Progressive Secret Image Sharing Using Multiplicative Primitive Irreducible Polynomials

Abstract

Progressive secret image sharing (PSIS) encodes a secret into $n$ shadows, enabling the secret to be gradually revealed as more shared data becomes available. However, current techniques suffer from non-uniform progressive recovery and inferior visual quality of low-level decoded images. To address these issues, we combine multiplicative primitive irreducible polynomials with polynomial ring Chinese Remainder Theorem (CRT) to establish a PSIS. A method with 2 progressive levels is presented, where two polynomials $f_{1}(\alpha)$ and $f_{2}(\alpha)$ determine a main polynomial $F(\alpha) = f_{1}(\alpha) \times f_{2}(\alpha)$ . A $(k-1)$ -degree polynomial over $(\bmod \; F(\alpha))$ is constructed for shadow generation. The secret can be losslessly recovered by Lagrange interpolation over $(\bmod \; F(\alpha))$ , or gradually recovered through modified shadows using Lagrange interpolation and polynomial ring CRT. The scheme is further extended to $t$ progressive levels, and a bit rearrangement technique is devised to enhance progressive recovery performance. Experiments and comparisons illustrate the effectiveness of the proposed technique, including uniform progressive recovery and superior low-level image quality.