Lattice Sieving via Quantum Random Walks

Raffaella Carbone, Yan Pautrat

Swarnadeep Majumder, Austen Couvertier

Beomgeun Cho, Minki Hhan, Taehyun Kim +2 more

In this paper, we study the requirement for quantum random access memory (QRAM) in quantum lattice sieving, a fundamental algorithm for lattice-based cryptanalysis. First, we obtain a lower bound on the cost of quantum lattice sieving with a bounded size QRAM. We do so in a new query model encompassing a wide range of lattice sieving algorithms similar to those in the classical sieving lower bound by Kirshanova and Laarhoven [CRYPTO 21]. This implies that, under reasonable assumptions, quantum speedups in lattice sieving require the use of QRAM. In particular, no quantum speedup is possible without QRAM. Second, we investigate the trade-off between the size of QRAM and the quantum speedup. We obtain a new interpolation between classical and quantum lattice sieving. Moreover, we show that further improvements require a novel way to use the QRAM by proving the optimality of some subroutines. An important caveat is that this trade-off requires a strong assumption on the efficient replacement of QRAM data, indicating that even speedups with a small QRAM are already challenging. Finally, we provide a circuit for quantum lattice sieving without using QRAM. Our circuit has a better depth complexity than the best classical algorithms but requires an exponential amount of qubits. To the best of our knowledge, this is the first quantum speedup for lattice sieving without QRAM in the standard quantum circuit model. We explain why this circuit does not contradict our lower bound, which considers the query complexity.

Luigi Accardi, G. S. Watson

Jin-Fu Chen, Yu-Han Ma, Chang-Pu Sun

Quantum walk (QW), which is considered as the quantum counterpart of the classical random walk (CRW), is actually the quantum extension of CRW from the single-coin interpretation. The sequential unitary evolution engenders correlation between different steps in QW and leads to a non-binomial position distribution. In this paper, we propose an alternative quantum extension of CRW from the ensemble interpretation, named quantum random walk (QRW), where the walker has many unrelated coins, modeled as two-level systems, initially prepared in the same state. We calculate the walker’s position distribution in QRW for different initial coin states with the coin operator chosen as Hadamard matrix. In one-dimensional case, the walker’s position is the asymmetric binomial distribution. We further demonstrate that in QRW, coherence leads the walker to perform directional movement. For an initially decoherenced coin state, the walker’s position distribution is exactly the same as that of CRW. Moreover, we study QRW in 2D lattice, where the coherence plays a more diversified role in the walker’s position distribution.

Anwar Zaman, Rashid Ahmad, Safia Bibi +1 more