Frequency-Domain Concealment of Post-Quantum Syndromes: A Hybrid QC-LDPC Codes Based Niederreiter Cryptosystem With Discrete Cosine Transform (DCT) Image Steganographic Approach

Abstract

This paper proposes and empirically evaluates a hybrid system that couples a quasi-cyclic low-density paritycheck (QC-LDPC) instantiation of the Niederreiter cryptosystem with block-based discrete cosine transform (DCT) image steganography: a plaintext message is first encrypted into a syndrome ciphertext using a sparse QC-LDPC parity-check matrix masked by a random scrambler and permutation, and the resulting ciphertext bitstream is then embedded into the midfrequency DCT coefficients of a cover image via quantizationindex (odd/even) modulation. We implement the full pipeline quasi-cyclic key generation, constant-weight plaintext encoding, syndrome encryption, multi-restart Gallager bit-flipping decryption, block-DCT embedding and extraction as a reproducible Python prototype and evaluate it on eight standard grayscale test images (512×512) under a validated parameter set (n =192, r =96, t =4). Across 64 sweep configurations spanning payload densities from 4096 to 81920 bits, the scheme achieves PSNR of 38.9-51.3 dB and SSIM of 0.90-0.996, with an empirically measured Niederreiter decoding-failure rate of 9.5×10⁻⁴ per block and a DCT channel bit-error rate below 1.7×10⁻² in the worst observed case (zero in 51 of 64 runs). An equalbit-budget comparison against AES-256-CTR+DCT, unencrypted DCT embedding, and spatial LSB embedding shows that the cryptographic layer choice has no measurable effect on stegoimage fidelity (45.6–45.7 dB PSNR for all three DCT-domain variants) while raising the embedded payload’s Shannon entropy from 4.25 bits/byte (natural-language plaintext) to 7.87 bits/byte (near-uniform ciphertext); LSB embedding attains higher PSNR (64.4 dB) but a substantially higher chi-square pairs-of-values steganalysis statistic (2538 vs. 821-884 for the DCT-domain methods) and fails under JPEG re-compression even at quality 100 (9.2-9.4% BER), whereas the proposed method degrades gracefully (0% BER at quality ≥95, rising past 20% only below quality 70). We report the scheme’s true key-space and information-set-decoding limitations honestly: the demonstration parameters are far below post-quantum security margins and are not intended for deployment; a per-plaintext-bit ciphertext expansion factor of 3.84 (i.e., an information rate of 0.260) is measured and discussed as an inherent cost of the QC-LDPC code family used, rather than hidden or omitted. All reported numbers are drawn from executed experiments with the accompanying code, data, and figures released for reproducibility.