Analysis of the Correspondence Between DCT and Non-Classical Walsh-Hadamard Transforms for Code-Controlled Steganography

In the modern digital environment, where vast amounts of multimedia data are continuously exchanged across global networks, ensuring secure and covert communication has become a critical challenge. Steganography plays a crucial role in this context by allowing the imperceptible embedding of secret information into digital media, including images, audio, and video signals. However, most state-of-the-art steganographic methods rely on computationally intensive mathematical transforms, including the Discrete Cosine Transform (DCT), Singular Value Decomposition, and various wavelet-based or deep-learning-based approaches. While these techniques achieve high robustness and embedding capacity, their high computational complexity limits practical implementation on resource-constrained platforms such as mobile devices, embedded systems, and Internet-of-Things environments. A promising direction for overcoming these limitations is the development of code-controlled steganographic methods, which operate directly in the spatial domain. Their efficiency is based on establishing deterministic relationships between DCT coefficients and those of the Walsh-Hadamard Transform (WHT), enabling precise code-based control of the embedding process. To date, such research has been limited to classical Sylvester-type Hadamard matrices. However, multiple non-equivalent classes of Hadamard matrices exist, and exploring their properties may unlock new opportunities for efficient and secure information embedding. This paper presents a comprehensive analysis of the correspondence between DCT coefficients and WHT coefficients derived from five non-equivalent classes of Hadamard matrices of order 16. Large-scale experimental research was performed using 530 images from the NRCS public database. Each image was processed in 16×16 blocks, with controlled perturbations applied to individual DCT coefficients. The resulting changes in WHT coefficients were analyzed to determine the mappings between the transform domains. The results reveal that while the classical Sylvester matrix preserves a one-to-one correspondence with DCT, non-classical Hadamard matrix constructions exhibit overlapping mappings, particularly in the high-frequency range. This overlap, although reducing spectral resolution, introduces beneficial redundancy for steganographic applications, enhancing robustness and resistance to steganalysis. The findings provide a foundation for developing high-performance, spatial-domain steganographic schemes that exploit the structural diversity of non-classical Hadamard matrices for improved efficiency and security.