Lattice sieving via quantum random walks

Lattice-based cryptography is one of the leading proposals for post-quantum cryptography. The Shortest Vector Problem (SVP) is arguably the most important problem for the cryptanalysis of lattice-based cryptography, and many lattice-based schemes have security claims based on its hardness. The best quantum algorithm for the SVP is due to Laarhoven [Laa16 PhD] and runs in (heuristic) time $2^{0.2653d + o(d)}$. In this article, we present an improvement over Laarhoven\u27s result and present an algorithm that has a (heuristic) running time of $2^{0.2570 d + o(d)}$ where $d$ is the lattice dimension. We also present time-memory trade-offs where we quantify the amount of quantum memory and quantum random access memory of our algorithm. The core idea is to replace Grover\u27s algorithm used in [Laa16 PhD] in a key part of the sieving algorithm by a quantum random walk in which we add a layer of local sensitive filtering

Asymptotics of hybrid primal lattice attacks

The literature gives the impression that (1) existing heuristics accurately predict how effective lattice attacks are, (2) non-ternary lattice systems are not vulnerable to hybrid multi-decoding primal attacks, and (3) the asymptotic exponents of attacks against non-ternary systems have stabilized. This paper shows that 1 contradicts 2 and that 1 contradicts 3: the existing heuristics imply that hybrid primal key-recovery attacks are exponentially faster than standard non-hybrid primal key-recovery attacks against the LPR PKE with any constant error width. This is the first report since 2015 of an exponential speedup in heuristic non-quantum primal attacks against non-ternary LPR. Quantitatively, for dimension n, modulus n^{Q_0+o(1)}, and error width w, a surprisingly simple hybrid attack reduces heuristic costs from 2^{(ρ+o(1))n} to 2^{(ρ-ρ H_0+o(1))n}, where z_0=2Q_0/(Q_0+1/2)^2, ρ=z_0 log_4(3/2), and H_0=1/(1+(lg w)/0.057981z_0). This raises the questions of (1) what heuristic exponent is achieved by more sophisticated hybrid attacks and (2) what impact hybrid attacks have upon concrete cryptosystems whose security analyses have ignored hybrid attacks, such as Kyber-512