New quantum multi-image encryption and decryption protocol

Abstract

We disclose novel ways to encrypt / decrypt multiple images using a quantum computer. Our encryption / decryption schemes can also be used on classical computers. We invent two novel quantum representations for a gray-scale multi-image and two novel quantum representations for a color multi-image. They allow for accurate retrieval of the classical images. We operate a two-stage scrambling of the quantum multi-image using quantum baker maps: a scrambling of the images and of the bit planes on one hand and a scrambling of the pixel positions on the other hand. When the number of images to encrypt is large, the images are split into blocks of images in order to reduce the number of qubits needed. We disclose a general quantum circuit implementing the quantum baker map for any values of its parameters. Diffusion of the scrambled multi-image is realized using three different chaotic systems or maps, namely the sine chaotification of a chaotic system and some Chebyshev transforms.For all the schemes, an attackant has no way of knowing the precise number of images being transmitted.

Description

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[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 attackant 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 chao-tification 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. .

[0005] We gathered the novelties of our schemes in the five points below. 1. We disclose two novel quantum représentations for a gray-scale multi-image. For future reference, we call these two quantum représentations représentation A and re présentation B respectively. Suppose there are M gray-scale images of size 2" 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 repré sentation uses a total of jyl + । qubits, namely 2n qubits for the pixel positions, jyj qubits to encode the bit planes, pOg / yl 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 2^,z| Consequently, there are 2^, Z.] blocks , each containing ^[z^ Z.] images (counting also the blank images in the last block in the case when Mp- Thus, there are p^g _ p^ qubits encoding the blocks and p^g qubits encoding the images belonging to a given block, thus a total of 2n 4-14- [fog -I- [ / og Lj qubits for représentation B. 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 planes in p^g jjj qubits, instead of L qubits like in the well-known NEQR représentation, originates in [X. Liu and C. Liu, Quantum image encryption scheme using independent bit-plane permutation and Baker map, Quantum Inf. Process. 22262( 2023) ]• In [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, Quantum Inf. Process. (2018) 17:338], the authors also deal with multi-image quantum encryption / de- cryption, 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 adding n 2^1 extra qubits instead of simply extra qubits like in our quantum représentation A when 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 [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], 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 mea-surements. 2. 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. 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 [X. Liu and C. Liu, Quantum image encryption scheme using independent bit-plane permutation and Baker map, Quantum Inf. Process. 22262(2023) ], 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. 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 2Z> Hénon transform [ M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys. ( 1976) 50(1):69-77] (resp of a 5D hyperchaotic System, e.g. the one of [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)]), 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. 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 in-troduced in [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], 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. . The main novelty in our diffusion procès s 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 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 in-dependent diffusion like in scheme A. 3. It is known that the quantum baker map may be implemented using SWAP quantum gates and controlled SWAP quantum gates, see e.g. [X. Liu and C. Liu, Quantum image encryption scheme using independent bit-plane permutation and Baker map, Quantum Inf. Process. 22262( 2023) ]. 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. 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.

[0006] We now provide ail the technical details for the novel quantum représentations for a gray-scale multi-image, the quantum implémentation of the quantum baker map, the génération of the secret keys used for diffusion, the encryption / decryption schemes A and B and the generalized schemes C and D for a color multi-image.

[0007] Below, i dénotés the number of bit planes, M dénotés the number of images to encrypt / decrypt. Our own convention is that the O-th bit plane is the least significant bit plane.

[0008] I. Novel quantum représentations for a gray-scale multi-image Scheme A

[0009] The novel quantum représentation for a gray-scale multi-image is:

[0010] I y1 2^ AL 1 y2^'71 i y 2"-1 y 2" A p 4-m=0 4- / =0 Ly^y^jPmlxy

[0011] P mlxy is the bit value in the / -th bit plane of the w-th image at pixel (x, y ). With this représentation, it is understood that we set:

[0012] Pmlxy • ~ 0 when j 1 if < 2^°¾^

[0013] Pmlxy . = Ôm js O(jd when - ] if < 2^2^

[0014] where <5 dénotés the Kronecker Symbol. Of course m is odd could be replaced with m is even instead. 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> imposeM = L. Scheme B

[0015] The novel quantum représentation by block for a gray-scale multi-image is:

[0016] । y2'-1 f B > ~ tmxyl | / > | ^Z > J.V V -

[0017] with

[0018] 2: = .............*........—. A / ]

[0019] In this représentation, Ptmxyi is the bit value of the m-th image belonging to the r-th block at pixel (x, y) on the / -th bit plane. With this représentation, it is understood that we set: Ptmxyl • — 0 when j i < - 1 if L < 2^^

[0020] and [0021 ] Ptmxyi : = Ôm is odd when M - 2f M 1 + 2^^1 < m < 2^^2^1 -1 if M < 2^¾^

[0022] and t = 7-] ।

[0023] Again, in the Kronecker Symbol, m is odd could be replaced with m is even.

[0024] II. Quantum implémentation of the quantum baker map

[0025] The discrète baker map, as originally introduced in [J. Fridrich, Symmetric ciphers based on two-dimensional chaotic maps, Int. J. Bifurcation and Chaos ( 1998) 8, 1259-1284], is defined on a square of size 2” X 2” with

[0026] 2^ = 2^+ ... + 2^

[0027] whcrc the 7, ’s are the parameters of the map,

[0028]

[0029] as: N o = 0 and N t = 2^ 1 + ... + 2 9 ' and x = +1, ..., -1 / , / ir \ j \-vmod2 ïf \ (x, y) = 12 (x-N^) + y mod 2 \ + —I

[0030] As noticed by the authors of [C. Hou and X. Liu, Quantum image scrambling algorithm based on discrète Baker map, Modem Physics Letters A ( 2020)], 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] M s (x, y) = (2"’x mod 2 tl +y mod 2^, —fx-x mod 2'j

[0032] has a quantum implémentation. (ii) Each subfunction on {A^4, ..., Nj - 1} of the discrète baker map is Mq. if and onlyif2sp^+ ... + 2^- A discrète baker map with parameters satisfying to the divisibility condition of ( U ) is called a quantum baker map. Its parameters are secret keys in our schemes. In [X. Liu and C. Liu, Quantum image encryption scheme using independent bit-plane permutation and Baker map, Quantum Inf. Process. 22262(2023) ], 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 B(q with parameters ( z? q ) such that 2” = 2?! + + and divides 9^1 + + for each i

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[0044] with 2 < i < k. Let / 1 be the subfunction defined on |jq 2^1 - 1| j • And so we hâve / i = Mqv Explicitly, f j ( y ) = ( ----¾ y^^ ■ • • yff -Vi • • • - • • y^ ) Let Sj be the subfunction of the quantum baker map defined for the integers À with 2^1 + ... + £ x £ 2^ + ... + tlA + -1. / 5 will be our base function from which the other ^,’s are built. We call / . the function such that The quantum circuit for f y is only composed of SWAP gates, whereas the quantum circuits for the other f .’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 f ? -.., f k sequentially. If 9,- " 9[, then is Mq^ — f Therefore, f{ is simply the identity. Otherwise, write + + 2^1 = x ] Xo 'n ^ase and 'ct t>e the set of indices i such that in this binary writing = 1. Then, the Controls on the SWAP gates of / . are defined like follows: (i) Suppose 2 < i < k - 1, then r = H rM cW * IjcJ .y =1 11 c (=0 (ii) c = n '-'k ■ 1 lyez^^l where the braces indicate the conditions to be satisfied for the control to happen and the products signify conjunction of Controls. Claim 1. Quantum circuit for the quantum baker map. Quantum circuit for f r if AqB-^2 ■ • • if """y""" s 7+1 with 2 S l s n-1 / r = wiïh3<l<n-l CQ ifq^n-1 Quantum circuit for f r with 2<r<k. 'e*1’ if Cf "V -y c)2) 'lf<lr=‘2 e)3ie)4' g(3)g(5) Ü-- p A^ p 2^ — 2(3)Ô(6) Æ»- p A^ p 2q if — < qr < qx 1 41 Q^Q{^ p p if 4] < 4,. < 2-4[ Q(r)Q{9- A. -p p if = e(7)g(10) if 4^¾

[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 .... 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. A o : = 11; .7 vu 1 / ’! v . ? ) b» - ■— flj-o yiq-(1-2) n-l-i ) H=o O^q y^ ,( / .2),,,,-.,+,) c ( , : = (n^s^u+A0)(n^

[0046] Quantum gates for f r-

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

[0048] e;'0:=n^'C;.w^^

[0049] e<2). = -v)

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[0068] y.)^' r 'c,SWAP(y f x^) 2,' 41 : = nu ; nu Q? : = nX'c^WAP(x^^ y^A} e' : =nu - HV C^AP(y,^ z O / _q C r SWAP\ X„^ | ■ )Oy = o n^.f) C t SWAP(x n - q ^ Xn-q t +q-iA+j) = Q, - -0 / -0 C fSW AP ^X n .q+i, jOj = ()O / _Q C r SWAP^Xn-qp+j, Xn-qçi-l+j'j .A 11, GSWMP(y^^^^ ej” : = & ' 0 * : = nU'C^WAP ( ?w> +i ) nX'c^SWAP ( X,,, +f , ) III. Génération of the secret keys used for diffusion Scheme A One of the classical chaotic Systems is the 2-dimensional Hénon discrète dynamical System: X +1 =l-û ^ + T W y ,-b x n F n+1 It exhibits a chaotic behavior when the two parameters are respectively set to a = 1.4 and b — 0.3. It was introduced by Michel Hénon in the 1970’s, see [M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys. ( 1976) 50(1): 69-77], We now define a new two-dimensional chaotic System which is a sine chaotification of the Hénon System: ( X / + i = sin 7TÀ m ^1-1.4 xj+y.. ) ) [y. +1 = (0.3 x,)} For each image m, we pick different parameters jp”) > । and ^0«) > p These pa rameters are secret keys. The sine chaotification increases the two Lyapunov exponents

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[0081] by ) and lu ) respectively. 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 _ V 0<a<2"-1 V.™n A0. -À, (K^-l ’ with M the total number of images to be encrypted (without counting the blank images) and Vxym the pixel value of the / n-th image at pixel (x, y) . Then we set: y0- = where Tk is the A-th Chebyshev polynomial. The Chebyshev polynomial can be im-plemented directly by: W)=L^(l)(x2-i)V21 or inductively by: (T^L T^X U*+2(X)=2XT M (X)-T lt (X) With the seed depending on the multi-image plaintext, the diffusion process is not only related to some secret keys, but also related to the multi-image plaintext. From the seed, iterate the discrète dynamical System as long as is necessary to obtain a sequence of 2" distinct values • • ■, for x and a sequence of 2” distinct values ' ’ ' ’ m 2"-1 f°r Giscard the first hundred itérations so as to avoid transient effects. Sort the two pseudo-random lists by increasing order and assign to the horizontal i-th (resp vertical J-th) pixel coordinate the integer (resp denoting the index position of (resp in the respective sorted lists. For a pixel ( i, j ), compute [r......(vJMW x io**] mod 2*"*. with for instance > 7^0¾ ^1- Obtain a ^pognM]-qubit secret key: Scheme B According to [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)], the 5-D hyperchaotic System

[0082] ' x3 = a ( x2 - X5 ) + x2x3x4 x2 = b ( x ] + x2 ) + -¾ - xJX3X4 x3 = - cx2 - dx3 - ex4 + x3x2x4 x4- -fx4+x 5X2X3 a5 = -SÙ1 + X2)

[0083] with control parameters set to a — 30, b = 10, c — 15.7, J — 5, e — 2.5, / = 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 [S. Sun, A novel hyperchaotic image en- cryption scheme based on DNA encoding, pixel-level scrambling and bit-level scrambling, IEE Photonics Journal, Vol. 10, No.2(2018)]. We define a 5-D hyperchaotic System which is a sine chaotification of the previous System:

[0084] x = sin ( ttX4 ( 30 ( y - x ) + y zi ) ) y = sin ( æ22 ( 10 ( x + y ) + u - xzt ) ) z - sin(^3( -15.7y - 5z- 2.5 / + xyt) ) t = sin ( jtà4 ( - 4.45 / + xyz ) ) ü = sin ( ?U 5 ( - 3 8.5 ( x + y ) ) )

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[0087] 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 :(0) [Mo410a(0),1]x1<F] 102

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[0090] These two numbers belong to the interval [0, []. The initial values y(0) and t(0) are then obtained by applying some Chebyshev transformation to respectively x(0) and z ( 0 ) • Explicitly, x(0) : ■ 7 M2 12-i with Vxym the pixel value of the image m at pixel ( x, y ) • As for y ( 0 ) and / ( 0 ), define for instance two plaintext dépendent integers:

[0091] a\ = 1 jE i

[0092] And

[0093] b- = 2-;: v / T ^JE

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[0105] 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: y(0) := 7^(0)) KO) := O(z(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 < 2^5, m\. The total intensity of the k - th block with (Kt.v<2"-Ï xym is: Whereas, the total intensity of the last block gets defined as: Further define when m] < 2[ / ^ / .] PS Now set: m(0) : = Lx2""x( y2l h

[0106] or when 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 2^, e] blocks in the case when the number of blocks is greater than k]. For each such image, we compute the sum of ail the bit values on ail the bit planes. We obtain Min 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),y(0),<:(0), / (0), m ( 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:

[0107] -¼ -% --- 5 ¾]

[0108] Then sort the four lists by increasing order and assign to the i-th (resp J-th) pixel co-ordinate (resp m-th image, resp 1 h block) the index position of xi (resp ^j, resp zm, resp in the sorted sequences. Obtain in turn four sequences of integers:

[0109] «O ...,¾ ..., k^

[0110] For an image m belonging to a block b and a pixel ( i, j ) on that image, compute: 101111 x 1O 1 mod

[0112] with for instance Obtain a ^[ / og^Lpqubit secret key

[0113] > •

[0114] IV . Encryption and Decryption

[0115] 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 pa-rameters 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

[0116] Scheme A

[0117] 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 (j, y ), we apply a quantum baker map on the two qubits > and \l > . The parameters, as well as the number of itérations of the QBM dépend on the pixel. We obtain:

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[0129] lw> ) 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: 12Mli w=o >\ml> ([vy > ) 1 vIl r," i 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 2^ + |^2^ CCNOT q ua nt U m gates, so as to obtain the quantum multi-image ciphertext: |C > = —-r—0 > ImZxv > 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 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: 2"-i y \P lmxy i> \tmlxy> We then fix a bit plane and an image belonging to a certain block and perform in-dependent scramblings of the pixel positions. It yields: 101301 | / > = .V2"-1 v2pf’°-LUv2^^ i ; " ^^x.^O^mJ^O ^1=0 I ^tmxyl > i >

[0131] From there, the scrambled quantum multi-image is ready to be diffused.

[0132] Préparé qubit ancillas encoding the secret keys in order to perform

[0133] jwMfog l] r’irixrr'A'T «- » 2 n ^2 n °2 i CCNOT quantum gates,

[0134] so as to obtain the quantum multi-image ciphertext:

[0135] ,v2"4 v2N3tiv2^^ mn l , C > =2jL ruL / n L, n [K, ®P Tmxv i>\tmlxV> 1 B ^xy=&—tmJ=A) r l^j^y imxyi |

[0136] 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.

[0137] Last, the quantum ciphertext is converted to its classical version using projective measurements and sent to the receiver. Decryption

[0138] 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, using the same secret keys. 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 less 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.

[0139] V . Generalized schemes for a color multi-image

[0140] 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 [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] 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 .

[0141]

[0142]

[0143]

[0144]

[0145]

[0146]

[0147]

[0148] it is necessary. We first provide the bit plane quantum représentation for a color multi-image as with 4 like already defined before. Pmlxv (resP Pmixv’ Pmh^ 's value in the Z-th bit plane of the / n-th image at pixel (x, y) for color R (resp G, B). If p < p the P^y^ where C e G, B}, are ail set to the bit value 0 when £ < ^ < ^los^ zj p If set 'Ol instance P(M+k ) Ixy — ^{À-0 mod. 3} G P (M+k) Ixy — ^{A—1 mod 3} ,P(M+.k) Ixy — &{k=2 mod 3} That is, add red, green and blue images in alternance. In our quantum encryption / de-cryption 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> 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: tmxylPtmxyl > jtW.Zxy > with 4 defined like before. , (resp p? „ pŸ ,) is the bit value in the / -th bit plane of the / 7 / -th image belonging to the '-th block at pixel (x, y) for color R (resp G, B). If £ 2^¾ Z-] p the P^,ïy / s, where C e {7?, G, B}, are ail set to the bit value 0 when T . J / op d ,. Finally, if . , then set for instance

[0149]

[0150] G=7r*2 h - 1 }xyl " ^=0mod 3] P J ’i 4], 4^8, G — <5(t—1 mod 3! . )xy / ^{k-2 mod 3} for when M _ + m <

[0151] We will call the encryption / decryption scheme using this quantum représentation for a color multi-image scheme D.

[0152] 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.

[0153] For the diffusion stage, we keep the same chaotic discrète dynamical Systems and adapt the initial conditions.

[0154] 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 T is a Chebyshev polynomial:

[0155] [ _ jX° " 3(2^-1)22¾ v = Ty Pc (a'o)

[0156] From the seed, iterate the discrète dynamical System as long as is necessary to obtain a sequence of 2” distinct values • • • ’ *mf-l for A and a sequence of 2” distinct values ?mff ■ • • ’ 3^2"-[ for -4 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 Mh (resp vertical J-th) pixel coordinate the integer (resp Imj) denoting the index position of (resp in the respective sorted lists.

[0157] For a pixel ( j ), compute [01581

[0159] with for instance >3 2^2^' Obtain a secret 3 namely:

[0160]

[0161] 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: ioi«i k^qp^j

[0163] 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.

[0164] 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.

[0165] We now proceed scheme D.

[0166] We keep the sine chaotification model of the 5D hyperchaotic System of scheme B and take five initial conditions as follows.

[0167] [ ffA • _ 1^'4 10a(0),1]x102| 14°) : = *-----n?-----

[0168] x(0) 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 / i) to be the integer part of the average of these integers (resp squared integers).

[0169] And so,

[0170] i y y1 ç 22" 1 Ll 1

[0172] fi- =

[0171] and 1 y y pC hmxyl \ Ce{R,G£}

[0173] Then, we set:

[0174] y(0) / (0) := MdO))

[0175] For the fifth and last initial condition, we compute intensities by block. The total intensity of the k-th block with 0 < k < L] _ 1ls:

[0176] {E,V.y£2"-l

[0177] Whereas, the total intensity of the last block gets defined as:

[0178] or when

[0180]

[0181]

[0182]

[0183]

[0184]

[0185]

[0186]

[0187]

[0188]

[0189]

[0190]

[0191]

[0192] Pk Now set: u(Q) Ce]ÆG3} l] C 1=0 P k ( 2K r-U-1 ) xvl ,, T,, ifd<[ / og, MW / og, ^2¾^ 12 1 1 2 1 ' 2 1 if l] 12 I I 2 I From the seed x ( 0 ), y ( 0 ), z ( 0 ), t ( 0 ), U ( 0 ), iterate the dynamical System and generate four sequences of distinct numbers: -¾ ...,x / 4 >0’ ---¾ H)’ •.., } / q, ..., When doing so, ignore the first hundred itérations so as to avoid transient effects. Then sort the four lists by increasing order and assign to the i-th (resp J-th) pixel co-ordinate (resp m-th image, resp è-th block) the index position of xi (resp resp zm, resp tb) in the sorted sequences. Obtain in turn four sequences of integers: - - - ^4 For an image m belonging to a block b and a pixel ( / , j ) on that image, compute: mod with for instance > 3 9¾. Obtain a secret $ 9^¾^]-qubit, namely: Préparé this ancilla and at the end of the scrambling process 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

[0193] bmi jb

[0194] for ail possible values of C, Z, 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.

[0195] The decryption process for schemes C and D is precisely the same as the decryption procès s in schemes A and B.< / m>

Claims

Revendications [Revendication 1] The use of any of the four novel quantum représentations for a multi-image disclosed here. [Revendication 2] 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. [Revendication 3] The implémentation of a general quantum baker map in ternis of quantum circuit, as disclosed here, or as a similar variant. [Revendication 4] Cascading the sine chaotification of a chaotic discrète dynamical System and some Chebyshev transformations for generating pseudo-random bits used for diffusion. [Revendication 5] 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 en-cryption / decryption, using a classical computer or using a quantum computer.