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arXiv:2605.24798

Improved Dual Attack and Trapdoor Sampling via Quantum Rejection Sampling

Cong Ling, Hao Yan, Nicholas Zhao

quant-phcs.DSSubmitted May 24, 2026• Announced May 26, 2026• 0 citations
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Abstract

In this work, we revisit the dual attack and GPV trapdoor sampling, focusing on the lattice Gaussian sampling term, which can be a significant bottleneck in the overall complexity. We show that this sampling step can be quantumly accelerated by combining the lower bound underlying Wang and Ling's analysis of Klein's algorithm with the quantum rejection sampling (QRS) framework proposed by Ozols et al. Specifically, this lower bound gives precisely the pointwise domination condition required for quantum rejection sampling when given coherent oracle access to a truncated Klein proposal distribution, which yields a quantum procedure for preparing the truncated dual $q$-ary lattice Gaussian with a quadratic reduction in the sampling complexity. The truncation radius is chosen so that the truncated distribution is negligibly close to the full lattice Gaussian in total variation distance. Substituting this sampler into the dual attack framework results in reduced overall attack-cost estimates. Compared with Pouly and Shen's modern dual attack under the same parameter choices, our estimates reduce the attack cost by \(9\), \(4\), and \(13\) bits for Kyber-512, Kyber-768, and Kyber-1024, respectively. We also report the corresponding estimates with modulus switching. Finally, by replacing the Markov chain Monte Carlo (MCMC) sampler with the QRS algorithm, we achieve a similar quadratic speedup in the GPV signing process.

Comments
25 pages, 3 figures

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Digital signatures and encryption schemes constitute arguably an integral part of cryptographic schemes with the goal to meet the security needs of present and future private and business applications. However, almost all public key cryptosystems applied in practice are put at risk due to its vulnerability to quantum attacks as a result of Shor's quantum algorithm. The magnitude of economic and social impact is tremendous inherently asking for alternatives replacing classical schemes in case large-scale quantum computers are built. Lattice-based cryptography emerged as a powerful candidate attracting lots of attention not only due to its conjectured resistance against quantum attacks, but also because of its unique security guarantee to provide worst-case hardness of average-case instances. Hence, the requirement of imposing further assumptions on the hardness of randomly chosen instances disappears, resulting in more efficient instantiations of cryptographic schemes. 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•Jun 10, 2015•0 citations
In this work, we revisit the dual attack and GPV trapdoor sampling, focusing on the lattice Gaussian sampling term, which can be a significant bottleneck in the overall complexity. We show that this sampling step can be quantumly accelerated by combining the lower bound underlying Wang and Ling's analysis of Klein's algorithm with the quantum rejection sampling (QRS) framework proposed by Ozols et al. Specifically, this lower bound gives precisely the pointwise domination condition required for quantum rejection sampling when given coherent oracle access to a truncated Klein proposal distribution, which yields a quantum procedure for preparing the truncated dual qqq-ary lattice Gaussian with a quadratic reduction in the sampling complexity. The truncation radius is chosen so that the truncated distribution is negligibly close to the full lattice Gaussian in total variation distance. Substituting this sampler into the dual attack framework results in reduced overall attack-cost estimates. Compared with Pouly and Shen's modern dual attack under the same parameter choices, our estimates reduce the attack cost by (9), (4), and (13) bits for Kyber-512, Kyber-768, and Kyber-1024, respectively. We also report the corresponding estimates with modulus switching. Finally, by replacing the Markov chain Monte Carlo (MCMC) sampler with the QRS algorithm, we achieve a similar quadratic speedup in the GPV signing process.
Discrete Gaussian Sampling on lattices is a fundamental problem in lattice-based cryptography. It appears both in basic cryptographic primitives such as digital signatures and as an important cryptanalysis building block for solving hard lattice problems. In this paper, we show a quantum algorithm based on the quantum rejection sampling technique whose complexity is asymptotically quadratically faster than its classical counterpart in [Wang & Ling, IEEE Trans. Inf. Theory 2019]. Our sampler outputs a quantum state which can either be measured to get the desired distribution or be used directly as such in other quantum algorithms. By doing so, we derive two versions of quantum dual attacks that improve upon the previous ones in [Pouly & Shen, EUROCRYPT 2024]. The two versions are incomparable, each having distinct advantages (speed vs memory requirement). The second version is particularly interesting as it requires only polynomial classical and quantum memory, excluding the classical memory used in the preprocessing step of the Discrete Gaussian sampler. Our quantum Discrete Gaussian sampler can also be used to speed up the algorithm for solving the Short Integer Solution problem, in any norm, of [Bollauf, Pouly & Shen, ePrint 2026/225].
Rejection sampling technology is a core tool in the design of lattice-based signatures with ‘Fiat–Shamir with Aborts’ structure, and it is related to signing efficiency and signature, size as well as security. In the rejection sampling theorem proposed by Lyubashevsky, the masking vector of rejection sampling is chosen from discrete Gaussian distribution. However, in practical designs, the masking vector is more likely to be chosen from bounded uniform distribution due to better efficiency and simpler implementation. Besides, as one of the third-round candidate signatures in the NIST postquantum cryptography standardization process, the 3rd round version of CRYSTALS-Dilithium has proposed a new method to decrease the rejection probability in order to achieve better efficiency and smaller signature size by decreasing the number of nonzero coefficients of the challenge polynomial according to the security levels. However, it is seen that small entropies in this new method may lead to higher risk of forgery attack compared with former schemes proposed in its 2nd version. Thus, in this paper, we first analyze the complexity of forgery attack for small entropies and then introduce a new method to decrease the rejection probability without loss of security including the security against forgery attack. This method is achieved by introducing a new rejection sampling theorem with tighter bound by utilizing Rényi divergence where masking vector follows uniform distribution. By observing large gaps between the security claim and actual security bound in CRYSTALS-Dilithium, we propose two series of adapted parameters for CRYSTALS-Dilithium. The first set can improve the efficiency of the signing process in CRYSTALS-Dilithium by factors of 61.7 % and 41.7 % , according to the security levels, and ensure the security against known attacks, including forgery attack. And, the second set can reduce the signature size by a factor of 14.09 % with small improvements in efficiency at the same security level.
Sampling tasks are a natural class of problems for quantum computers due to the probabilistic nature of the Born rule. Sampling from useful distributions on noisy quantum hardware remains a challenging problem. A recent paper [D. Layden et al., Nature (London) 619, 282 (2023).] proposed a quantum-enhanced Markov-chain Monte Carlo algorithm where moves are generated by a quantum device and accepted or rejected by a classical algorithm. While this procedure is robust to noise and control imperfections, its potential for quantum advantage is unclear. Here we show that there is no speedup over classical sampling on a worst-case unstructured sampling problem. We present an upper bound to the Markov gap that rules out a speedup for any unital quantum proposal.
Sampling tasks are a natural class of problems for quantum computers due to the probabilistic nature of the Born rule. Sampling from useful distributions on noisy quantum hardware remains a challenging problem. A recent paper [Layden, D. et al. Nature 619, 282-287 (2023)] proposed a quantum-enhanced Markov chain Monte Carlo algorithm where moves are generated by a quantum device and accepted or rejected by a classical algorithm. While this procedure is robust to noise and control imperfections, its potential for quantum advantage is unclear. Here we show that there is no speedup over classical sampling on a worst-case unstructured sampling problem. We present an upper bound to the Markov gap that rules out a speedup for any unital quantum proposal.
Rejection samplings are the essential building blocks to design lattice-based digital signatures under the Fiat–Shamir paradigm. Up to now most of them have been built by Gaussian samplings or uniform samplings. Gaussian-based rejection sampling signatures such as the BLISS scheme have very short signature sizes but are vulnerable to timing attacks, whereas the uniform-based rejection sampling signatures such as ML-DSA are allowed to be simply implemented but have larger signature sizes. This work intends to use a new probability distribution, rather than Gaussian or uniform distributions, to build the rejection sampling in Fiat–Shamir signatures, and aims to achieve short signature sizes while avoiding the cost of Gaussian sampling. To this end, we choose centered binomial distribution as a replacement, and build a new and compact rejection sampling that has the properties of both high-precision sampling and semi-uniform operation. As an application, we combine this rejection sampling with Lyubashevsky’s signature scheme, then propose the first lattice-based Fiat–Shamir signature scheme from centered binomial distribution. The proposed scheme not only avoids Gaussian sampling, but also is very efficient in terms of the signature sizes.
Digital signatures and encryption schemes constitute arguably an integral part of cryptographic schemes with the goal to meet the security needs of present and future private and business applications. However, almost all public key cryptosystems applied in practice are put at risk due to its vulnerability to quantum attacks as a result of Shor's quantum algorithm. The magnitude of economic and social impact is tremendous inherently asking for alternatives replacing classical schemes in case large-scale quantum computers are built. Lattice-based cryptography emerged as a powerful candidate attracting lots of attention not only due to its conjectured resistance against quantum attacks, but also because of its unique security guarantee to provide worst-case hardness of average-case instances. Hence, the requirement of imposing further assumptions on the hardness of randomly chosen instances disappears, resulting in more efficient instantiations of cryptographic schemes. The best known lattice attack algorithms run in exponential time. \nIn this thesis we contribute to a smooth transition into a world with practically efficient lattice-based cryptographic schemes. This is indeed accomplished by designing new algorithms and cryptographic schemes as well as improving existing ones. Our contributions are threefold. \nFirst, we construct new encryption schemes that fully exploit the error term in LWE instances. To this end, we introduce a novel computational problem that we call Augmented LWE (A-LWE), differing from the original LWE problem only in the way the error term is produced. In fact, we embed arbitrary data into the error term without changing the target distributions. Following this, we prove that A-LWE instances are indistinguishable from LWE samples. This allows to build powerful encryption schemes on top of the A-LWE problem that are simple in its representations and efficient in practice while encrypting huge amounts of data realizing message expansion factors close to 1. This improves, to our knowledge, upon all existing encryption schemes. Due to the versatility of the error term, we further add various security features such as CCA and RCCA security or even plug lattice-based signatures into parts of the error term, thus providing an additional mechanism to authenticate encrypted data. Based on the methodology to embed arbitrary data into the error term while keeping the target distributions, we realize a novel CDT-like discrete Gaussian sampler that beats the best known samplers such as Knuth-Yao or the standard CDT sampler in terms of running time. At run time the table size amounting to 44 elements is constant for every discrete Gaussian parameter and the total space requirements are exactly as large as for the standard CDT sampler. Further results include a very efficient inversion algorithm for ring elements in special classes of cyclotomic rings. In fact, by use of the NTT it is possible to efficiently check for invertibility and deduce a representation of the corresponding unit group. Moreover, we generalize the LWE inversion algorithm for the trapdoor candidate of Micciancio and Peikert from power of two moduli to arbitrary composed integers using a different approach. \nIn the second part of this thesis, we present an efficient trapdoor construction for ideal lattices and an associated description of the GPV signature scheme. Furthermore, we improve the signing step using a different representation of the involved perturbation matrix leading to enhanced memory usage and running times. Subsequently, we introduce an advanced compression algorithm for GPV signatures, which previously suffered from huge signature sizes as a result of the construction or due to the requirement of the security proof. We circumvent this problem by introducing the notion of public and secret randomness for signatures. In particular, we generate the public portion of a signature from a short uniform random seed without violating the previous conditions. This concept is subsequently transferred to the multi-signer setting which increases the efficiency of the compression scheme in presence of multiple signers. Finally in this part, we propose the first lattice-based sequential aggregate signature scheme that enables a group of signers to sequentially generate an aggregate signature of reduced storage size such that the verifier is still able to check that each signer indeed signed a message. This approach is realized based on lattice-based trapdoor functions and has many application areas such as wireless sensor networks. \nIn the final part of this thesis, we extend the theoretical foundations of lattices and propose new representations of lattice problems by use of Cauchy integrals. Considering lattice points as simple poles of some complex functions allows to operate on lattice points via Cauchy integrals and its generalizations. For instance, we can deduce for the one-dimensional and two-dimensional case simple expressions for the number of lattice points inside a domain using trigonometric or elliptic functions.