Hardness of the Shortest Vector Problem on Ideal Lattices under Quantum Reductions

The security of efficient post-quantum cryptographic primitives, particularly those based on the Ring Learning With Errors (Ring-LWE) problem, relies fundamentally on the presumed intractability of the Shortest Vector Problem (SVP) over ideal lattices. This monograph investigates the hardness of Ideal-SVP in the presence of quantum adversaries. We rigorously analyze the worst-case to average-case reductions that underpin the security of ideal lattice-based schemes, extending classical Gaussian smoothing arguments to the quantum setting. Specifically, we examine the impact of quantum sieving algorithms and the Quantum Fourier Transform (QFT) on the algebraic structure of number fields. While quantum algorithms provide a quadratic speedup for general lattice sieving, we argue that the additional algebraic structure of ideal lattices does not yield a catastrophic collapse in hardness under current quantum reduction models. We present a synthesis of recent results concerning the lower bounds of quantum query complexity for Ideal-SVP, establishing that for cyclotomic fields of degree n, the problem remains hard up to sub-exponential factors, thereby validating the asymptotic security of structured lattice cryptography.