Solving for Security: A New Approach to Post-Quantum Cryptography

Author: Denis Avetisyan

Researchers are exploring the use of the notoriously difficult SAT problem to build cryptographic systems resilient to attacks from future quantum computers.

This review details a novel post-quantum cryptography algorithm based on the hardness of solving equilibrium SAT instances, offering a potential defense against quantum-based attacks on public-key cryptography.

The relentless increase in computational power, particularly with the advent of quantum computing, threatens the security of currently deployed public-key cryptography. Addressing this critical challenge, the research detailed in ‘Equilibrium SAT based PQC: New aegis against quantum computing’ introduces a novel post-quantum cryptographic algorithm grounded in the inherent difficulty of solving specifically constructed Satisfiability (SAT) problems. This approach generates ciphertexts through subset selection, offering a potentially faster and more broadly applicable solution that avoids reliance on computationally expensive large numbers. Could this SAT-based framework represent a viable path towards securing future digital communications against quantum attacks and beyond?

The Inevitable Quantum Reckoning: Why We Need Post-Quantum Cryptography

The foundation of modern digital security, public-key cryptography – encompassing algorithms like RSA and ECC – faces an existential threat from the anticipated arrival of large-scale quantum computers. These algorithms rely on the mathematical difficulty of factoring large numbers or solving the discrete logarithm problem; however, Shor’s algorithm, a quantum algorithm, can efficiently solve these problems, effectively breaking the encryption. This isn’t a hypothetical concern; while currently beyond the reach of existing quantum hardware, the development of sufficiently powerful quantum computers is considered inevitable. Consequently, data encrypted today using vulnerable public-key methods could be decrypted retroactively, jeopardizing sensitive information like financial transactions, government secrets, and personal communications. The vulnerability stems from the fundamental mathematical principles underpinning these widely-used cryptographic systems, making them inherently susceptible to quantum attacks, and highlighting the urgent need for a cryptographic overhaul.

The accelerating development of quantum computing presents a fundamental challenge to modern data security, demanding a proactive shift toward Post-Quantum Cryptography (PQC). Current encryption standards, such as RSA and ECC, are based on mathematical problems easily solvable by sufficiently powerful quantum computers employing Shor’s algorithm, rendering sensitive information vulnerable to decryption. Consequently, PQC focuses on algorithms believed to be resistant to attacks from both classical and quantum computers, safeguarding digital communications and data storage against future threats. This isn’t simply a matter of upgrading software; it requires a complete overhaul of cryptographic infrastructure, anticipating a world where current security measures are effectively obsolete and ensuring the confidentiality and integrity of information for decades to come. The transition to PQC is therefore not a hypothetical exercise, but a critical necessity for maintaining trust in digital systems.

Post-Quantum Cryptography represents a crucial shift in cryptographic design, moving beyond algorithms like RSA and ECC that underpin much of modern digital security. Recognizing the potential of quantum computers to break these established systems, researchers are actively developing new approaches resistant to both classical computing power and the unique capabilities of quantum algorithms. This diversification isn’t simply about creating algorithms that are hard to break; it’s about exploring fundamentally different mathematical problems – lattice-based cryptography, multivariate cryptography, code-based cryptography, and hash-based signatures – that are believed to offer security even against a future quantum threat. The National Institute of Standards and Technology (NIST) is currently leading a standardization process to identify and validate these next-generation algorithms, ensuring a smooth transition to a quantum-resistant infrastructure and safeguarding sensitive data for years to come.

PQC Candidates: A Landscape of Trade-offs and Challenges

The National Institute of Standards and Technology (NIST) is currently in the process of standardizing several Post-Quantum Cryptography (PQC) families to address the potential threat of quantum computers to current cryptographic systems. Notable candidates include code-based cryptography, exemplified by the Classic McEliece scheme, which relies on the difficulty of decoding general linear codes. Lattice-based cryptography is also a primary focus, with algorithms like CRYSTALS-Kyber, NTRU, and SABER under consideration. These lattice-based schemes are based on well-studied problems in lattice theory, such as the Shortest Vector Problem (SVP) and Learning With Errors (LWE). The standardization process involves rigorous evaluation of security, performance, and implementation characteristics to ensure these algorithms are suitable for widespread adoption.

Post-quantum cryptography (PQC) algorithms derive their security from the difficulty of solving specific mathematical problems. Code-based schemes, such as Classic McEliece, rely on the hardness of decoding general linear codes, while lattice-based cryptography, including CRYSTALS-Kyber, NTRU, and SABER, is based on problems like the Shortest Vector Problem (SVP) and Learning With Errors (LWE). The presumed hardness of these problems dictates the algorithm’s resistance to attack; however, the computational complexity of solving these problems also impacts performance characteristics. Different PQC families exhibit trade-offs between key sizes, encryption/decryption speeds, and the level of security they provide, with some algorithms prioritizing smaller key sizes at the potential expense of increased computational overhead, and vice-versa. The selection of an appropriate PQC algorithm depends on the specific application’s security requirements and resource constraints.

Post-quantum cryptography (PQC) schemes, while demonstrating significant advancement, currently face obstacles regarding practical implementation. Many candidate algorithms exhibit substantially larger key sizes compared to widely deployed classical cryptography like RSA or ECC. For example, some code-based schemes require keys exceeding 1MB, presenting storage and bandwidth challenges. Furthermore, computational efficiency remains a concern; while some lattice-based schemes like CRYSTALS-Kyber offer reasonable performance, others demand significant computational resources for key generation, encryption, and decryption. Optimizing these schemes to reduce key sizes and improve processing speed is crucial for widespread adoption, particularly in resource-constrained environments such as embedded systems and IoT devices. Current research focuses on algorithmic improvements, hardware acceleration, and parameter selection to address these limitations.

A Novel Approach: Leveraging SAT and Random Subsets for Post-Quantum Security

The proposed post-quantum cryptographic (PQC) algorithm leverages the computational complexity associated with solving instances of Equilibrium SAT. Unlike standard SAT, which seeks a truth assignment satisfying a Boolean formula, Equilibrium SAT requires finding an assignment where each variable is assigned to true or false with roughly equal probability, adding a probabilistic constraint. This variation is believed to retain the NP-hardness of classic SAT while potentially resisting known quantum algorithms designed to accelerate SAT solving. The algorithm’s security is therefore predicated on the assumption that finding a solution to a sufficiently large and carefully constructed Equilibrium SAT instance is computationally infeasible for both classical and quantum computers, providing a foundation for secure encryption and decryption processes.

The encryption process begins with a multiset, $M$, of size $n$. A random subset, $S$, is extracted from $M$, and the elements of $S$ are used to populate a ciphertext array, $C$. The selection of elements for $C$ is probabilistic, governed by a uniform distribution across all possible subsets of $M$. This array, $C$, is then combined with a publicly known transformation, $T$, and the private key, $k$, to generate the final ciphertext. The relationship between the elements within $S$, and thus $C$, is directly linked to the satisfying assignment of the underlying Equilibrium SAT instance used in key generation; knowing $k$ and $C$ does not reveal the SAT solution without solving the associated problem.

The public key generation process centers on a 2k-SAT instance, a boolean satisfiability problem with clauses containing exactly two literals. This instance is constructed such that the solution – a variable assignment satisfying all clauses – is concealed within the encryption key. Decryption necessitates identifying this specific solution to the 2k-SAT instance, as the solution directly unlocks the ciphertext. The computational difficulty arises from the NP-completeness of SAT; finding a solution requires, in the worst case, an exhaustive search of the solution space, which scales exponentially with the number of variables. Therefore, without knowledge of the solution, decryption becomes computationally intractable, providing the security foundation for the proposed post-quantum cryptographic scheme.

Algorithm Specifics and Security Considerations

The construction of the SAT instance relies on representing the cryptographic problem as a set of clauses, specifically utilizing $k$-TRUE-clauses, where each clause contains exactly $k$ literals. The Literal Spectrum, which describes the distribution of literals within the SAT instance, directly influences the instance’s complexity and, consequently, its solvability by SAT solvers. A carefully designed Literal Spectrum is crucial; imbalances can lead to solver inefficiencies or vulnerabilities. The number of clauses and variables are directly related to the key size and security parameters, and the algorithm’s performance is sensitive to the choice of $k$ and the resulting clause distribution. This approach allows for a trade-off between instance size, solver performance, and the security level of the cryptographic scheme.

The private key in this system is directly represented by the solution vector to the generated Satisfiability (SAT) instance. Each element of the solution vector corresponds to a variable in the SAT problem; a value of ‘true’ or ‘1’ indicates the variable is part of the solution, and ‘false’ or ‘0’ indicates it is not. Decryption is achieved by applying this solution vector to the ciphertext array. Specifically, each element of the ciphertext is either decrypted directly if the corresponding variable in the solution vector is ‘true’, or a transformation is applied if the variable is ‘false’. This mapping between the solution vector and the ciphertext elements allows for the recovery of the original plaintext when the correct private key – the solution to the SAT instance – is used.

This cryptosystem is designed to prioritize reduced private key size at the expense of increased public key and ciphertext sizes. Specifically, the private key is projected to be 25 to 50 times smaller than that used in the CRYSTALS-Kyber scheme. This reduction in private key size is achieved through algorithmic choices detailed elsewhere, while the corresponding public key and ciphertext sizes are expected to be 10 to 40 times larger than those of CRYSTALS-Kyber. This trade-off allows for potential benefits in constrained environments where private key storage or transmission is a limiting factor, although it necessitates increased bandwidth or storage capacity for public keys and ciphertexts.

Looking Ahead: Impact and the Future of Secure Communication

This research introduces a distinctly different paradigm for post-quantum cryptography (PQC), moving beyond reliance on established mathematical problems like lattice structures or code-based systems. By leveraging the principles of Boolean satisfiability (SAT) solving, it establishes a novel framework for encryption and decryption, offering a complementary approach to algorithms currently undergoing standardization. This isn’t intended to replace existing PQC candidates, but rather to diversify the cryptographic landscape, potentially bolstering overall security by providing an alternative layer of defense against unforeseen attacks. The inherent flexibility of SAT-based cryptography allows for exploration of unique security-performance trade-offs, and could prove invaluable in specialized applications or as a component within hybrid cryptographic systems designed to withstand the evolving threats of the quantum computing era. Further investigation into this direction may unlock new avenues for constructing robust and adaptable cryptographic solutions.

The realization of a fully implemented and rigorously validated cryptographic system based on this new approach promises a significant bolstering of data security infrastructure. Current encryption methods, while effective, often rely on a limited number of underlying algorithms, creating potential vulnerabilities should one be compromised. Diversifying these systems with novel techniques, such as this method, introduces redundancy and resilience. Successful validation would not merely add another tool to the cybersecurity arsenal, but fundamentally strengthen the overall protective landscape for sensitive data – encompassing financial transactions, personal communications, and critical infrastructure – against both current and future threats, including those posed by quantum computing.

A significant enhancement offered by Method 3 lies in its ability to dramatically reduce ciphertext length, achieving a 2.5 to 3x compression compared to traditional bit-by-bit encryption schemes. This efficiency is realized by processing information in units of $b$ bits simultaneously, rather than individually. Conventional cryptographic methods often treat each bit as a separate entity during encryption and decryption; however, Method 3 leverages the inherent parallelism of $b$-bit operations. This not only reduces the overall data volume requiring storage and transmission, but also accelerates cryptographic processes, potentially offering substantial performance gains in bandwidth-constrained environments and resource-limited devices. The reduction in ciphertext size directly translates to lower communication costs and faster data transfer rates, representing a considerable advancement in practical cryptographic implementations.

The emerging field of SAT-based cryptography presents a compelling avenue for future advancements in secure communication, particularly as the threat from quantum computers looms. Unlike many post-quantum cryptography (PQC) schemes reliant on complex mathematical structures, SAT-based approaches leverage the well-understood problem of Boolean satisfiability – determining if a logical formula can be satisfied with certain variable assignments. This foundation allows for the construction of cryptographic primitives where security rests on the hardness of solving SAT instances, a problem demonstrably difficult even for quantum algorithms. Further research into optimized SAT solvers, novel encoding techniques, and the development of practical cryptographic protocols based on SAT promises not only enhanced data privacy but also the potential for more efficient and versatile cryptographic systems capable of adapting to evolving security needs in the post-quantum era. The inherent flexibility of SAT-based constructions could also facilitate the creation of specialized cryptographic solutions tailored to specific applications and hardware platforms.

The pursuit of cryptographic agility feels less like innovation and more like delaying the inevitable. This paper, wrestling with the SAT problem for post-quantum security, exemplifies the cycle. It’s a valiant attempt to build a fortress against future threats, yet one suspects that even enhanced conditions on subset selection won’t hold forever. As Claude Shannon observed, “Communication is the conveyance of meaning, not simply the transmission of information.” This rings true here; the ‘meaning’ of security is constantly shifting, and the ‘transmission’ of a complex algorithm is only as strong as the weakest link in its implementation. Better one thoroughly vetted, understandable system than a hundred brittle, ‘scalable’ solutions, each promising invulnerability and inevitably revealing a flaw.

What’s Next?

The appeal of mapping cryptographic security to the Satisfiability problem is, predictably, its apparent simplicity. The field has a long history of elegant theories collapsing under the weight of production realities. This work, while interesting, merely shifts the battleground. Someone, somewhere, is already optimizing a SAT solver, or discovering a new instance class that renders these ‘enhanced conditions’ surprisingly tractable. The NP-hardness guarantee is a comforting thought, but it doesn’t account for the clever shortcuts that always emerge.

Future research will inevitably focus on instantiation. How does one build a practical public-key system on top of this foundation? The devil, as always, will be in the parameters. Subset selection problems are notoriously sensitive to structure. It’s a safe bet that initial implementations will be slow, bulky, and vulnerable to side-channel attacks – the usual suspects. The truly hard part isn’t proving security; it’s surviving deployment.

One suspects this approach, like many before it, will become another layer of the cryptographic onion. A useful, perhaps even necessary, component, but hardly a silver bullet. If the code looks perfect, no one has deployed it yet. The real test will come when someone attempts to break it with actual, adversarial traffic, and that’s a future rarely predicted by theoretical guarantees.

Original article: https://arxiv.org/pdf/2512.02598.pdf

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