In this paper we investigate the impact of decryption failures on the chosen-ciphertext security of (Ring/Module)-Learning With Errors and (Ring/Module)-Learning with Rounding based primitives. Our analysis is split in three parts: First, we use a technique to increase the failure rate of these schemes called failure boosting. Based on this technique we investigate the minimal effort for an adversary to obtain a failure in 3 cases: when he has access to a quantum computer, when he mounts a multi-target attack and when he can only perform a limited number of oracle queries. Secondly, we examine the amount of information that an adversary can derive from failing ciphertexts. Finally, these techniques are combined in an attack on (Ring/Module)-LWE and (Ring/Module)-LWR based schemes with decryption failures. We provide both a theoretical analysis as well as an implementation to calculate the security impact and show that an attacker can significantly reduce the security of (Ring/Module)-LWE/LWR based schemes that have a relatively high failure rate. However, for the candidates of the NIST post-quantum standardization process that we assessed, the number of required oracle queries is above practical limits due to their conservative parameter choices.
In this paper we investigate the impact of decryption failures on the chosen-ciphertext security of (Ring/Module)-Learning With Errors and (Ring/Module)-Learning with Rounding based primitives. Our analysis is split in three parts: First, we use a technique to increase the failure rate of these schemes called failure boosting. Based on this technique we investigate the minimal effort for an adversary to obtain a failure in 3 cases: when he has access to a quantum computer, when he mounts a multi-target attack and when he can only perform a limited number of oracle queries. Secondly, we examine the amount of information that an adversary can derive from failing ciphertexts. Finally, these techniques are combined in an attack on (Ring/Module)-LWE and (Ring/Module)-LWR based schemes with decryption failures. We provide both a theoretical analysis as well as an implementation to calculate the security impact and show that an attacker can significantly reduce the security of (Ring/Module)-LWE/LWR based schemes that have a relatively high failure rate. However, for the candidates of the NIST post-quantum standardization process that we assessed, the number of required oracle queries is above practical limits due to their conservative parameter choices.
Quantum computing is a significant threat to classical public-key cryptography. In strong "quantum access" security models, numerous symmetric-key cryptosystems are also vulnerable. We consider classical encryption in a model which grants the adversary quantum oracle access to encryption and decryption, but where the latter is restricted to non-adaptive (i.e., pre-challenge) queries only. We define this model formally using appropriate notions of ciphertext indistinguishability and semantic security (which are equivalent by standard arguments) and call it QCCA1 in analogy to the classical CCA1 security model. Using a bound on quantum random-access codes, we show that the standard PRF-based encryption schemes are QCCA1-secure when instantiated with quantum-secure primitives. We then revisit standard IND-CPA-secure Learning with Errors (LWE) encryption and show that leaking just one quantum decryption query (and no other queries or leakage of any kind) allows the adversary to recover the full secret key with constant success probability. In the classical setting, by contrast, recovering the key requires a linear number of decryption queries. The algorithm at the core of our attack is a (large-modulus version of) the well-known Bernstein-Vazirani algorithm. We emphasize that our results should not be interpreted as a weakness of these cryptosystems in their stated security setting (i.e., post-quantum chosen-plaintext secrecy). Rather, our results mean that, if these cryptosystems are exposed to chosen-ciphertext attacks (e.g., as a result of deployment in an inappropriate real-world setting) then quantum attacks are even more devastating than classical ones.
Large-scale quantum computing poses a major threat to classical public-key cryptography. Recently, strong “quantum access” security models have shown that numerous symmetric-key cryptosystems are also vulnerable. In this paper, we consider classical encryption in a model that grants the adversary quantum oracle access to encryption and decryption, but where we restrict the latter to non-adaptive (i.e., pre-challenge) queries only. We formalize this model using appropriate notions of ciphertext indistinguishability and semantic security (which are equivalent by standard arguments) and call it QCCA 1 in analogy to the classical CCA 1 security model. We show that the standard pseudorandom function ( PRF )-based encryption schemes are QCCA 1 -secure when instantiated with quantum-secure primitives. Our security proofs use a strong bound on quantum random-access codes with shared randomness. Revisiting plain IND − CPA -secure Learning with Errors ( LWE ) encryption, we show that leaking only a single quantum decryption query (and no other leakage or queries of any kind) allows the adversary to recover the full secret key with constant success probability. Information-theoretically, full recovery of the key in the classical setting requires at least a linear number of decryption queries. Our results thus challenge the notion that LWE is unconditionally “just as secure” quantumly as it is classically. The algorithm at the core of our attack is a new variant of the well-known Bernstein–Vazirani algorithm. Finally, we emphasize that our results should not be interpreted as a weakness of these cryptosystems in their stated security setting (i.e., post-quantum chosen-plaintext secrecy). Rather, our results mean that, if these cryptosystems are exposed to chosen-ciphertext attacks (e.g., as a result of deployment in an inappropriate real-world setting) then quantum attacks are even more devastating than classical ones.
The Learning with Rounding (LWR) problem, introduced as a deterministic variant of Learning with Errors (LWE), has become a promising foundation for post-quantum cryptography. This Systematization of Knowledge (SoK) article presents a comprehensive survey of the theoretical foundations, algorithmic developments, and practical implementations of LWR-based cryptographic schemes. We introduce LWR within the broader landscape of lattice-based cryptography and post-quantum security, highlighting its advantages such as reduced randomness, improved efficiency, and enhanced side-channel resistance. We explore the evolution of security reductions from LWR to LWE, including recent advances that support practical parameter regimes and address challenges in both bounded and unbounded sample settings. This article systematically reviews existing LWR-based schemes — including Saber, Lizard, Florete, Espada, Sable, and SMAUG — analyzing their design choices, parameter sets, and performance tradeoffs. Furthermore, we examine the impact of LWR on side-channel resistance, failure probabilities, and masking efficiency, demonstrating its suitability for secure and efficient implementations. By consolidating the research spanning theory and practice, this SoK aims at guiding future cryptographic design and standardization efforts leveraging LWR.
The ongoing transition to Post-Quantum Cryptography (PQC) has highlighted the need for cryptographic schemes that offer high security, strong performance, and fine-grained parameter selection. In lattice-based cryptography, particularly for the popular module variants of learning with errors (Module-LWE) and learning with rounding (Module-LWR) schemes based on power-of-two cyclotomics, existing constructions often force parameter choices that either overshoot or undershoot desired security levels due to structural constraints. In this work, we introduce a new class of techniques that are the advantages of both worlds: structured Module-LWE (or LWR) embeds more algebraic structure than a module such that it significantly improves performance, yet less structure than a power-of-two cyclotomic ring such that it still enables more flexible and efficient parameter selection. We present two concrete instances: a construction based on a radical extension of a two-power cyclotomic field denoted radical Ring-LWE (RR-LWE) or Ring-LWR (RR-LWR), and a cyclotomic block-ring module lattice approach (BRM-LWE or BRM-LWR). These new structured Module-LWE and LWR reduce the required number of uniformly random bytes in its matrix by a factor up to the module rank and allow efficient NTT implementations while enabling more granular security-performance trade-offs. We analyze the security of these constructions, provide practical parameter sets, and present implementation results demonstrating a performance improvement of up to 37% compared to an optimized implementation of ML-KEM. Our techniques apply to both key encapsulation mechanisms and digital signature schemes, offering a pathway to more adaptable and performant PQC standards.
Ring-LWE (Ring Learning With Errors) is a post-quantum cryptography algorithm based on mathematical problems in number theory and algebra. It is an extension of LWE (Learning With Errors) first introduced by Oded Regev and is used for secure data encryption from quantum computer attacks. The process consists of three main steps: key generation, encryption, and decryption. Key generation uses random polynomials and noise to create public and private keys. While encryption produces ciphertext in the form of polynomial pairs, decryption uses the private key to return the ciphertext to the original message. To conduct the experiment, the modulus value q = 17, the modulus polynomial f(x) = x2 + 1, and the private key and random polynomial have been determined. The results of the experiment show that the message can be accurately decrypted using the private key, thus proving the success of this algorithm in maintaining data security. With its security properties that are resistant to quantum attacks, Ring-LWE is a promising alternative for future cryptography.
Research on the security of lattice-based public-key encryption schemes against misuse attacks is an important part of the cryptographic assessment of the National Institute of Standards and Technology (NIST) post-quantum cryptography (PQC) standardization process. In particular, many NIST-PQC cryptosystems follow the same meta-cryptosystem. At EUROCRYPT 2019, Ba˘etu et al. mounted a classical key recovery under plaintext checking attacks (KR-PCA) and a quantum key recovery under chosen ciphertext attacks (KR-CCA). They analyzed the security of the weak version of nine submissions to NIST. In this paper, we focus on learning with error (LWE)-based FrodoPKE, whose IND-CPA security is tightly related to the hardness of plain LWE problems. We first review the meta-cryptosystem and quantum algorithm for solving quantum LWE problems. Then, we consider the case where the noise follows a discrete Gaussian distribution and recompute the success probability for quantum LWE by using Hoeffding bound. Finally, we give a quantum key recovery algorithm based on LWE under CCA attack and analyze the security of Frodo. Compared with the existing work of Ba˘etu et al., our method reduces the number of queries from 22 to 1 with the same success probability.