Abstract. The learning with rounding (LWR) problem, introduced by Banerjee, Peikert and Rosen [BPR12] at EUROCRYPT ’12, is a variant of learning with errors (LWE), where one replaces random errors with deterministic rounding. The LWR problem was shown to be as hard as LWE for a setting of parameters where the modulus and modulus-to-error ratio are super-polynomial. In this work we resolve the main open problem of [BPR12] and give a new reduction that works for a larger range of parameters, allowing for a polynomial modulus and modulus-to-error ratio. In particular, a smaller modulus gives us greater efficiency, and a smaller modulus-to-error ratio gives us greater security, which now follows from the worst-case hardness of GapSVP with polynomial (rather than super-polynomial) approximation factors. As a tool in the reduction, we show that there is a “lossy mode ” for the LWR problem, in which LWR samples only reveal partial information about the secret. This property gives us several interesting new applications, including a proof that LWR remains secure with weakly random secrets of sufficient minentropy, and very simple new constructions of deterministic encryption, lossy trapdoor functions and reusable extractors. Our approach is inspired by a technique of Goldwasser et al. [GKPV10] from ICS ’10, which implicitly showed the existence of a “lossy mode ” for LWE. By refining this technique, we also improve on the parameters of that work to only requiring a polynomial (instead of super-polynomial) modulus and modulus-to-error ratio.Abstract. The learning with rounding (LWR) problem, introduced by Banerjee, Peikert and Rosen [BPR12] at EUROCRYPT ’12, is a variant of learning with errors (LWE), where one replaces random errors with deterministic rounding. The LWR problem was shown to be as hard as LWE for a setting of parameters where the modulus and modulus-to-error ratio are super-polynomial. In this work we resolve the main open problem of [BPR12] and give a new reduction that works for a larger range of parameters, allowing for a polynomial modulus and modulus-to-error ratio. In particular, a smaller modulus gives us greater efficiency, and a smaller modulus-to-error ratio gives us greater security, which now follows from the worst-case hardness of GapSVP with polynomial (rather than super-polynomial) approximation factors. As a tool in the reduction, we show that there is a “lossy mode ” for the LWR problem, in which LWR samples only reveal partial information about the secret. This property gives us several interesting new applications, including a proof that LWR remains secure with weakly random secrets of sufficient minentropy, and very simple new constructions of deterministic encryption, lossy trapdoor functions and reusable extractors. Our approach is inspired by a technique of Goldwasser et al. [GKPV10] from ICS ’10, which implicitly showed the existence of a “lossy mode ” for LWE. By refining this technique, we also improve on the parameters of that work to only requiring a polynomial (instead of super-polynomial) modulus and modulus-to-error ratio.

International Cryptology ConferenceJan 1, 201361 citations

Direct Reduction for Learning with Rounding and its Applications

Leyou Zhang, Zhuang Shan, Qing Wu +1 more

Jan 1, 20240 citations

On the Hardness of Learning with Rounding over Small Modulus∗

Andrej Bogdanov, Siyao Guo, Daniel Masny +2 more

We show the following reductions from the learning with errors problem (LWE) to the learning with rounding problem (LWR): (1) Learning the secret and (2) distinguishing samples from random strings is at least as hard for LWR as it is for LWE for efficient algorithms if the number of samples is no larger than O(q/Bp), where q is the LWR modulus, p is the rounding modulus, and the noise is sampled from any distribution supported over the set {−B,..., B}. Our second result generalizes a theorem of Alwen, Krenn, Pietrzak, and Wichs (CRYPTO 2013) and provides an alternate proof of it. Unlike Alwen et al., we do not impose any number theoretic restrictions on the modulus q. The first result also extends to variants of LWR and LWE over polynomial rings. As additional results we show that (3) distinguishing any number of LWR samples from random strings is of equivalent hardness to LWE whose noise distribution is uniform over the integers in the range [−q/2p,..., q/2p) provided q is a multiple of p and (4) the “noise flooding” technique for converting faulty LWE noise to a discrete Gaussian distribution can be applied whenever q = Ω(BWe show the following reductions from the learning with errors problem (LWE) to the learning with rounding problem (LWR): (1) Learning the secret and (2) distinguishing samples from random strings is at least as hard for LWR as it is for LWE for efficient algorithms if the number of samples is no larger than O(q/Bp), where q is the LWR modulus, p is the rounding modulus, and the noise is sampled from any distribution supported over the set {−B,..., B}. Our second result generalizes a theorem of Alwen, Krenn, Pietrzak, and Wichs (CRYPTO 2013) and provides an alternate proof of it. Unlike Alwen et al., we do not impose any number theoretic restrictions on the modulus q. The first result also extends to variants of LWR and LWE over polynomial rings. As additional results we show that (3) distinguishing any number of LWR samples from random strings is of equivalent hardness to LWE whose noise distribution is uniform over the integers in the range [−q/2p,..., q/2p) provided q is a multiple of p and (4) the “noise flooding” technique for converting faulty LWE noise to a discrete Gaussian distribution can be applied whenever q = Ω(B

Feb 16, 20168 citations

Designs for practical SHE schemes based on Ring-LWR

Madalina Bolboceanu, Anamaria Costache, Erin Hales +3 more

The Learning with Errors problem (LWE) and its variants are among the most popular assumptions underlying lattice-based cryptography. The Learning with Rounding problem (LWR) can be thought of as a deterministic variant of LWE. In this work, we present a thorough study of Somewhat Homomorphic Encryption schemes based on Ring-LWR that are the analogue of the Ring-LWE-based BFV scheme. Our main contribution is to present two new schemes, in the LPR and Regev paradigms, and give a thorough analysis of their security (provable and concrete). The technical tools we developed in the process may be of independent interest to the community. Our schemes inherit the many benefits of being based on LWR, including avoiding the need for expensive Gaussian sampling and improved ciphertext size. Indeed, we give a detailed comparison showing that our schemes marginally outperform the BFV scheme in terms of ciphertext size. Moreover, we show that both our schemes support RNS variants. Our Regev-type scheme can be seen as an improved generalisation of the only prior work in this direction (Costache-Smart, 2017). In particular, our scheme resolves the tangled modulus issue in the Costache-Smart proposal that led to unmanageable noise growth, and achieves a factor n improvement in the size of the public key.The Learning with Errors problem (LWE) and its variants are among the most popular assumptions underlying lattice-based cryptography. The Learning with Rounding problem (LWR) can be thought of as a deterministic variant of LWE. In this work, we present a thorough study of Somewhat Homomorphic Encryption schemes based on Ring-LWR that are the analogue of the Ring-LWE-based BFV scheme. Our main contribution is to present two new schemes, in the LPR and Regev paradigms, and give a thorough analysis of their security (provable and concrete). The technical tools we developed in the process may be of independent interest to the community. Our schemes inherit the many benefits of being based on LWR, including avoiding the need for expensive Gaussian sampling and improved ciphertext size. Indeed, we give a detailed comparison showing that our schemes marginally outperform the BFV scheme in terms of ciphertext size. Moreover, we show that both our schemes support RNS variants. Our Regev-type scheme can be seen as an improved generalisation of the only prior work in this direction (Costache-Smart, 2017). In particular, our scheme resolves the tangled modulus issue in the Costache-Smart proposal that led to unmanageable noise growth, and achieves a factor n improvement in the size of the public key.

IACR Communications in CryptologyApr 8, 20254 citations

Verifiable quantum advantage in extremely low depth

Alexandru Gheorghiu

We give a sampling problem that is solvable by shallow quantum circuits, hard for polynomial-time classical algorithms under lattice-based assumptions, and efficiently verifiable by a classical computer. The quantum sampler admits two implementations: one uses log-logarithmic-depth quantum circuits with one- and two-qubit gates, i.e., $\mathsf{QNC}^0[\log\log]$ circuits, while the other uses constant-depth quantum circuits with unbounded fan-in gates, i.e., $\mathsf{QAC}^0$ circuits. Our construction can be seen as compiling the Learning with Errors (LWE)-based single-round proof of quantumness of Arabadjieva et al. (2025) to very low depth. The price paid for this compilation is the reliance on less standard, though well-motivated, assumptions: in addition to the lattice knowledge assumption used by Arabadjieva et al. (2025), we require a strengthened variant of the adaptive-hardcore-bit property of LWE, for which we provide supporting evidence. Unlike previous low-depth proofs of quantumness, the quantum computation here requires no mid-circuit measurements or feed-forward: it consists only of running a shallow circuit and sampling from its output distribution. This shows that shallow quantum circuits have sufficient structure to solve certain classically hard tasks whose solutions can be verified efficiently.We give a sampling problem that is solvable by shallow quantum circuits, hard for polynomial-time classical algorithms under lattice-based assumptions, and efficiently verifiable by a classical computer. The quantum sampler admits two implementations: one uses log-logarithmic-depth quantum circuits with one- and two-qubit gates, i.e., $\mathsf{QNC}^0[\log\log]$ circuits, while the other uses constant-depth quantum circuits with unbounded fan-in gates, i.e., $\mathsf{QAC}^0$ circuits. Our construction can be seen as compiling the Learning with Errors (LWE)-based single-round proof of quantumness of Arabadjieva et al. (2025) to very low depth. The price paid for this compilation is the reliance on less standard, though well-motivated, assumptions: in addition to the lattice knowledge assumption used by Arabadjieva et al. (2025), we require a strengthened variant of the adaptive-hardcore-bit property of LWE, for which we provide supporting evidence. Unlike previous low-depth proofs of quantumness, the quantum computation here requires no mid-circuit measurements or feed-forward: it consists only of running a shallow circuit and sampling from its output distribution. This shows that shallow quantum circuits have sufficient structure to solve certain classically hard tasks whose solutions can be verified efficiently.

quant-pharXiv:2609.01448Sep 2, 20260 citations

\mathsf{QNC}^0[\log\log]

circuits, while the other uses constant-depth quantum circuits with unbounded fan-in gates, i.e.,

\mathsf{QAC}^0

circuits. Our construction can be seen as compiling the Learning with Errors (LWE)-based single-round proof of quantumness of Arabadjieva et al. (2025) to very low depth. The price paid for this compilation is the reliance on less standard, though well-motivated, assumptions: in addition to the lattice knowledge assumption used by Arabadjieva et al. (2025), we require a strengthened variant of the adaptive-hardcore-bit property of LWE, for which we provide supporting evidence. Unlike previous low-depth proofs of quantumness, the quantum computation here requires no mid-circuit measurements or feed-forward: it consists only of running a shallow circuit and sampling from its output distribution. This shows that shallow quantum circuits have sufficient structure to solve certain classically hard tasks whose solutions can be verified efficiently.