Multi-image quantum encryption scheme using blocks of bit planes and images

We present a multi-image quantum encryption/decryption scheme based on blocks of bit planes and images. We provide a quantum circuit for the quantum baker map.

💡 Research Summary

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The paper proposes a novel quantum encryption scheme for simultaneously transmitting a large set of images. The core contribution is a new quantum representation called QBBRMI (Quantum Block‑Bit‑Plane Representation for Multi‑Image), which groups the M images into blocks, each block containing up to 2⌈log₂L⌉ images where L is the number of bit‑planes (normally 8). By encoding the block index, the intra‑block image index, the pixel coordinates, and the bit‑plane index, the total number of qubits required for an 2ⁿ×2ⁿ image becomes

  2ⁿ + ⌈log₂L⌉ + ⌈log₂M⌉

instead of the 2ⁿ + 2⌈log₂M⌉ required by earlier approaches. For example, encoding 200 gray‑scale 256×256 images (n = 8, L = 8) needs only 28 qubits with the proposed method, whereas the reference scheme needs 33 qubits.

Encryption proceeds in two major stages. First, a quantum baker map is employed to scramble the data. The baker map is a piecewise linear chaotic transformation that can be implemented using only SWAP and controlled‑SWAP gates when its parameters (q₁,…,q_k) satisfy certain divisibility conditions. The authors apply the baker map twice: (i) a block‑independent scrambling that simultaneously permutes block identifiers, bit‑plane identifiers, and image identifiers; (ii) a second scrambling that, for a fixed block‑image‑bit‑plane triple, independently permutes the pixel coordinates. This double‑layer scrambling destroys all statistical correlations among pixels, bit‑planes, and images.

The second stage is diffusion, realized by a five‑dimensional hyper‑chaotic system whose equations are taken from a known chaotic model. To increase key sensitivity and enlarge the key space, each equation is “sine‑chaotified”: the output of the original map is passed through a sine function with a scaling factor λ ≥ 1. This operation raises the Lyapunov exponent, making the system more sensitive to initial conditions. The initial state of the chaotic system is derived from the plaintext: the total intensity of all images determines x(0), while two plaintext‑dependent integers α and β (averages of the number of ‘1’ bits and their squares across all images) are transformed by Chebyshev polynomials to obtain y(0) and t(0). Consequently, an attacker who does not know the exact number of images or their content cannot reconstruct the chaotic sequences.

After discarding a transient of 100 iterations, four pseudo‑random sequences are generated: one for pixel positions, one for image‑block ordering, and two for bit‑plane and block indices. These sequences are sorted, and the rank of each element provides permutation indices (n_i, k_i, r_i, s_i). For each pixel (i, j) of image m in block b, a secret key K_{l,m,b,i,j} is computed as a modular combination of the four sequences, producing a 2⌈log₂L⌉‑qubit value. The key bits are XORed with the scrambled image bits, and the diffusion is completed by applying 2^{2n+⌈log₂M⌉+⌈log₂L⌉} CCNOT gates using ancilla qubits that store the secret keys. The resulting quantum ciphertext is measured to obtain a classical bit‑string, transmitted, and later decrypted by reversing the entire quantum circuit and performing a final set of projective measurements.

The authors present a detailed quantum circuit for the general quantum baker map. The map is decomposed into sub‑functions f₁,…,f_k, each realized by a combination of SWAP, controlled‑SWAP, and CNOT gates. The circuit depth grows linearly with the number of blocks and with the chosen parameters, but because the total qubit count is reduced, the overall resource requirement is lower than in prior work. The diffusion stage’s CCNOT count is explicitly given as 2^{2n+⌈log₂M⌉+⌈log₂L⌉}, which, while exponential in n, is comparable to the size of the data being encrypted and is feasible for near‑term fault‑tolerant quantum computers.

Security analysis highlights three main strengths: (1) Key sensitivity – the sine‑chaotified hyper‑chaotic system yields a very large Lyapunov exponent, so a minute change in the plaintext or in any λ_i leads to a completely different key stream; (2) Infinite key space – the scaling parameters λ_i can take any real value ≥ 1, effectively providing an unbounded key space; (3) Plaintext‑dependent initialization – because the chaotic seeds are derived from the total image intensity and bit‑statistics, chosen‑plaintext or chosen‑ciphertext attacks are thwarted.

The paper concludes that the block‑bit‑plane representation together with the quantum baker map and high‑dimensional chaotic diffusion offers a more qubit‑efficient and secure method for multi‑image quantum encryption. It acknowledges that practical implementation will require careful selection of baker‑map parameters to match hardware constraints and that experimental validation on real quantum processors is an important direction for future work.