High-Memory Masked Convolutional Codes

• High-memory masked convolutional codes are error-correcting codes that integrate advanced algebraic structures with enhanced memory and systematic masking to boost reliability.
• They employ probabilistic optimization techniques and cycle avoidance methods to minimize short-cycle errors and improve iterative decoding performance.
• Their tunable design supports applications in post-quantum cryptography, streaming data, and storage systems, offering scalable and secure error correction.

High-memory masked convolutional codes are a class of error-correcting codes that combine the high error-correction capability and flexible structure of convolutional codes with masking, memory augmentation, and advanced combinatorial optimization. These codes are motivated by applications in modern communication systems, storage technologies, and cryptography, particularly post-quantum cryptography. The defining feature is the integration of high code memory—expanded through algebraic or probabilistic design—and systematic masking that conceals algebraic and structural properties. This class unifies deep results from algebraic code construction, probabilistic combinatorics, and code-based cryptography, providing both strong error-correction and security guarantees.

1. Algebraic Construction and Design Principles

High-memory masked convolutional codes generalize classical convolutional codes by adopting advanced algebraic and structural constructions. The baseline method uses invertible unit schemes: let $U$ be an invertible $n \times n$ matrix over a finite field, with inverse $V$, so $UV = I$. One forms the generator matrix as a polynomial in the delay operator $z$: $G[z] = E_0 + E_1 z + \ldots + E_s z^s$ where each $E_i$ is an $r \times n$ submatrix of $U$, constructed by carefully selecting rows to ensure the generator is noncatastrophic and has a polynomial right inverse (as in Lemma 1 of (Hurley, 2014)).

Masking introduces additional randomization or selective obfuscation in the structure. In post-quantum cryptographic constructions, masking is implemented as

$G = S (G_{pQ} + \mathcal{G}) R$

where $G_{pQ}$ is a high-memory (long-polynomial) generator matrix, $\mathcal{G}$ is a low-rank masking matrix, $S$ is a random nonsingular matrix, and $R$ is a permutation matrix (Ariel, 17 Oct 2025). This yields a dense, random-like generator that conceals the algebraic structure from attackers.

Block or edge-masked designs further augment code memory by spreading the support of generator entries across multiple instants or by applying probabilistic partitioning, tailoring the memory and structure to optimize performance metrics or security margins (Yang et al., 2021, Huang, 18 Jul 2025).

2. Memory Augmentation and Probabilistic Optimization

The memory of a convolutional code—defined as the maximal delay in its generator polynomial—directly impacts its error-correcting potential and structure. High-memory is achieved by increasing the degree of constituent polynomials or by coupling/masking across larger windows.

In spatially-coupled and masked convolutional code design, probabilistic optimization is employed to minimize cycles and detrimental combinatorial structures. Gradient-based methods (GRADE-AO) optimize the distribution $\mathbf{p}$ of ones in the partitioning (mask) to minimize the expected number of cycles, leveraging the coupling polynomial $$f(X; \mathbf{a}, \mathbf{p}) = \sum_{i} p_i X^{a_i}$$ and its higher-order moments to predict the survival of harmful structures (e.g., cycles of length 6) (Yang et al., 2021).

The Clique Lovász Local Lemma (CLLL) and Moser–Tardos algorithm provide rigorous bounds and constructive algorithms for obtaining masking and lifting patterns that probabilistically eliminate all short cycles or trapping sets (Huang, 18 Jul 2025). Explicitly, the bound for a cycle to survive in the masked (protograph) code after eliminating 4-cycles is

$P_{LLL}[c_6] \leq P_{\Omega}[c_6] \cdot e^{8/3}$

demonstrating control over harmful substructures and enabling finite, predictable design iterations for high-memory codes.

3. Error-Correcting Performance and Decoding

The error-correction capabilities of high-memory masked convolutional codes reflect both their algebraic roots and their memory/masking design:

• The code free distance $d_{\text{free}}$ satisfies

$$d_{\text{free}} \leq (n-k) \left(\left\lfloor \frac{\delta}{k}\right\rfloor + 1 \right) + \delta + 1$$

with carefully designed high-memory schemes achieving or nearly reaching the Singleton bound (Hurley, 2014).

• By eliminating or greatly reducing the incidence of short cycles (notably 4- and 6-cycles) and trapping sets, the error floor is suppressed, yielding improved iterative and algebraic decoding performance (Huang, 18 Jul 2025, Yang et al., 2021).

Decoding approaches include:

• Algebraic decoding using structure-induced check matrices (for designs with explicit right inverses) (Hurley, 2014).
• Iterative/min-sum decoding or belief-propagation (particularly for LDPC-convolutional or SC designs with low-density masking) (Hurley, 2014).
• Parallel Viterbi decoding for cryptographic constructions, where the decryption process branches across multiple masking candidates and selects the output closest to the expected error profile (Ariel, 17 Oct 2025).

In cryptographic settings, the decryption (decoding) process is designed for linear-time scalability, with unmasking and polynomial division introducing additional error that further obfuscates the true codeword, thwarting syndromic and ISD-type attacks.

4. Masking, Density Distribution, and Cycle Avoidance

Masking, both deterministic and probabilistic, is leveraged to optimize code structure and eliminate harmful cycles:

• In LDPC and SC code settings, the optimization problem centers on how to assign edge connections (or nonzero generator matrix entries) so that the aggregate number of unwanted subgraphs is minimized.
• The density distribution $\mathbf{p}$ determines the frequency and locations of ones in the partition/mask, and its optimality is attained via gradient descent to satisfy necessary conditions imposed by the code's coupling pattern (Yang et al., 2021).
• The constraint for cycle-6 avoidance, for example, is given via

$$P_6(\mathbf{a}, \mathbf{p}) = [f^3(X) f^3(X^{-1})]_0$$

where elimination of lower-length cycles (e.g., 4-cycles) is handled simultaneously.

The design must balance aggressive cycle elimination with the probabilistically quantifiable side effect that longer cycles may become somewhat more likely, but upper bounds (such as the factor $e^{8/3}$ for 6-cycles after 4-cycle removal) guarantee that this increase is controlled (Huang, 18 Jul 2025).

5. Cryptographic Applications and Security Margins

Recent proposals for post-quantum cryptosystems (PQC) exploit high-memory masked convolutional codes as the central primitive (Ariel, 17 Oct 2025). Their security and efficiency stem from several complementary mechanisms:

• High-memory generator matrices, when masked and permuted, produce dense public keys indistinguishable from random codes, with all revealing algebraic features erased by a semi-invertible transformation.
• Random error injection is performed at higher rates than in block-code McEliece-type systems, and additional noise is introduced via polynomial division in the decryption process, dramatically increasing the effective error rate and precluding effective ISD or algebraic attacks.
• The cryptanalytic complexity (syndrome decoding attack cost) routinely exceeds previous code-based schemes (including McEliece) by factors exceeding 2100× in quantum settings, reflecting the increased randomness, error resilience, and obfuscation.
• The decryption protocol, featuring parallel Viterbi decoders running on different unmasking candidates, is engineered for scalability and hardware efficiency, making the scheme suitable for practical deployment.

6. Practical Implementation and Application Domains

High-memory masked convolutional codes are applicable in diverse domains:

• Quantum-safe public key cryptography, where scalability, security margin, and efficient decryption are essential (Ariel, 17 Oct 2025).
• Streaming data and storage systems, with spatially-coupled high-memory codes providing threshold saturation and low-rate error floors, and windowed decoding for low-latency operation (Yang et al., 2021).
• Communication systems and channels with memory, where convolutional polar codes and masked variants outperform classical block codes and exhibit robustness against burst errors, as demonstrated by improved noise suppression of up to 5 dB in relevant simulation regimes (Bourassa et al., 2018).

The underlying algebraic and combinatorial design frameworks (unit schemes, probabilistic masking, CLLL, Moser–Tardos) generalize across applications, enabling the fine-tuning of code parameters (memory, block length, masking density) to meet target performance or security requirements.

7. Summary of Design Methodologies and Theoretical Bounds

The overarching methodology integrates algebraic code construction (from unit schemes and generator polynomials), probabilistic combinatorial optimization (density-distribution and clique-LLL-based masking), and cryptographic transformations:

• Generator matrices are constructed to maximize free distance, ensure noncatastrophic structure, and enable efficient algebraic or iterative decoding (Hurley, 2014).
• Masking and density optimization are achieved via probabilistic frameworks (GRADE-AO, CLLL), quantifiable through explicit polynomial constraints and upper bounds on cycle survival (Yang et al., 2021, Huang, 18 Jul 2025).
• Cryptographic masking utilizes semi-invertible basis transformation and permutation, random error and noise injection, and parallelizable decryption structures (Ariel, 17 Oct 2025).
• Sufficient and necessary conditions for cycle and trapping set avoidance are provided by explicit partitioning formulas and probabilistic bounds, allowing the code designer to guarantee low error floors and robust decoding under high-memory configurations (Huang, 18 Jul 2025).

This comprehensive theoretical and applied framework positions high-memory masked convolutional codes at the intersection of modern coding theory, combinatorial optimization, and post-quantum cryptography, offering a tunable design space for robust, efficient, and secure communication and storage systems.