Matrix embedding in steganography with binary Reed–Muller codes

This study presents a modified majority-logic decoding algorithm of Reed–Muller (RM) codes for matrix embedding (ME) in steganography. An ME algorithm uses linear block code to improve the embedding efficiency in steganography. The optimal embedding algorithm in steganography is equivalent to the maximum likelihood decoding (MLD) algorithm in error-correcting codes. The main disadvantage of ME is that the equivalent MLD algorithm of lengthy embedding codes requires highly complex embedding. This study used RM codes to embed data in binary host images. The authors propose a novel low-complexity embedding algorithm that uses a modified majority-logic algorithm to decode RM codes, in which a message-passing algorithm (i.e. sum-product, min-sum, or bias propagation) is performed on the highest order of information bits in the RM codes. The experimental results indicate that integrating bias propagation into the proposed scheme achieves superior embedding efficiency (relative to when the sum-product or min-sum algorithm is used) and can even achieve the embedding bound of RM codes.