A fast quantum image encryption algorithm based on affine transform and fractional-order Lorenz-like chaotic dynamical system

The arithmetic of finite fields with characteristic 2 is central in many cryptographic algorithms. Based on the normal basis representation for elements of the finite field \(\mathbb {F}_{2^n}\), quantum circuits for addition and multiplication have complexities O(n) and \(O(n^3)\), respectively. However, the complexity of the quantum circuit performing multiplication in \(\mathbb {F}_{2^n}\) can be reduced to \(O(n^2)\) provided that \(\mathbb {F}_{2^n}\) contains an optimal normal basis. In this paper, a lossless quantum circuit for the affine transforms on \(\mathbb {F}_{2^n}\) is designed with the quadratic time complexity for some values of n. Moreover, each constituent of the affine transforms on \(\mathbb {F}_{2^n}\) is a unitary operation, and accordingly, the domain of these transforms can be extended to the \(2^n\)-dimensional Hilbert space \(\mathbb {H}^{\otimes n}\). In addition, these transforms are one–one mappings from the set of computational basis states, of \(\mathbb {H}^{\otimes n}\), to itself. This property of affine transforms confirms that these transforms can be used in the encryption algorithms for quantum images because a quantum image uses the computational basis states for storing the color and positional information of pixels. Accordingly, a fast quantum image encryption algorithm combining affine transforms with fractional-order Lorenz-like system is presented for the novel quantum representation of color digital images. A selected pair of affine transforms is used to scramble the position information of the quantum image, while the color information of red, blue, and green layers of the scrambled quantum image is encrypted through controlled right translations determined by the fractional-order Lorenz-like system. Analyses of the time complexity reveal that the proposed quantum image scrambling scheme is exponentially faster than some state-of-the-art schemes. Simulation results confirm that the newly suggested encryption algorithm is fast, robust, and reliable.