Robust Secret Image Sharing Scheme Based on Polynomial <i>k</i> -Consistency

The (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>)-threshold Secret Image Sharing (SIS) is a naturally fault-tolerant technique for image privacy protection. A secret image is processed through secret sharing to generate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> shadow images, which are then distributed to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> different recipients. During the recovery phase, the complete secret image can be reconstructed by any <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> out of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> shadow images. Although (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>)-threshold SIS itself allows for the loss of up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n-k$ </tex-math></inline-formula> shadow images, if there are pixel errors in the remaining <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> shadow images, the recovery of the secret image will be declared a failure. Therefore, Robust Secret Image Sharing (RSIS) has been proposed to address the issue. However, the current proposed RSIS schemes only demonstrated limited robustness against noise attacks. This paper presents a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i>-consistency-based RSIS scheme to resist malicious attacks, including noise, JPEG compression, tampering, and cropping. In the sharing phase, a dual-SIS mechanism is first designed to perform two rounds of secret sharing on the secret image. In the recovery phase, high-quality secret image can be reconstructed based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i>-consistency after attacking. The experimental results demonstrated that our scheme not only provides comprehensive robustness but also allows for flexible adjustment of shadow images’ sizes, ensuring both security and efficiency during image sharing.