Cracking the Code: A New Threat to Post-Quantum Signatures

Author: Denis Avetisyan

A detailed security analysis reveals vulnerabilities in integer learning with errors, a core component of emerging post-quantum digital signature schemes.

Rejection sampling implementations of integer learning with errors are susceptible to least squares attacks, demanding careful parameter selection and distribution design.

While lattice-based cryptography promises post-quantum security, practical implementations remain vulnerable to sophisticated attacks exploiting subtle weaknesses in parameter selection. This paper, ‘Security Analysis of Integer Learning with Errors with Rejection Sampling’, presents a comprehensive theoretical and experimental study of a linear least squares attack against integer learning with errors (ILWE) constructions, as utilized in digital signature schemes like CRYSTALS-Dilithium. Our analysis, focusing on instances derived solely from observed signatures, reveals the effectiveness of the attack and highlights the critical role of rejection sampling in mitigating risk. Ultimately, these findings contribute to a more nuanced understanding of ILWE-based security and prompt further investigation into optimal parameter choices for robust cryptographic systems.

The Inevitable Quantum Reckoning

The digital infrastructure underpinning modern communication relies heavily on public-key cryptosystems, notably RSA and Elliptic Curve Digital Signature Algorithm (ECDSA). These systems secure everything from online banking and e-commerce to governmental communications by leveraging the mathematical difficulty of certain problems – like factoring large numbers or solving the elliptic curve discrete logarithm problem. However, the anticipated arrival of sufficiently powerful quantum computers poses a significant threat to these foundational algorithms. Quantum computers, harnessing the principles of quantum mechanics, are capable of executing algorithms that render these previously intractable mathematical problems solvable in a reasonable timeframe. Specifically, the security of RSA stems from the difficulty of factoring large numbers into their prime components; a task that becomes exponentially easier with $Grover’s$ and $Shor’s$ algorithms on a quantum computer. This vulnerability extends to ECDSA, which relies on the difficulty of the elliptic curve discrete logarithm problem, also susceptible to quantum acceleration. Consequently, the widespread adoption of quantum computing necessitates a proactive shift towards cryptographic solutions designed to withstand these emerging quantum threats, safeguarding the confidentiality and integrity of digital information.

The bedrock of much modern digital security, the RSA encryption algorithm, relies on the computational difficulty of factoring extremely large numbers into their prime components. However, this assumption of hardness is fundamentally challenged by Shor’s algorithm, a quantum algorithm developed by Peter Shor in 1994. Unlike classical algorithms which require exponential time to factor large numbers, Shor’s algorithm achieves this task in polynomial time, effectively rendering RSA insecure in the face of sufficiently powerful quantum computers. The algorithm leverages the principles of quantum superposition and quantum Fourier transforms to identify the period of a mathematical function related to the number being factored, a process dramatically faster than any known classical method. This breakthrough doesn’t merely offer a speedup; it represents a fundamental shift in computational complexity, transforming a previously intractable problem into a solvable one and posing a significant threat to the confidentiality and integrity of data currently protected by RSA.

The relentless advancement of quantum computing presents a clear and present danger to currently deployed cryptographic systems. While still in its nascent stages, the projected scalability of quantum processors compels a proactive shift towards post-quantum cryptography. Existing public-key algorithms, such as RSA and those based on elliptic curves, rely on the computational difficulty of problems like integer factorization and the discrete logarithm – problems that Shor’s algorithm efficiently solves on a quantum computer. Consequently, sensitive data transmitted today could be decrypted in the future once sufficiently powerful quantum computers become available. The cryptographic community is actively researching and standardizing new algorithms, based on mathematical problems believed to be resistant to both classical and quantum attacks – including lattice-based cryptography, multivariate cryptography, code-based cryptography, and hash-based signatures – to ensure continued secure communication in a post-quantum world. This transition is not merely a technical upgrade, but a fundamental reassessment of the foundations of digital security.

A New Foundation: Beyond Brute Force

Post-Quantum Cryptography (PQC) represents a proactive approach to cryptographic security, addressing the potential threat posed by the development of large-scale quantum computers. Current public-key cryptographic algorithms, such as RSA and ECC, are vulnerable to Shor’s algorithm, which can efficiently factor large numbers and solve the discrete logarithm problem on a quantum computer, thereby compromising their security. PQC focuses on developing algorithms that are believed to be resistant to attacks from both classical and quantum computers, relying on mathematical problems that are not known to be efficiently solvable by either type of computer. This includes exploring alternative mathematical structures and algorithms that offer a different security foundation than those currently used, ensuring continued confidentiality and integrity of digital communications and data even in a post-quantum world.

Lattice-based cryptography derives its security from the computational difficulty of solving mathematical problems defined on lattices – specifically, problems like the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP). These problems involve finding the shortest or closest vector within a lattice, a discrete subgroup of $\mathbb{R}^n$. The presumed hardness stems from the lack of known polynomial-time algorithms to solve these problems, even with quantum computers. Lattice-based schemes offer strong security guarantees based on well-studied mathematical foundations and exhibit favorable performance characteristics, making them a leading candidate for standardization in post-quantum cryptography. Different lattice constructions, such as those utilizing ideal lattices, further enhance security and efficiency.

In 2022, the National Institute of Standards and Technology (NIST) concluded its multi-year Post-Quantum Cryptography (PQC) standardization process, selecting four algorithms for initial standardization. Among these are CRYSTALS-Dilithium, a digital signature algorithm based on Module-LWE and Module-KEM; FALCON, a digital signature algorithm utilizing the NTRU lattice; and SPHINCS+, a stateless hash-based signature scheme. These algorithms were chosen following a rigorous evaluation of candidate submissions based on their security, performance, and implementation characteristics. The standardization aims to provide organizations with cryptographic tools resistant to attacks from both classical and future quantum computers, with the selected algorithms intended to replace currently used algorithms vulnerable to quantum attacks like Shor’s algorithm.

The Geometry of Security

Lattice-based cryptography derives its security from the computational hardness of problems defined on mathematical lattices. A lattice is formed by a discrete additive subgroup of $\mathbb{R}^m$, where $m$ represents the dimension of the lattice. Essentially, this means a lattice consists of all integer linear combinations of a set of linearly independent vectors, called a basis, in $m$-dimensional space. The difficulty arises from the fact that certain problems, such as the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP), are believed to be computationally intractable for high-dimensional lattices. Specifically, finding the shortest non-zero vector, or the vector closest to a given target vector, within a lattice becomes exponentially harder as the dimension $m$ increases, forming the foundation for cryptographic security.

qq-ary lattices are constructed over rings of the form $ℤ_q[x]/(f(x))$, where $q$ is a prime power and $f(x)$ is an irreducible polynomial. This ring structure allows for modular arithmetic, which facilitates faster computations compared to lattices defined over real numbers. The use of a prime power modulus enables the application of Number Theoretic Transform (NTT) algorithms, providing significant speedups in polynomial multiplication – a core operation in lattice-based schemes. Furthermore, the ring structure inherently provides a level of algebraic protection, making certain attacks, such as those relying on continuous approximations, less effective. The choice of parameters, including $q$ and $f(x)$, directly impacts both the security level and the computational efficiency of algorithms operating on these lattices.

CRYSTALS-Dilithium employs Hash to Ball and Rejection Sampling to generate the necessary random values for its cryptographic operations and maintain a secure distribution of these values. However, recent cryptanalysis has revealed vulnerabilities in the underlying Integer Learning with Errors (ILWE) constructions. Specifically, a linear least squares (LSM) attack can successfully recover secret values with an $L_1$ norm error ranging from 139453 to 6144515. The precise error range is contingent upon the specific parameters used within the ILWE construction and the number of samples utilized in the attack, demonstrating a quantifiable security margin that must be considered in parameter selection and implementation.

Beyond the Math: The Real-World Threat

Cryptographic systems are often assessed for mathematical robustness, yet practical security also hinges on implementation details. Side-channel attacks represent a significant threat by exploiting unintended information leakage during computations. Rather than targeting the mathematical flaws of an algorithm, these attacks analyze physical characteristics like power consumption, electromagnetic radiation, or even timing variations to deduce secret keys. For instance, a processor using more power while processing a ‘1’ bit versus a ‘0’ bit can reveal information about the key being used in an encryption process. Similarly, slight variations in the time taken to perform different operations, even if imperceptible to a user, can be statistically analyzed to recover sensitive data. These attacks demonstrate that even mathematically secure algorithms are vulnerable if not carefully implemented and protected against such physical leakage.

Despite the theoretical security of modern cryptographic algorithms like BLISS – a lattice-based signature scheme gaining traction in post-quantum cryptography – practical implementations are susceptible to side-channel attacks. These attacks don’t target flaws in the mathematical principles of BLISS itself, but rather exploit unintentional information leakage during computation, such as variations in power consumption or execution time. This demonstrates that a secure algorithm on paper doesn’t guarantee a secure system in practice; meticulous implementation is paramount. The vulnerability of BLISS underscores a critical need for developers to proactively consider and address potential side-channel leakage points during the design and deployment phases, ensuring that theoretical security translates into real-world resilience against increasingly sophisticated attacks.

Securing cryptographic systems against side-channel attacks necessitates a dual approach, addressing weaknesses both within the mathematical algorithms and the physical hardware on which they operate. Research demonstrates that the effectiveness of attacks like the Local Side-channel Mapping (LSM) attack is significantly amplified when sampling distributions are non-uniform; variations in timing or power consumption become more pronounced and easier to exploit. Consequently, practical implementations of cryptographic schemes must prioritize uniform sampling techniques – ensuring that each operation takes a consistent amount of time and consumes a predictable amount of power – to obscure potentially revealing data and substantially bolster resistance against these subtle, yet powerful, attacks. This emphasis on uniformity isn’t merely a theoretical consideration, but a crucial engineering practice for deploying genuinely secure cryptographic solutions.

The pursuit of post-quantum cryptography, as evidenced by this analysis of integer learning with errors, feels predictably fraught. It’s a delicate dance between mathematical elegance and the brutal realities of implementation. This paper dissects vulnerabilities in ILWE constructions, specifically regarding rejection sampling and least squares attacks – proving, once again, that theoretical security isn’t the same as practical resilience. As John McCarthy observed, “It is better to be vaguely right than precisely wrong,” a sentiment that rings true here. The researchers demonstrate how seemingly robust parameter choices can succumb to attack, reminding everyone that any system, no matter how cleverly designed, will eventually reveal its weaknesses under sufficient scrutiny. Better one carefully vetted parameter set than a hundred untested configurations, it seems.

What’s Next?

The analysis presented here, while detailing vulnerabilities in specific ILWE parameter sets, predictably confirms a longstanding truth: cryptographic elegance rarely survives contact with practical implementation. The demonstrated susceptibility to least squares attacks isn’t a flaw in the concept of ILWE, but rather a reminder that parameter selection is less a mathematical exercise and more a game of anticipating the inevitable ingenuity of attackers. It’s a temporary reprieve, naturally. One can already envision optimizations to the attack, or entirely new vectors exploiting the same underlying weaknesses.

Future work will undoubtedly focus on ‘harder’ distributions – those with properties that appear to resist known attacks. This will lead to more complex, and therefore more brittle, constructions. The field seems perpetually locked in a cycle of proposing complex schemes, then discovering the simple attacks that bypass them. The claim of ‘post-quantum’ security, in particular, feels increasingly optimistic. It’s not that quantum computers are the only threat; it’s simply that they provide a conveniently distant deadline for problems that will surface long before then.

One anticipates a resurgence of interest in side-channel resistance, not as an afterthought, but as a fundamental design principle. However, even perfect masking and shuffling offer only temporary security. The history of cryptography is littered with schemes declared ‘immune’ to side-channels, only to fall prey to increasingly subtle timing or power analysis techniques. It’s not about eliminating the signal; it’s about making it expensive to measure. And expense, of course, is relative.

Original article: https://arxiv.org/pdf/2512.08172.pdf

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

The Inevitable Quantum Reckoning

A New Foundation: Beyond Brute Force

The Geometry of Security

Beyond the Math: The Real-World Threat

What’s Next?

See also: