Paper 2025/394

Reducing the Number of Qubits in Solving LWE

Abstract

At Crypto 2021, May presented an algorithm solving the ternary Learning-With-Error problem, where the solution is a ternary vector $s\in\{0,\pm 1\}^{n}$ with a known number of $(+1)$ and $(-1)$ entries. This attack significantly improved the time complexity of $\mathcal{S}^{0.5}$ from previously known algorithms to $\mathcal{S}^{0.25}$, where $\mathcal{S}$ is the size of the key space. Therefore, May exploited that using more representations, i.e., allowing ternary interim results with additional $(+1)$ and $(-1)$ entries, reduces the overall time complexity. Later, van Hoof et al. (PQCrypto 2021) combined May's algorithm with quantum walks to a new attack that performs in time $\mathcal{S}^{0.19}$. However, this quantum attack requires an exponential amount of qubits. This work investigates whether the ternary LWE problem can also be solved using only $\mathcal{O}(n)$ qubits. Therefore, we look closely into Dicke states, which are an equal superposition over all binary vectors with a fixed Hamming weight. Generalizing Dicke states to ternary vectors makes these states applicable to the ternary LWE problem. Bärtschi and Eidenbenz (FCT 2019) proposed a quantum circuit to prepare binary Dicke states deterministically in linear time $\mathcal{O}(n)$. Their procedure benefits from the inductive structure of Dicke states, i.e., that a Dicke state of a particular dimension can be built from Dicke states of lower dimensions. Our work proves that this inductive structure is also present in generalized Dicke states with an underlying set other than $\{0,1\}^{n}$. Utilizing this structure, we introduce a new algorithm that deterministically prepares generalized Dicke states in linear time, for which we also provide an implementation in Qiskit. Finally, we apply our generalized Dicke states to the ternary LWE problem, and construct an algorithm that requires $\mathcal{O}(n)$ qubits and classical memory space up to $\mathcal{S}^{0.22}$. We achieve $\mathcal{S}^{0.379}$ as best obtainable time complexity.