Scrambled Codes, Stronger Security

Author: Denis Avetisyan

A new approach to constructing pseudorandom codes offers improved resilience against tampering and enhanced protection for sensitive data.

This work introduces a novel construction of pseudorandom codes with provable subexponential security and adaptive robustness against edit distance attacks, demonstrating applications to robust watermarking for language models.

Despite recent advances in watermarking techniques for identifying AI-generated content, existing pseudorandom codes (PRCs) remain vulnerable to attack and lack robustness necessary for practical large language model applications. This work, ‘Improved Pseudorandom Codes from Permuted Puzzles’, introduces a novel construction of PRCs that achieves provable subexponential security, resilience to worst-case edits over a binary alphabet, and robustness even against adversaries possessing the detection key. These codes are built upon a new assumption-the permuted codes conjecture-linking it to established results in private information retrieval. Could this approach unlock truly adaptive and reliable watermarking schemes capable of safeguarding against increasingly sophisticated content manipulation?

The Inevitable Decay of Trust: Charting the Provenance Challenge

The exponential increase in digitally generated content – from text and images to audio and video – presents a significant challenge to establishing trust and authenticity online. As synthetic media becomes increasingly sophisticated and readily available, discerning between genuine and fabricated material becomes difficult, if not impossible, with conventional methods. This proliferation demands the development of robust provenance techniques capable of verifying the origin and integrity of digital assets. Without reliable mechanisms to trace content back to its source, the potential for misinformation, manipulation, and malicious use escalates dramatically, impacting everything from news and political discourse to personal reputations and financial systems. Consequently, research focuses on creating systems that can demonstrably link content to its creator, providing a verifiable audit trail and fostering greater accountability in the digital landscape.

Current methods for establishing content provenance frequently exhibit critical weaknesses, rendering them susceptible to manipulation and undermining trust in digital information. Many existing techniques rely on easily altered metadata or centralized authorities, creating single points of failure and opportunities for malicious actors to falsely claim authorship or tamper with records. Watermarking and digital signatures, while useful, can often be removed or forged with sufficient effort, particularly as generative AI tools become more sophisticated. Moreover, systems designed for verification often lack the cryptographic robustness needed to withstand advanced adversarial attacks, where subtle modifications can bypass detection mechanisms. This inherent fragility poses a significant challenge in an era where the authenticity of online content is increasingly difficult to ascertain, demanding the development of more resilient and secure provenance solutions.

The pursuit of trustworthy content authentication hinges on the development of subtle, robust signaling mechanisms. These signals, ideally imperceptible to human observation, must be woven into the fabric of digital content – images, audio, or text – to establish a verifiable chain of custody. Crucially, this embedding process needs to withstand deliberate manipulation; any attempt to remove or alter these provenance indicators should either be detectable or render the content demonstrably compromised. Researchers are exploring techniques like imperceptible watermarking, cryptographic hashing, and even leveraging the inherent noise characteristics of digital media to achieve this resilience. The challenge lies in balancing the need for strong security with the preservation of content quality and user experience, ensuring that authentication doesn’t come at the cost of usability or artistic integrity. Successful implementation of such techniques promises a future where the origin and authenticity of digital information can be confidently established, combating the spread of misinformation and fostering trust in the digital realm.

Constructing Resilience: The Architecture of Secure Codes

The Pseudorandom Code (PRC) functions as a core component by integrating cryptographic secrecy with the ability to tolerate and correct data corruption. This is achieved through the construction of codes that are not only capable of encoding information but also of introducing redundancy designed to withstand errors introduced during transmission or storage. The resulting encoded data, while appearing random to an observer without the decoding key, retains sufficient structure to allow for accurate reconstruction of the original embedded information even in the presence of partial data loss or modification. This dual functionality – security and error resilience – is critical for applications where data integrity and confidentiality must be maintained simultaneously.

Pseudorandom Codes (PRCs) utilize the mathematical properties of established error-correcting codes, specifically Reed-Solomon codes, as a foundation for embedding watermarks. Reed-Solomon codes are well-understood for their ability to reconstruct data from partially corrupted or erased segments; PRCs adapt this capability by encoding watermark information within the code’s redundancy. This involves a modification of the standard encoding process to interleave watermark bits with the data being protected, effectively hiding the watermark within the error-correction structure. The selection of Reed-Solomon codes is based on their maximum distance separable (MDS) property, which provides optimal error correction for a given level of redundancy, and their suitability for systematic encoding, enabling efficient watermark extraction.

Folded Reed-Solomon Codes represent a modification of standard Reed-Solomon error correction that allows for a configurable trade-off between computational efficiency and robustness against data corruption. Traditional Reed-Solomon codes, while powerful, can be computationally expensive, particularly during encoding and decoding. Folding reduces the size of the finite field used in the calculations – typically $GF(2^n)$ – to a smaller field, $GF(2^m)$, where $m < n$. This reduction lowers the computational cost but also diminishes the code’s error-correcting capability. The degree of folding is a key parameter, allowing designers to prioritize either speed and lower resource utilization or a greater capacity to recover from errors in the watermarked data.

The utilization of error-correcting codes, specifically Reed-Solomon and Folded Reed-Solomon variants, ensures watermark recovery even with data corruption. This is achieved by distributing watermark information across multiple tokens, allowing for reconstruction despite partial loss or alteration of individual tokens. Crucially, this design maintains a consistent level of entropy – measured as bits per token – regardless of the degree of corruption. This constant per-token entropy is vital for watermarking schemes, as it guarantees a predictable and reliable signal strength for detection and verification, even under adverse conditions, and prevents information leakage due to fluctuating watermark robustness.

Underlying Principles: Formalizing Security Assumptions

The Permuted Codes Assumption (PCA) is foundational to the security of the Proposed Rate-compatible Code (PRC). This assumption posits that applying a random permutation to a codeword generated by the PRC results in a sequence indistinguishable from a truly random string. Specifically, given a valid codeword $c$, and a random permutation $\pi$, the permuted codeword $\pi(c)$ should exhibit no discernible statistical patterns that could be exploited by an adversary attempting to determine the original message or forge a valid watermark. The validity of the PCA directly impacts the robustness of the PRC against attacks aiming to recover embedded data by analyzing the statistical properties of the scrambled signal; if the assumption fails, adversaries could potentially differentiate between valid and invalid watermarks, compromising the system’s security.

The Permuted Puzzles Assumption builds upon the core principle of the Permuted Codes Assumption to specifically address scenarios where data is incomplete or lost, such as erasures or partial data corruption. This extension posits that even with a proportion of codeword bits missing, the remaining visible bits continue to appear statistically indistinguishable from random noise. The assumption enables analysis of the watermarking scheme’s resilience to attacks that attempt to deduce the hidden message from incomplete or damaged data, effectively modeling scenarios where an adversary has limited access to the full watermarked signal. Rigorous mathematical treatment of this assumption allows for quantifiable security guarantees against such attacks, providing a basis for assessing the watermarking system’s robustness in practical deployment conditions.

The validity of the Permuted Codes and Permuted Puzzles Assumptions enables a formal security analysis of the watermarking scheme, specifically against adaptive adversaries. This means that, given these computational hardness assumptions, we can mathematically prove bounds on an adversary’s ability to successfully remove or forge watermarks, even when the adversary can dynamically choose its attacks based on observed responses from the system. The analysis considers all possible adaptive strategies available to the adversary within the defined computational constraints, providing quantifiable guarantees about the scheme’s resilience and allowing for a rigorous assessment of its security parameters. This formal approach differs from heuristic security arguments and allows for precise statements about the scheme’s robustness.

The system’s architecture incorporates defenses against attacks targeting weaknesses in the underlying code structure. Specifically, the design aims to maintain security even when subjected to analysis by distinguishers operating within subexponential time complexity, denoted as $2^{o(\sqrt{n})}$ where $n$ represents the input size. This resilience is achieved through careful code construction and parameter selection, mitigating the potential for successful attacks based on exploiting structural vulnerabilities. While complete immunity to all future attacks is not guaranteed, the chosen parameters and construction methods plausibly render attacks within this complexity class computationally infeasible.

Beyond Robustness: Demonstrating Real-World Resilience

The watermarking scheme distinguishes itself through its Strong Adaptive Robustness, a critical feature in an era of increasingly sophisticated adversarial attacks. Unlike traditional watermarks vulnerable when the embedding key becomes known, this system maintains its integrity even under complete key compromise. This resilience is achieved not through obscurity, but through a carefully constructed embedding process that distributes the watermark information across the data in a manner resistant to targeted manipulation. The design anticipates and neutralizes attacks that attempt to remove or alter the watermark by exploiting knowledge of the key, ensuring reliable authentication and provenance tracking even in hostile environments. This proactive defense mechanism represents a substantial advancement in watermark security, offering a level of protection previously unattainable.

The system’s reliability hinges on its ability to avoid false positives – incorrectly identifying a watermark where none exists – and this is ensured through the implementation of Product-of-Random-Codes (PRC) error correction. This technique introduces a level of redundancy that allows the system to confidently distinguish between genuine watermarks and naturally occurring patterns or intentional noise. By distributing the watermark information across multiple, randomly generated codes, the PRC significantly reduces the probability of a false detection to a negligible level, effectively guaranteeing Soundness. This is critical for practical applications, as even a small rate of false positives could undermine trust and usability, particularly in scenarios involving content authentication or copyright protection. The robust error correction capability means the system operates with high confidence, minimizing the risk of incorrectly flagging unmodified content as watermarked.

The watermarking scheme demonstrates notable resilience when subjected to a substitution channel, a common model for data transmission errors where symbols are replaced with others. This means even if minor alterations – such as single character swaps or symbol substitutions – are introduced into the watermarked data, the watermark remains reliably detectable. The system effectively mitigates the impact of these minor manipulations by encoding the watermark information in a way that is robust to localized changes. This resilience is crucial for practical applications where data may be subject to unintentional errors during storage or transmission, or deliberate, yet subtle, attempts at removal by adversaries who only make minor alterations to avoid detection. Consequently, the watermark’s integrity is maintained despite the presence of these distortions, ensuring continued functionality and reliability.

The watermark recovery system leverages list decoding, a powerful technique allowing it to function effectively even with noisy or corrupted data. Unlike prior watermarking schemes that demanded a superconstant amount of entropy to resist attacks – essentially requiring exponentially increasing security measures against even minor alterations – this approach achieves robustness with a constant edit distance rate. This means the watermark can reliably be recovered even if a significant, but limited, number of bits are changed or corrupted, offering a substantial improvement in resilience. By identifying a list of likely watermarks rather than a single definitive one, the system gracefully handles imperfections and intentional manipulations, ensuring a high probability of successful detection despite adversarial efforts. This constant-rate resistance represents a significant leap forward in practical watermark security and reliability.

Toward Optimized Structures: The Path Forward

The Folded Reed-Solomon Code’s utility stems significantly from its inherent dual distance property, a characteristic that directly influences the balance between security and computational efficiency. This dual distance, essentially a measure of the code’s ability to detect and correct errors, allows designers to finely tune watermarking and data protection schemes. A larger dual distance enhances the code’s resilience against attacks and data corruption, but simultaneously increases the complexity of encoding and decoding processes. Conversely, a smaller dual distance offers faster processing but potentially compromises security. Consequently, optimizing this property is paramount; a well-chosen dual distance ensures robust error correction without incurring excessive computational overhead, making the Folded Reed-Solomon Code a practical solution for applications demanding both security and speed, particularly when dealing with $n$-dimensional data streams.

The effectiveness of an error-correcting code hinges on its capacity to not only identify when data has been corrupted, but also to reconstruct the original, accurate information. This ability is directly linked to the code’s dual distance – a measure of how far a corrupted codeword must be from a valid codeword. A larger dual distance signifies a greater ‘separation’ between valid and invalid codewords, meaning the code can reliably detect and correct a higher number of errors. Essentially, a code with a substantial dual distance possesses increased resilience; even if a significant portion of the data is altered or lost, the original message can still be recovered with a high degree of certainty. This principle is foundational in applications demanding data integrity, such as digital storage, communication systems, and, critically, the development of secure watermarking techniques where subtle alterations must not compromise the embedded information. The greater the dual distance, the more robust the code becomes against both accidental errors and malicious attempts at data manipulation.

Watermarking schemes, designed to protect digital content, benefit significantly from the application of codes possessing a polynomial dual distance. These codes enable the creation of watermarks that are demonstrably robust against common signal processing attacks – such as compression, cropping, and noise addition – while remaining imperceptible to the average user. The key lies in the code’s ability to distribute watermark information across a wider spectrum of the data, making it exceptionally difficult for an adversary to remove or disable the mark without causing significant damage to the content itself. This technique moves beyond theoretical feasibility, offering a pathway to practical, real-world applications in areas like copyright protection, content authentication, and tamper detection, ultimately safeguarding digital assets with a blend of security and usability.

Despite the robustness of certain cryptographic constructions, those dependent on planted structures of size $O(log n)$ exhibit a critical vulnerability: distinguishability in quasipolynomial time, specifically $n^{O(log n)}$. This means that while seemingly secure against many conventional attacks, an adversary can, with reasonable computational resources, determine whether a constructed watermark or code is genuinely planted or merely a random artifact. The limitation arises from the inherent predictability introduced by these smaller structures, offering a potential foothold for advanced attacks that exploit the statistical properties of the planted data. Consequently, designs relying on such constructions require careful consideration of the trade-off between efficiency and security, as they remain susceptible to detection in scenarios involving computationally capable adversaries.

The pursuit of robust systems, as demonstrated in this construction of pseudorandom codes, echoes a fundamental truth about all complex architectures. This paper’s focus on adaptive robustness against edits and subexponential security isn’t merely about building better watermarking schemes; it’s an acknowledgement that every system, even one meticulously designed with provable guarantees, exists within a constant state of flux. As Alan Turing observed, “Sometimes people who are unhappy tend to look for a person to blame.” This sentiment, while seemingly unrelated, reflects the inherent imperfection within systems and the tendency to seek fault when they inevitably degrade. The codes presented here strive not to prevent decay, but to manage it, to build a structure that ages gracefully even under duress, understanding that improvements themselves are transient.

What Lies Ahead?

The construction of pseudorandom codes, as demonstrated, is not about achieving perfect immunity to alteration-all systems learn to age gracefully. Rather, it is about understanding the contours of that decay, mapping the boundaries where meaningful signal degrades into noise. The presented work offers a refinement of those boundaries, a sharper definition of robustness against adversarial edits, but it does not eliminate the inevitable erosion of information. Future efforts will likely focus less on building impregnable defenses and more on developing systems that can intelligently adapt to imperfection, gracefully recovering from predictable distortions.

The application to watermarking language models is a natural extension, yet also highlights a fundamental tension. The desire for robust attribution clashes with the inherent fluidity of language itself. A watermark, however cleverly embedded, is still an imposition upon a system designed for open-ended expression. Perhaps the most fruitful avenue of research lies in exploring watermarks that are not static signatures, but dynamic patterns that evolve alongside the underlying text, becoming indistinguishable from its natural variations.

Ultimately, the field may find that sometimes observing the process of degradation is more valuable than striving to accelerate invulnerability. The true challenge isn’t to build codes that resist change, but to understand how systems respond to it, and to design architectures that can anticipate-and even benefit from-the inevitable passage of time.

Original article: https://arxiv.org/pdf/2512.08918.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/