Concrete Analysis of Quantum Lattice Enumeration

Lattice reduction algorithms such as BKZ (Block-Korkine-Zolotarev) play a central role in estimating the security of lattice-based cryptography. The subroutine in BKZ which finds the shortest vector in a projected sublattice can be instantiated with enumeration algorithms. The enumeration procedure can be seen as a depth-first search on some "enumeration tree" whose nodes denote a partial assignment of the coefficients, corresponding to lattice points as a linear combination of the lattice basis with the coefficients. This work provides a concrete analysis for the cost of quantum lattice enumeration based on Montanaro's quantum tree backtracking algorithm. More precisely, we give a concrete implementation in the quantum circuit model. We also show how to optimize the circuit depth by parallelizing the components. Based on the circuit designed, we discuss the concrete quantum resource estimates required for lattice enumeration.