Approximate short vectors in ideal lattices of Q(ζpe) with Precomputation of Cl (OK)

Abstract

Let a, b be constants such that (Formula presented). We present a classical heuristic PIP resolution method that finds a generator of any input I such that (Formula presented) in time (Formula presented) given a one time classical precomputation of cost and size (Formula presented). We also present a quantum variant of this PIP algorithm with precomputation. Let 1/3 < a< 1/2. With a quantum coprocessor running Shor’s algorithm, our algorithm solves the γ -ideal-SVP for (Formula presented) in time (Formula presented) using O~ (n2-a) qubits and a one time classical precomputation on (Formula presented) of cost (Formula presented). This is a superpolynomial improvement over the best classical method relying on the BKZ reduction, and it uses asymptotically fewer qubit than the quantum polynomial time method relying on the PIP algorithm of [BS16] which requires Ω(n3) qubits.