PDF Steganography based on Chinese Remainder Theorem

We propose different approaches of PDF files based steganography, essentially based on the Chinese Remainder Theorem. Here, after a cover PDF document has been released from unnecessary A0, a secret message is hidden in it using one of the proposed approaches, making it invisible to common PDF readers, and the file is then transmitted through a non-secure communication channel. Where each of our methods, ensure the condition that the number of inserted A0 is less than the number of characters of the secret message s.

💡 Research Summary

The paper presents novel PDF‑based steganographic schemes that exploit the Chinese Remainder Theorem (CRT) to embed secret data using the non‑breaking space character (ASCII A0). Existing PDF steganography, particularly the “null‑space coding” approach introduced by I‑Shi et al., hides data by inserting sequences of A0 characters between characters in a PDF. While this method is invisible to most PDF readers, it suffers from a critical drawback: the number of inserted A0 symbols often exceeds the length of the secret message, leading to noticeable file‑size inflation and increased detection risk.

To overcome this limitation, the authors propose two CRT‑driven algorithms that guarantee the number of inserted A0 symbols is never larger than the number of bits in the secret message (plus a small constant). Both schemes assume that the sender and receiver share a secret key (either a block size k or a parameter m) through a secure channel before communication.

Pre‑processing
Before embedding, the cover PDF is “cleaned”: any existing A0 characters are either removed or replaced by ordinary spaces (ASCII 20) depending on their width. This ensures that only the newly inserted A0 symbols belong to the steganographic payload, preventing ambiguity during extraction.

Mathematical foundation
The paper recalls the classic CRT: given pairwise coprime moduli (n_i) and residues (a_i), there exists a unique integer (x) modulo (N = \prod n_i) satisfying all congruences. Garner’s algorithm is presented as an efficient method to reconstruct (x) from the residues. These tools are the core of the embedding and extraction processes.

First approach (two‑modulus scheme)

1. The secret binary string (s) is divided into blocks of length (k) (the secret key).
2. Two coprime primes are derived: (p_1 = 2^{\lceil k/2\rceil}) and (p_2 = p_1 + 1).
3. Each block is interpreted as a decimal number (dec