Abstract. This work presents a study of the complexity of the Blum-Kalai-Wasserman (BKW) algorithm when applied to the Learning with Errors (LWE) problem, by providing refined estimates for the data and computational effort requirements for solving concrete instances of the LWE problem. We apply this refined analysis to suggested parameters for various LWE-based cryptographic schemes from the literature and compare with alternative approaches based on lattice reduction. As a result, we provide new upper bounds for the concrete hardness of these LWE-based schemes. Rather surprisingly, it appears that BKW algorithm outperforms known estimates for lattice reduction algorithms starting in dimension n ≈ 250 when LWE is reduced to SIS. However, this assumes access to an unbounded number of LWE samples.

We optically drive a microsphere at constant speed through entangled actin networks of 0.2 -1.4 mg/ml at rates faster than the critical rate controlling the onset of a nonlinear response. By measuring the resistive force exerted on the microsphere during and following strain we reveal a critical concentration c * 0.4 mg/ml for nonlinear features to emerge. For c > c * , entangled actin stiffens at short times with the degree of stiffening S and corresponding timescale t sti f f scaling with the entanglement tube density, i.e. S ∼ t sti f f ∼ d

Composites of flexible and rigid polymers are ubiquitous in biology and industry alike, yet the physical principles determining their mechanical properties are far from understood. Here, we couple force spectroscopy with large-scale Brownian Dynamics simulations to elucidate the unique viscoelastic properties of custom-engineered blends of entangled flexible DNA molecules and semiflexible actin filaments. We show that composites exhibit enhanced stress-stiffening and prolonged mechano-memory compared to systems of actin or DNA alone, and that these nonlinear features display a surprising nonmonotonic dependence on the fraction of actin in the composite. Simulations reveal that these counterintuitive results arise from synergistic microscale interactions between the two biopolymers. Namely, DNA entropically drives actin filaments to form bundles that stiffen the network but reduce the entanglement density, while a uniform well-connected actin network is required to reinforce the DNA network against yielding and flow. The competition between bundling and connectivity triggers an unexpected stress response that leads equal mass DNA-actin composites to exhibit the most pronounced stress-stiffening and the most long-lived entanglements.Mixing polymers with distinct structural features and stiffnesses endows composite materials with unique macroscopic properties such as high strength and resilience coupled with low weight and malleability [1][2][3][4]. These versatile materials, ranging from carbon nanotube-polymer nanocomposites and liquid crystals to cytoskeleton and mucus, have numerous applications from tissue engineering to high-performance energystorage [2,[5][6][7][8][9][10][11][12]. Compared to single-constituent materials, polymer composites offer a wider dynamic range and increased control over mechanical properties by tuning the relative concentrations and properties of the different species. Importantly, the unique mechanics that emerge in composites often cannot be deduced from those of the corresponding single-component systems [3,[13][14][15][16][17]. However, the physical principles that couple structural interactions to mechanics in composites remain elusive.Over the past two decades, DNA and actin have been extensively studied as model polymer systems [18][19][20][21][22]. While the contour lengths of each biopolymer can be comparable (L≈10-50 µm), actin is much stiffer than DNA with a persistence length lp of ~10 µm compared to lp≈50 nm for DNA. When sufficiently long, both polymers form entangled networks over similar concentrations (c≈0.1-2.5 mg/ml), with actin forming nematic domains above 2.5 mg/ml [18]. Despite their wide use as model systems, very few studies have examined composites of actin and DNA, focusing solely on steady-state structure at concentrations above the nematic crossover or under microscale confinement [23][24][25]. These studies reported large-scale phase separation such that DNA and actin polymers were rarely interacting. Co-entangled systems of DNA and actin have yet to be i...

Abstract. Some recent constructions based on LWE do not sample the secret uniformly at random but rather from some distribution which produces small entries. The most prominent of these is the binary-LWE problem where the secret vector is sampled from {0, 1} * or {−1, 0, 1} * . We present a variant of the BKW algorithm for binary-LWE and other small secret variants and show that this variant reduces the complexity for solving binary-LWE. We also give estimates for the cost of solving binary-LWE instances in this setting and demonstrate the advantage of this BKW variant over standard BKW and lattice reduction techniques applied to the SIS problem. Our variant can be seen as a combination of the BKW algorithm with a lazy variant of modulus switching which might be of independent interest.

Abstract. We present a study of the concrete complexity of solving instances of the unique shortest vector problem (uSVP). In particular, we study the complexity of solving the Learning with Errors (LWE) problem by reducing the Bounded-Distance Decoding (BDD) problem to uSVP and attempting to solve such instances using the 'embedding' approach. We experimentally derive a model for the success of the approach, compare to alternative methods and demonstrate that for the LWE instances considered in this work, reducing to uSVP and solving via embedding compares favorably to other approaches.