RDHEI with CRTP-SS and Adaptive Coding — MATLAB Implementation

MATLAB implementation of:

G. Fang, F. Wang, C. Zhao, C. Qin, C.-C. Chang, C.-C. Chang, "Reversible Data Hiding With Secret Encrypted Image Sharing and Adaptive Coding" IEEE Internet of Things Journal, vol. 12, no. 13, pp. 23933–23945, July 2025. DOI: 10.1109/JIOT.2025.3555380

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Overview

A multi-party RDHEI scheme with two key components:

1. CRTP-SS encryption — secret sharing based on the Chinese Remainder Theorem for polynomials over GF(2⁸), using irreducible degree-8 polynomials as moduli. Unlike integer-CRT schemes, encrypted pixels are uniformly distributed over the full 0–255 range, and the in-block XOR structure is preserved (Eq. 8), vacating room for embedding with no preprocessing.
2. Adaptive-coding (AC) embedding — XOR-preservation (XORP) compresses each 2×2 block against its reference pixel; per-block thresholds λ are labeled with Huffman codes (≤ 5 bits, guaranteed decodable on the fly).

The original image is recovered losslessly from any k of n marked shares; secrets embed/extract bit-exact per data hider.

Content owner : scramble (Key0) ──> CRTP-SS (k,n) ──> n encrypted shares
Data hider i  : AC embed (Key3_i) ──> marked share i
Receiver      : any k shares ──> AC extract + restore ──> CRT-P ──> unscramble

Repo Structure

File	Implements
src/gf2_{deg,mul,divmod,mod,inv}.m	GF(2)[x] polynomial arithmetic (uint64 bitmasks)
src/irreducible_polys.m	Paper's moduli q₀…q₄ + all 30 irreducible degree-8 polys
src/scramble_image.m / unscramble_image.m	Key₀ block + pixel scrambling (Sec. IV-A)
src/crtp_share.m	CRTP-SS encryption, Eqs. 5–6
src/crtp_reconstruct.m	CRT-P reconstruction, Eq. 14
src/huffman_label.m	λ label codes + threshold-merge rule (Sec. IV-B)
src/ac_embed.m	Adaptive-coding embedding (Fig. 4)
src/ac_extract.m	Extraction + share restoration (Fig. 5)
utils/image_metrics.m	Entropy (Eq. 17), NPCR, UACI, PSNR
main_demo.m	Full (3,4)-threshold pipeline demo
run_all_tests.m	9 test groups incl. the paper's worked example
run_experiments.m	Tables II/VI/VII + Fig. 10 equivalents
validate_algorithm.py	Python pre-validation of the algorithm design

Quick Start

addpath(genpath('src')); addpath(genpath('utils'));

k = 3; n = 4;
[q0, q_list] = irreducible_polys(n);
img = imread('house.png');                 % 8-bit grayscale, even dims

% Content owner
scr    = scramble_image(img, [11 22]);     % Key0
shares = crtp_share(scr, n, k, 33, 44, q_list, q0);   % Key1, Key2

% Data hider i (independent)
[~, ~, st] = ac_embed(shares{1}, [], 101); % query capacity
secret = randi([0 1], 1, st.pureEC);
[marked, side] = ac_embed(shares{1}, secret, 101);    % Key3

% Receiver (any k shares)
[restored, secret_out] = ac_extract(marked, side, 101);
% ... collect k restored shares, then:
s_total = numel(img) / 4;
rec = unscramble_image( ...
        crtp_reconstruct({restored, r2, r3}, q_list([1 2 3]), q0, 33, s_total), ...
        [11 22]);
isequal(rec, img)        % 1 — lossless

Run main_demo, run_all_tests, or run_experiments directly.

Verified Against the Paper

run_all_tests.m T2 checks the paper's worked example (Sec. IV-A): block [192 191; 195 180] with R1=10000000, R2=…0001, q₀=111110101, q₁=100011011 encrypts to share-1 block [174 209; 173 218] — bit-exact. The same example and the full pipeline were independently validated in validate_algorithm.py (all 6 checks pass, including 2000 random pixels × all C(4,3) reconstruction combos).

Algorithm Notes

• λ classification (Table I): λ = 0 if Dmax = 0, else ⌈log₂(Dmax+1)⌉; B1 iff Dmax ≤ 63. Capacity per B1 block = 3·(8−λ) − 1 bits.
• Label decodability: max Huffman code length is forced ≤ 5 bits (= minimum B1 capacity), so the receiver always holds block j's label before reaching block j. When the tree depth would be 6 (7 active thresholds), the min-weight threshold is merged into the next one.
• R2 width = min(16, 8(k−1)) bits so deg Y < 8k, guaranteeing a unique CRT solution (paper's (3,4) setting → 16 bits).
• Side info (side_info struct): Huffman table, first-block λ, secret length — the paper's "informed" auxiliary information.

Requirements

MATLAB R2019b+ (or recent Octave). No toolboxes required.

Companion Repo

mrdhcbi-matlab — Chen et al., "Multi-Party Reversible Data Hiding in Ciphertext Binary Images Based on Visual Cryptography", IEEE SPL 2025.

Reference

@article{fang2025rdhei,
  author  = {Fang, Guangtian and Wang, Feng and Zhao, Chenbin and Qin, Chuan
             and Chang, Ching-Chun and Chang, Chin-Chen},
  title   = {Reversible Data Hiding With Secret Encrypted Image Sharing
             and Adaptive Coding},
  journal = {IEEE Internet of Things Journal},
  year    = {2025},
  volume  = {12},
  number  = {13},
  pages   = {23933--23945},
  doi     = {10.1109/JIOT.2025.3555380}
}