A Monoid Ring Approach to Color Visual Cryptography
Maximilian Reif, Jens Zumbrägel
TL;DR
This work extends color visual cryptography by modeling colors as elements of a finite commutative monoid $M$ and recovering the secret via the monoid ring, thereby generalizing the KI11 multivariate-polynomial framework to color images. It formalizes the color model, the corresponding visual-cryptography schemes, and the polynomial description, with a differential operator $ abla$ (represented as $ abla = d_a + d_{c_1} + abla ext{…}$ in the text) guiding subset restrictions; the relation $p^{(ullet)}$ connects polynomials to subset-specific basis matrices. The authors present constructions of basis matrices for various $(t,n)$-threshold schemes using the extended color monoid (e.g., $igl\\{0,\, ext{C},\, ext{M},\, ext{B},\,1igr\\}$) and explicit polynomials that achieve reduced pixel expansion and improved contrast, including monoid-ring interpretations of stacking. The monoid-ring approach captures color stacking algebraically and suggests future work on broader monoids and grayscale-like behavior, with potential impact on more realistic color reconstruction in visual cryptography.
Abstract
A visual cryptography scheme is a secret sharing scheme in which the secret information is an image and the shares are printed on transparencies, so that the secret image can be recovered by simply stacking the shares on top of each other. Such schemes do therefore not require any knowledge of cryptography tools to recover the secret, and they have widespread applications, for example, when sharing QR codes or medical images. In this work we deal with visual cryptography threshold schemes for color images. Our color model differs from most previous work by allowing arbitrary colors to be stacked, resulting in a possibly different color. This more general color monoid model enables us to achieve shorter pixel expansion and higher contrast than comparable schemes. We revisit the polynomial framework of Koga and Ishihara for constructing visual cryptography schemes and apply the monoid ring to obtain new schemes for color visual cryptography.