International Association
for Cryptologic Research

We present cryptanalysis of the inhomogenous short integer solution (ISIS) problem for anomalously small moduli q by exploiting the geometry of BKZ reduced bases of q-ary lattices. We apply this cryptanalysis to examples from the literature where taking such small moduli has been suggested. A recent work [Espitau--Tibouchi--Wallet--Yu, CRYPTO 2022] suggests small q versions of the lattice signature scheme Falcon and its variant Mitaka. For one small q parametrisation of Falcon we reduce the estimated security against signature forgery by approximately 26 bits. For one small q parametrisation of Mitaka we successfully forge a signature in 15 seconds.

We propose the signature scheme Hawk, a concrete instantiation of proposals to use the Lattice Isomorphism Problem (LIP) as a foundation for cryptography that focuses on simplicity. This simplicity stems from LIP, which allows the use of lattices such as $\mathbb{Z}^n$, leading to signature algorithms with no floats, no rejection sampling, and compact precomputed distributions. Such design features are desirable for constrained devices, and when computing signatures inside FHE or MPC. The most significant change from recent LIP proposals is the use of module lattices, reusing algorithms and ideas from NTRUSign and Falcon. Its simplicity makes Hawk competitive. We provide cryptanalysis with experimental evidence for the design of Hawk and implement two parameter sets, Hawk-512 and Hawk-1024. Signing using Hawk-512 and Hawk-1024 is four times faster than Falcon on x86 architectures, produces signatures that are about 15% more compact, and is slightly more secure against forgeries by lattice reduction attacks. When floating-points are unavailable, Hawk signs 15 times faster than Falcon. We provide a worst case to average case reduction for module LIP. For certain parametrisations of Hawk this applies to secret key recovery and we reduce signature forgery in the random oracle model to a new problem called the one more short vector problem.

As lattice-based key encapsulation, digital signature, and fully homomorphic encryption schemes near standardisation, ever more focus is being directed to the precise estimation of the security of these schemes. The primal attack reduces key recovery against such schemes to instances of the unique Shortest Vector Problem (uSVP). Dachman-Soled et al. (Crypto 2020) recently proposed a new approach for fine-grained estimation of the cost of the primal attack when using Progressive BKZ for lattice reduction. In this paper we review and extend their technique to BKZ 2.0 and provide extensive experimental evidence of its accuracy. Using this technique we also explain results from previous primal attack experiments by Albrecht et al. (Asiacrypt 2017) where attacks succeeded with smaller than expected block sizes. Finally, we use our simulators to reestimate the cost of attacking the three lattice KEM finalists of the NIST Post Quantum Standardisation Process.

Quantum variants of lattice sieve algorithms are routinely used to assess the security of lattice based cryptographic constructions. In this work we provide a heuristic, non-asymptotic, analysis of the cost of several algorithms for near neighbour search on high dimensional spheres. These algorithms are key components of lattice sieves. We design quantum circuits for near neighbour search algorithms and provide software that numerically optimises algorithm parameters according to various cost metrics. Using this software we estimate the cost of classical and quantum near neighbour search on spheres. For the most performant near neighbour search algorithm that we analyse we find a small quantum speedup in dimensions of cryptanalytic interest. Achieving this speedup requires several optimistic physical and algorithmic assumptions.

We propose the General Sieve Kernel (G6K, pronounced / e.si.ka/), an abstract stateful machine supporting a wide variety of lattice reduction strategies based on sieving algorithms. Using the basic instruction set of this abstract stateful machine, we first give concise formulations of previous sieving strategies from the literature and then propose new ones. We then also give a light variant of BKZ exploiting the features of our abstract stateful machine. This encapsulates several recent suggestions (Ducas at Eurocrypt 2018; Laarhoven and Mariano at PQCrypto 2018) to move beyond treating sieving as a blackbox SVP oracle and to utilise strong lattice reduction as preprocessing for sieving. Furthermore, we propose new tricks to minimise the sieving computation required for a given reduction quality with mechanisms such as recycling vectors between sieves, on-the-fly lifting and flexible insertions akin to Deep LLL and recent variants of Random Sampling Reduction.Moreover, we provide a highly optimised, multi-threaded and tweakable implementation of this machine which we make open-source. We then illustrate the performance of this implementation of our sieving strategies by applying G6K to various lattice challenges. In particular, our approach allows us to solve previously unsolved instances of the Darmstadt SVP (151, 153, 155) and LWE (e.g. (75, 0.005)) challenges. Our solution for the SVP-151 challenge was found 400 times faster than the time reported for the SVP-150 challenge, the previous record. For exact-SVP, we observe a performance crossover between G6K and FPLLL’s state of the art implementation of enumeration at dimension 70.

The Shortest Vector Problem (SVP) is one of the mathematical foundations of lattice based cryptography. Lattice sieve algorithms are amongst the foremost methods of solving SVP. The asymptotically fastest known classical and quantum sieves solve SVP in a d-dimensional lattice in $$2^{\mathsf {c}d + o(d)}$$ time steps with $$2^{\mathsf {c}' d + o(d)}$$ memory for constants $$c, c'$$ . In this work, we give various quantum sieving algorithms that trade computational steps for memory.We first give a quantum analogue of the classical k-Sieve algorithm [Herold–Kirshanova–Laarhoven, PKC’18] in the Quantum Random Access Memory (QRAM) model, achieving an algorithm that heuristically solves SVP in $$2^{0.2989d + o(d)}$$ time steps using $$2^{0.1395d + o(d)}$$ memory. This should be compared to the state-of-the-art algorithm [Laarhoven, Ph.D Thesis, 2015] which, in the same model, solves SVP in $$2^{0.2653d + o(d)}$$ time steps and memory. In the QRAM model these algorithms can be implemented using $$\mathrm {poly}(d)$$ width quantum circuits.Secondly, we frame the k-Sieve as the problem of k-clique listing in a graph and apply quantum k-clique finding techniques to the k-Sieve.Finally, we explore the large quantum memory regime by adapting parallel quantum search [Beals et al., Proc. Roy. Soc. A’13] to the 2-Sieve, and give an analysis in the quantum circuit model. We show how to solve SVP in $$2^{0.1037d + o(d)}$$ time steps using $$2^{0.2075d + o(d)}$$ quantum memory.