Information is uploaded and downloaded through the Internet day in and day out ever since we immersed ourselves in the Internet. Data security becomes an area demanding high attention and one of the efficient techniques to protect data is data hiding. Among various data hiding schemes, more applications prefer the schemes with reversibility. In recent reversible data hiding studies, the indices of a codebook can be reordered to hide secret bits. The hiding capacity of the codeword index reordering scheme increases when the size of the codebook gets larger. Since the codewords in the codebook are not modified, the visual performance of compressed images is retained. We propose a novel scheme making use of the fundamental principle of codeword index reordering technique to hide secret data in encrypted images. By observing our experimental results, the embedding capacity is significantly larger. Secret data can be extracted when a receiver owns a data hiding key and image can be recovered when a receiver owns an encryption key.

Jacobian-free Newton Krylov (JFNK) is an attractive method to solve nonlinear equations in nuclear engineering systems, and has been successfully applied to steady-state neutron diffusion (k-eigenvalue) problems and multi-physics coupling problems. Preconditioning technique plays an important role in JFNK algorithm, significantly affecting computational efficiency. The key point is how to automatically construct a high-quality preconditioning matrix that can improve the convergence rate, and perform the preconditioning matrix factorization efficiently. An efficient reordering-based ILU(k) preconditioner is proposed to achieve the above objectives. In detail, the finite difference technique combined with the coloring algorithm is utilized to construct an efficient preconditioning matrix with low computational cost automatically. Furthermore, the reordering algorithm is employed for the ILU(k) to reduce the additional non-zero elements and pursue robust computational performance. A two-dimensional LRA neutron steady-state benchmark problem is used to evaluate the performance of the proposed preconditioning technique, and steady-state neutron diffusion problem with thermal-hydraulic feedback is also utilized as a supplement. The results show that coloring algorithms can efficiently and automatically construct the preconditioning matrix. The reordering-based ILU(k) preconditioner shows excellent robustness, avoiding the effect of fill-in levels k choice in incomplete LU factorization.