A Recursive Threshold Visual Cryptography Scheme

This paper presents a recursive hiding scheme for 2 out of 3 secret sharing. In recursive hiding of secrets, the user encodes additional information about smaller secrets in the shares of a larger secret without an expansion in the size of the latter, thereby increasing the efficiency of secret sharing. We present applications of our proposed protocol to images as well as text.

💡 Research Summary

The paper introduces a novel recursive hiding technique applied to a 2‑out‑of‑3 visual secret sharing scheme, aiming to dramatically improve the information efficiency of both text‑based and image‑based secret sharing. Traditional (k,n) secret sharing expands a secret of b bits into n shares each of at least b bits, so each share conveys at most 1/n bits of the original secret. Visual cryptography, which splits an image into multiple shares that reveal the secret when a predetermined number of them are stacked, suffers from a similar inefficiency: a single pixel often becomes several sub‑pixels across the shares, inflating the data size by a factor of three to nine.

The authors first describe a comparison‑based 2‑out‑of‑3 scheme for binary text. A secret bit is represented by three symbols p₁, p₂, p₃ drawn from the set {0, 1, 2}. To encode a ‘0’, all three symbols are identical (e.g., 000, 111, 222); to encode a ‘1’, the three symbols must be a permutation of 0, 1, 2 (e.g., 012, 021, 102, 120, 201, 210). This yields three possible encodings for ‘0’ and six for ‘1’, providing flexibility while preserving the 2‑out‑of‑3 threshold property. The paper gives a concrete example with a 27‑bit message, showing how the three shares are constructed and noting that, when the ternary symbols are further encoded with a prefix code (0→0, 1→10, 2→11), each secret bit expands to five transmitted bits, giving an overall efficiency of 20 %.

Recursive hiding is then introduced to overcome this inefficiency. The key idea is to embed smaller secrets within the shares of a larger secret, with the size of the secrets increasing by a factor of three at each recursion level. The authors present a table (Table 1) that demonstrates how three smaller messages (M₁, M₂, M₃) are hidden inside the three shares of a larger message M. Crucially, the shares of the smaller messages are distributed among the three participants so that no single participant holds all the shares of any smaller secret; at least two participants must collaborate, preserving the 2‑out‑of‑3 security guarantee. This distribution contrasts with earlier work