyi : = 5m is odd when M. 2^ + 2^ ^<m< 2^ -1 if M <

[0023] and t = l] _ j

[0024] Again, in the Kronecker Symbol, m is odd could be replaced with m is even. 1. Quantum implémentation of the quantum baker map

[0025] The discrète baker map is defined on a square of size 2” x 2W with

[0026] 2" = 2*’+...+2*\

[0027] where the ^’s are the parameters of the map,

[0028] as:

[0029] (tvo = O and ^- = 2^+...+2^ and x = N^,Nn +1, ...,Nr 1 ■ » .x I , .. < , , y-y mod 2 ' I (x,_y) = (2 ‘(x-NiA)+y mod 2 + ^—--J

[0030] As noticed by the authors of [3], under some additional conditions on its parameters, the discrète baker map has a quantum implémentation. Their resuit is as follows: (i) The map defined by

[0031] ms(x, y) = [ 2n'sx mod 2” + y mod 2^, + x-x mod 2S)

[0032] has a quantum implémentation. (ii) Each subfunction on py. ^y. _ || of the discrète baker map is Mq. if and only iW’T ... + / -• A discrète baker map with parameters satisfying to the divisibility condition of ( il ) is called a quantum baker map. Its parameters are secret keys in our schemes. In [5], the authors exhibit a quantum circuit for some spécial values of the parameters but they do not treat the general case. Here, we disclose a general quantum circuit to implement the quantum baker map. In what follows, we consider a quantum baker map ? ) with parameters .... q^ j such that 2” = 2?14- 4-9^ and 2^ divides 2^4- 4-for eachz with 2 < i < k. Let f be the subfunction defined on [ | q, , 29‘ -1|] ■ And so we hâve = Mq.. Explicitly,

[0033] f}(x,y) = (xq-l...x0 yn^_l...yQ,xn_l...xq] y^.-.y,^)

[0034] Let g, be the subfunction of the quantum baker map defined for the integers x with

[0035] 2’1 + ... + 2’“ £ x < 2,[ + ... + 2^ + 2^ - L

[0036] will be our base function from which the other ^(’s are built. We call f. the function such that

[0037] g^fffA

[0038] The quantum circuit for f ( is only composed of SWAP gates, whereas the quantum circuits for the other / 3 s are composed of controlled SWAP gates in such a way that the quantum circuit for the quantum baker map consists of applying the quantum circuits for / p .. •, fk sequentially. If Q, = then is Mqx = f Therefore, f. is simply the identity. Otherwise, write + + 2^’ — % % in base 2 and let 7 be the set of indices ’ such that in this binary writing x{ = 1. Then, the Controls on the SWAP gates of f. are defined like follows: (i) Suppose 2 < i < k -1, then 100391 c: = rUcfel i A 76 / yj- 1 1 1 =0 ^Xn^ys-0 seJ

[0040]

[0041]

[0042] where the braces indicate the conditions to be satisfied for the control to happen and the products signify conjunction of Controls.

[0043] Claim 1. Quantum circuit for the quantum baker map. Quantum circuit for f Y AA) if 2 A^^ ... Spll+l if < q} < with 2< l < n -1 / r = if = with3<Z<n-l Co if^ra-l

[0044] Quantum circuit for fr withZ ^r<k. if qr< h 2 ïîqr = 21 2 gPgP T if - < . 3 O O if = 3 çpçA •r 2?’ if — < I if = SV Xf'p ^p if 9r< if = ■^p ^p if^> 2^i

[0045] In the above, I dénotés the identity matrix (meaning do not act on the ^n-qubit). To implement the quantum baker map, it suffices to apply in this order f? ---, fk with ail the quantum gates provided below. AU the quantum gates hâve been written in the format of a quantum circuit as far as matrix multiplication, that is they must be applied from left to right in the order in which they appear. Quantum gates forf p

[0046] A := Bo := y n-q^ y i ) B; : = 1^^ ^ yn-q-i+f> yn-q^l+j^ S{ : = n;=0 SWAP( J^^^ 1^ : = li^-g $WAP X^y, ylq -l—( / -l)q-( / -2)a-l > lï / a) Ilfao SWAPfy^^),,.^;^^ Co : =

[0047] Quantum gates for fr with 2<r<k:

[0048] eœ. (^rWAP(x"^y^) (00491 g®. = n^C,S WAP ( x,^, ,t ) n^CSWAP ( yr ) rC,C^ WAP ( yf )

[0050] ¢5,. 100511 100521

[0053] ¢,. = n^-’lcrswAP(x^> jjn’i'nî'®' 'c>’w^(>'wv

[0054] g^nlJ'CSWAP^^

[0055] pj^’ C^WAP(xn.|_,,' CfSWAP^x,^^

[0056] ¢7,. j,;)

[0057] qW. = CrSWAP^n-q^fiX^ 100581

[0059] ¢,,.

[0060] ÿ0): -n^CpSWAP(ywi, 1. Génération of the secret keys used for diffusion#

[0061]

[0062]

[0063]

[0064]

[0065]

[0066]

[0067]

[0068]

[0069]

[0070]

[0071]

[0072]

[0073]

[0074] Scheme A One of the classical chaotic Systems is the 2-dimensional Hénon discrète dynamical System: xM+1=l-a xl + yn y < = b Xn l‘n+! n It exhibits a chaotic behavior when the two parameters are respectively set to « = 1.4 and b = 0.3. It was introduced by Michel Hénon in the 1970’s, see [2]. We now define a new two-dimensional chaotic System which is a sine chaotification of the Hénon System: xi+ ^sin^ (1-1.4 x? + y. ) ) y^sin^f (0.3 xf) ) For each image m, we pick different parameters jh") > [ and ^6”) > y These parameters are secret keys. The sine chaotification increases the two Lyapunov exponents by and ln{ resPechveV■ The secret bits for the diffusion stage are then obtained in the following manner. Iterate the dynamical System with a seed depending on the total intensity scaled between 0 and 1 of the set of images to be encrypted. A total intensity of 0 corresponds to ail white images, while a total intensity of 1 corresponds to ail black images. Thus, we set ECÜV5C2 -1 with M the total number of images to be encrypted (without counting the blank images) and VXym the pixel value of the m-th image at pixel (x, y ). Then we set: >o: = where Tk is the ^-th Chebyshev polynomial. The Chebyshev polynomial can be implemented directly by: or inductively by: [T0=l, T^X \T k+2 (X)-2XT k+i (X)-T k (X) With the seed depending on the plaintext multi-image, the diffusion process is not only related to some secret keys, but also related to the plaintext multi-image. From the seed, iterate the discrète dynamical System as long as is necessary to obtain a sequence of 2” distinct values xm.& • - •, xw,2"-] for x and a sequence of 2” distinct values JW)Q5 • • • ? for J. Discard the first hundred itérations so as to avoid transient effects. Sort the two pseudo-random lists by increasing order and assign to the horizontal f-th (resp vertical J-th) pixel coordinate the integer (resp ^mj) denoting the index position of xmi (resp ym^j) in the respective sorted lists. For a pixel ( i, j), compute

[0075]

[0076]

[0077]

[0078]

[0079] with for instance . Obtain a ^ / «^Ml-qubit secret key:

[0080]

[0081]

[0082]

[0083]

[0084]

[0085] Scheme B According to [9], the 5-D hyperchaotic System ' X{ - a ( X2 - *1 ) + X2X3X4 x2 = b ( x । + x2 ) + x5 - x jX3x4 x3 = - cx2 - dx3 - ex4 + x } x2x4 x4= - / x4+X1X2X3 x5= -g^ + Xo) with control parameters set to a = 30, b = 10, c = 15.7, d = 5, e = 2.5, f = 4.45 and g = 38.5 is in a chaotic State and could produce five chaotic sequences. Some phase portraits are displayed in Fig. 1 of [9]. We define a 5-D hyperchaotic System which is a sine chaotification of the previous System: + yzt) ) y - sin(7r^( 10(x + j) + u-xzt) )) 1 z = sîn(jïÀ3( -15Jy-5z-2.5t+xyt) ) t - sin ( tt24 ( - 4.45 / + xjz ) ) ,ù = sin(7TÀ5( -38.5(x + y) ) ) Our seed will be designed to dépend on the plaintext multi-image. A slight bit change will completely modify the ciphertext multi-image. Our seed will also carry the fact that the attachant has no way of knowing how many images are encrypted. We choose x ( 0 ) to be the total intensity of the images and then define . [Motif 1 (MO), l]x 10*J £ Ç (J _ __ These two numbers belong to the interval [0, 1 ]. The initial values j(0) and / (0) are then obtained by applying some Chebyshev transformation to respectively x(0) and Z ( 0 ). Explicitly,

[0086]

[0087]

[0088]

[0089]

[0090]

[0091]

[0092]

[0093]

[0094]

[0095]

[0096]

[0097]

[0098]

[0099]

[0100]

[0101] x ( 0 ) : = £^<2"- 0<m<M-ï ' with VXym the pixel value of the image m at pixel (x, j). As for y ( 0) and t ( 0 ), define for instance two plaintext dépendent integers: a: = And Pbmxyl 0<b<dl,>g2 XX Ll[ pbfflxyiy \ WK^ 4 s That is, for a pixel (x, y ), consider the total number of bit values equal to 1 for ail the images (including or not the blank ones in the last block). Then a is the integer part of the average of these integers and P is the integer part of the average of these squared integers. Now set: X0) := 7L(x(0)) / (0) := TXz(0)) We now define the intensity by block. We treat the last block aside as it may not be entirely filled with images in the case when ^log„ Af| The total intensity of the k-th block with „ . , 0<Æ< c 0<x,v<2nA Whereas, the total intensity Lli of the last block gets defined as: y (Kr.^-Î -------LZ------- Further define when pog2 L]<|"Zog2 £*| ■Q<k<d log 2 M ^ log 2 L l-1 or when [|*Zog2 M*|>2[zog2 l\ ( L1 - 2ÎX £] < k < ^log2 - ] 4i Pk(^l<’s2 XX L"\-k-l)xyl Now set: w(0) : Lx22"x2H tl| ^2^^ LXZ2’^^ i] ifpog7 L|<pog7 M^^Zog^ l\ if[zog2 w]>2pog2 £]

[0102]

[0103]

[0104]

[0105]

[0106]

[0107]

[0108]

[0109]

[0110] [OUI] For the latter initial condition, we consider the first image in the last block, the second image in the block before, and so on. We only consider the last ^log2 l] blocks in the case when the number of blocks is greater than ^log^ l| For each such image, we compute the sum of ail the bit values on ail the bit planes. We obtain Min (2^2 Mïï / og2 4 2^2 4) numbers. These numbers serve as weights in the sum of the intensities by block. We last divide adequately in order to get an initial condition belonging to the interval [0, 1]. From the seed (x(0),j(0),z(0), / (0),1 / (0)), iterate the dynamical System and ignore the first hundred itérations so as to avoid transient effects. Do so as long as is necessary to get four sequences of distinct numbers: Xq, . . . , X^n_ । 4* • • • , Z^2 £[[ ■ ■ • ’ 41 Then sort the four lists by increasing order and assign to the Mh (resp J-th) pixel coordinate (resp m-th image, resp ô-th block) the index position of xi (resp yj, resp zm, resp 4) in the sorted sequences. Obtain in tum four sequences of integers: ^0» • • •, ^2». j O ■ • • ’ t]. ] % • ■ • ’ 4i For an image m belonging to a block b and a pixel ( i- j ) on that image, compute: "1K 4,„) x 1O’J mod with for instance > ^logn l\- Obtain a ^p^Lpqubit secret key 41^'J • • • > • 1. Encryption and Decryption# The transmitter is in charge of picking an appropriate quantum représentation for the multi-image. He or she is in charge of generating ail the secret keys used for diffusion, that is ail the control parameters of the sine chaotification (or more generally ail the parameters of the chaotic Systems and maps used) and ail the necessary initial conditions to iterate the discrète dynamical System. Likewise, he or she is in charge of generating ail the quantum baker maps’control and itération parameters used for the scrambling. Then, the transmitter exchanges ail these secret keys with the receiver, using for instance some secure post-quantum cryptographie schemes. Amongst the secret keys could also figure data such as the number of images to decrypt, depending on the précisé protocol used. Encryption Scheme A

[0112] We operate a two-stage scrambling, each time using quantum baker maps. First, we do an independent pixel scrambling of the images and of the bit planes. Namely, for each pixel (x, y), we apply a quantum baker map on the two k" and i / > . The parameters, as well as the number of itérations of the QBM dépend on the pixel. We obtain:

[0113] ’a> = 1 y2”-!

[0114] We now fix the image and the bit plane and use the quantum baker map to scramble the pixel positions. This time, the parameters and the number of itérations of the QBM dépend on the couple ( m, l ). The expression for the new quantum multi-image after pixel scrambling is: ^A> “ i ndxy> > Q^^ml (lxT> ) _ 1 y2"l l | . “ \Pndxy> \mlxy>

[0116] The quantum multi-image is now ready for the second main phase of the quantum computation which deals with the independent image diffusion of the pixel values. Préparé qubit ancillas encoding the secret keys in order to perform

[0117] Ml) CCNOT quantum gates,

[0118] so as to obtain the quantum multi-image ciphertext:

[0119] ____i yf'-l ^1 i ^A > ~ ^logju] ^x\y=^mJ=Q ^Imjc.y ® Pmlxy > ÿnlxy >

[0120] The CNOT gâte is applied on the qubit of the secret key (control qubit) and the qubit of the pixel value (target qubit), with control qubits those of the images, the pixel positions and the bit planes. Finally, the quantum ciphertext is converted to its classical version using projective measurements and sent to the receiver. Scheme B

[0121]

[0122]

[0123]

[0124]

[0125]

[0126]

[0127]

[0128]

[0129]

[0130]

[0131]

[0132] We operate a two-stage scrambling, each time using quantum baker maps. Our scrambling is block independent, namely for each block the parameters of the quantum baker map and the number of itérations are picked differently. More explicitly, for each pixel (x, y) we do a block independent scrambling of the images and of the bit planes. We obtain: ^B^ “ |£xy> QBM X yf (\ml> ) «V2"’1 Ll) | „■ i, , ~ ^x,y=0^mJ=0 l^tmxyl> \tmlxy> We then fix a bit plane and an image belonging to a certain block and perform independent scramblings of the pixel positions. It yields: 4> = - 4-f=0 \Ptmxyl> \tmlxy> From there, the scrambled quantum multi-image is ready to be diffused. Préparé qubit ancillas encoding the secret keys in order to perform 2^k>g2 l] ccnOT quantum gates, so as to obtain the quantum multi-image ciphertext: kb > = *■> The CNOT gâte is applied on the qubit of the secret key (control qubit) and the qubit of the pixel value (target qubit), with control qubits those of the blocks, the images, the pixel positions and the bit planes. Last, the quantum ciphertext is converted to its classical version using projective measurements and sent to the receiver. Decryption The receiver converts the classical ciphertext into quantum représentation A or B and uses his or her quantum computer to perform ail the inverse quantum gates in the reverse order of the quantum encryption process. Projective measurements are performed in order to retrieve the transmitted classical images. Last, in scheme A, the blank images that were added in the case when the number of images is les s than the number of bit planes get discarded and in schemes B the blank images belonging to the last block, if any, get discarded. In case the images that are added are regular images, as part of the secret keys, the receiver knows how many images he or she should receive and accordingly discards the last images. 1. Generalized schemes for a color multi-image#

[0133]

[0134]

[0135]

[0136]

[0137]

[0138]

[0139]

[0140]

[0141] We introduce two novel quantum représentations for a color multi-image. They are based on the RGB color code and the bit plane représentation. One of these quantum représentations is designed by block in order to reduce the number of qubits when dealing with the encryption / decryption of a large set of color images. Again, and contrary for instance to the quantum représentation of [8] itself inspired from the FRQI représentation, our quantum représentation is designed so that the classical multi-image can be retrieved accurately by measurement instead of simply probabilistically. Again also, our own convention is that the O-th bit plane is the least significant bit plane. This convention is made in order to be able to conveniently add some blank bit planes in the quantum représentation of the plaintext multi-image when it is necessary. We first provide the bit plane quantum représentation for a color multi-image as r c — ^x=(r~W=() mlxy* mlxy* mlxy with 2 like already defined before. pR, (resp pG, , pG, ) is the bit value in the ;-th bit plane of the m-th image 2 mlxy v r 2 mlxy 2 mlxy7 r ° at pixel (x, y ) for color R (resp G, B). If p the where C G {R, G. fi}, are ail set to the bit value 0 when < < £1 1' M < set for instance when M<m< -f +k)lxy — &{k=0 mod 3} Ct P(M+R)lxy — mod 3} ,P(.M+k)lxy ~ ^{k-2 mod 3} That is, add red, green and blue images in alternance. In our quantum encryption / decryption scheme which we call scheme C for future reference, up to adding some blank images when M < L or adding some bit planes filled with bit values 0 when M > L, we impose that M = L. We next provide the bit plane quantum block représentation for a color multi-image whose number of images is greater than the number of bit planes. It reads: * U । jj—*y— with 2 defined like before. , (resp pG „ pG ,) is the bit value in the ;-th bit plane of the Wî-th image tmxyl ri tmxyl 2 tmxyl' r ° belonging to the z-th block at pixel (x, y) for color R (resp G, B). If £ < 2^ £1 - T the ^yi S’ where C e G, B], arc ail set to the bit value 0 when L<1<2 -r Finally, M< 2^1 , then set for instance

[0142]

[0143]

[0144]

[0145]

[0146]

[0147]

[0148]

[0149]

[0150]

[0151]

[0152]

[0153] £ _ - 1 'HtW? L\k)xyl ~ ^{*=0 mod 3] ' 4+^^ mod 3] L]*k)Xyl~ mod 3) for when 4. fyos2 - 1' We will call the encryption / decryption scheme using this quantum représentation for a color multi-image scheme D. The two-stage scrambling process for scheme C (resp D) is similar to the one of scheme A (resp scheme B), where the single qubit encoding the bit value must now be replaced with the 3-qubit encoding the three bit values corresponding to the three colors. For the diffusion stage, we keep the same chaotic discrète dynamical Systems and adapt the initial conditions. We treat scheme C first. The control parameters of the Hénon System are set like before. Again, the two parameters of the sine chaotification dépend on the images. We can take for seed, where M dénotés the total number of images to encrypt (without counting the blank images) and T1 is a Chebyshev polynomial: vfR .vG , * xym~r *■ xyrrï'y xym x<> = 3(2^1)2¾ Proceed with this seed exactly as in Section 3. in the framework of scheme A, except to obtain the secret bits at the very end, reduce modulo M] instead of reducing modulo 22^ That is, for a pixel (ï, j ), compute this time * 10’”J mod with for instance , where we used the same notations as before. Obtain a secret ^ / ogywl-qubit, namely: ’ 2110& ,W1 {jn / J QwJ 2log2 ‘ ^fnÎJ From there, préparé 3-qubit ancillas each encoding the secret color bits in order to perform three CNOT gates controlled by the qubits of the image, the bit plane and the pixel position. This quantum computation results in XORing for ail the colors, images, bit planes and pixel positions as follows.

[0154]

[0155] For each color, the CNOT gâte gets applied on the secret qubit ancilla serving as the control qubit and the bit value qubit of the scrambled quantum color multi-image which serves as target qubit, with control qubits those of the bit planes, the images and the pixel positions.

[0156] At the end of the quantum computation, the qubit ancillas are not entangled with the other qubits, hence they can simply vanish, leaving the color scrambled multi-image diffused.

[0157] We now proceed scheme D.

[0158] We keep the sine chaotification model of the 5D hyperchaotic System of Section 3. and take five initial conditions as follows.

[0159] [ . _ X(U) : - 3(2^1)2¾ [zW<«7( 10x( 0). l]x ! 0’ J z(0) ---

[0160] is the total intensity of the images. We will apply some Chebyschev transform to each of these numbers to obtain two other initial conditions. The indices of the Chebyshev polynomials will dépend on the plaintext multi-image. Explicitly, for a pixel (x y), consider the total number of bit values equal to 1 for the three colors for ail the images (including or not the blank ones in the last block). Define a (resp P) to be the integer part of the average of these integers (resp squared integers).

[0161] And so,

[0162] j y y 41 ç 22" 41 hmXy! Cg{R,gM

[0163] and

[0164] / 4 Q. _ J_ y y c ' 2^ 4K 4 ] bmxyl . \ c^{r,g,b} / .

[0165] Then, we set:

[0166] j(0) := Ta(x(0)) / (0) := ^(2(0))

[0167] For the fifth and last initial condition, we compute intensities by block. The total intensity of the £-th block with g < £ < ^log„ । is:

[0168]

[0169]

[0170]

[0171]

[0172]

[0173]

[0174]

[0175]

[0176]

[0177]

[0178]

[0179]

[0180]

[0181]

[0182] c - Y ^v<2a-\ * Whereas, the total intensity 4i of the last block gets defined as: y __ Further define when [ / OS2 L]<[tog2 M]<2[zog, o<^<^ Li-i or when |*Zog2 Ml>2[tog2 l| 2^^2 ^)^2 £] < k < 2^2 _ J _ y y y ^ApC Now set: m(0): = LOppg, k=O Pfi rl[ 3Lx22”x2^ 4 if 1^2 £ |<| ^2 ifP^2 M]>2['zog2 l] From the seedx(0),y(0),z(0), / (0),w(0), run the dynamical System and generate sequences like in Section 3, scheme B. Using the same notations as in Section 3. scheme B, for an image m belonging to a block b and a pixel ( i, j) on that image, compute: -H4,.^-.(¾^) * l0'J mod with for instance 2^1 ta'n a secret 3 namely: &2lH ^ll / nMZ ’ ■ * • ■ • aQ«Aü > Préparé this ^ / og^pq^it ancilla and at the end of the scrambling procès s perform three CNOT gates (one per color) with control qubits those of the block, the image, the bit plane and the pixel position. The quantum computation results in XORing Kïjnpi.j ® PbmiJb for ail possible values of C, l, m, b, i and j. For each color, the CNOT gâte gets applied on the secret qubit ancilla serving as the control qubit and the bit value qubit of the color scrambled quantum multi-image which serves as the target qubit, with control qubits those of the bit planes, the blocks, the images and the pixel positions.

[0183] The decryption process for schemes C and D is precisely the same as the decryption procès s in schemes A and B.

[0184] Bibliography

[0185] [1] J. Fridrich, Symmetric ciphers based on two-dimensional chaotic maps, Int. J. Bifurcation and Chaos (1998) 8, 1259-1284

[0186] [2] M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys. (1976) 50(1):69-77

[0187] [3] C. Hou and X. Liu, Quantum image scrambling algorithm based on discrète Baker map, Modem Physics Letters A ( 2020)

[0188] [4]Z. Hua, B. Zhou, Y. Zhou, Sine chaotification model for enhancing chaos and its hardware implémentation, IEEE Transactions on Industrial Electronics, Vol. 66 Issue 2 (2019) 1273-1284

[0189] [5]X. Liu and C. Liu, Quantum image encryption scheme using independent bit- plane permutation and Baker map, Quantum Inf. Process. 22 262 ( 2023)

[0190] [6]X. Liu, D. Xiao, W. Huang and C. Liu, Quantum block image encryption based on Arnold transform and sine chaotification model, IEE Access Vol. 7 ( 2019) 55188-55199

[0191] [7]J.Z. Sang, S. Wang and Q. Li, A novel quantum représentation of color digital images, Quantum Inf. Process. 16 (2017) 1-14

[0192] [8]X. Song, G. Chen and A.A. Abd El-Latif, Quantum color image encryption scheme based on géométrie transformation and intensity channel diffusion, Mathematics 10 (2022) 3038

[0193] [9]S. Sun, A novel hyperchaotic image encryption scheme based on DNA encoding, pixel-level scrambling and bit-level scrambling, IEE Photonics Journal, Vol. 10, No.2 (2018)

[0194]

[10] N.R. Zhou, X. Yan, H. Liang, X. Tao, G. Li, Multi-image encryption scheme based on quantum 3D Arnold transform and scaled Zhongtang chaotic System, QIP (2018) 17:338

Claims

[Revendication 1] [Revendication 2] [Revendication 3] [Revendication 4] [Revendication 5] Revendications The use of any of the four novel quantum représentations for a multi-image disclosed here. Realizing a two-stage scrambling using quantum baker maps, scrambling the pixel positions on one hand and scrambling the bit planes and the images on the other hand. The implémentation of a general quantum baker map in ternis of quantum circuit. Cascading the sine chaotification of a chaotic discrète dynamical System and some Chebyshev transformations for generating pseudo-random bits used for diffusion. The use of any of the schemes A,B, C or D or any of their variants, resulting for instance from changing the chaotic transformations or the seeds, the order of application of the scrambling and the diffusion or the order of application of the two stage scrambling, for multi-image encryption / decryption, either classically or using a quantum computer.