No abstract

No abstract

a b s t r a c tIterative solvers in combination with multi-grid have been used extensively to solve large algebraic systems. One of the best known is the Runge-Kutta iteration. We show that a generally used formulation [A. Jameson, Numerical solution of the Euler equations for compressible inviscid fluids, in: F. Angrand, A. Dervieux, J.A. Désidéri, R. Glowinski (Eds.), Numerical Methods for the Euler Equations of Fluid Dynamics, SIAM, Philadelphia, 1985, pp. 199-245] does not allow to form all possible polynomial transmittance functions and we propose a new formulation to remedy this, without using an excessive number of coefficients.After having converted the optimal parameters found in previous studies (e.g. [B. Van Leer, C.H. Tai, K.G. Powell, Design of optimally smoothing multi-stage schemes for the Euler equations, AIAA Paper 1989]) we compare them with those that we obtain when we optimize for an integrated 2-grid V -cycle and show that this results in superior performance using a low number of stages. We also propose a variant of our new formulation that roughly follows the idea of the Martinelli-Jameson scheme [A. Jameson, Analysis and design of numerical schemes for gas dynamics 1, artificial diffusion, upwind biasing, limiter and their effect on multigrid convergence, Int. J. Comput. Fluid Dyn. 4 (1995) 171-218; J.V. Lassaline, Optimal multistage relaxation coefficients for multigrid flow solvers. http://www.ryerson.ca/~jvl/papers/cfd2005.pdf] used on the advection-diffusion equation which that can be extended to other types. Gains in the order of 30%-50% have been shown with respect to classical iterative schemes on the advection equation. Better results were also obtained on the advection-diffusion equation than with the Martinelli-Jameson coefficients, but with less than half the number of matrix-vector multiplications.

No abstract

This paper investigates the power of SAT solvers in cryptanalysis. The contributions are twofold and are relevant to both theory and practice. First, we introduce an efficient, generic and automated method for generating SAT instances encoding a wide range of cryptographic computations. This method can be used to automate the first step of algebraic attacks, i.e. the generation of a system of algebraic equations. Second, we illustrate the limits of SAT solvers when attacking cryptographic algorithms, with the aim of finding weak keys in block ciphers and preimages in hash functions. SAT solvers allowed us to find, or prove the absence of, weak-key classes under both differential and linear attacks of full-round block ciphers based on the International Data Encryption Algorithm (IDEA), namely, WIDEA-n for n ∈ {4, 8}, and MESH-64(8). In summary: (i) we have found several classes of weak keys for WIDEA-n and (ii) proved that a particular class of weak keys does not exist in MESH-64(8). SAT solvers provided answers to two complementary open problems (presented in Fast Software Encryption 2009): the existence and non-existence of such weak keys. Although these problems were supposed to be difficult to answer, SAT solvers provided an efficient solution. We also report on experimental results about the performance of a modern SAT solver as the encoded cryptanalytic tasks become increasingly hard. The tasks correspond to preimage attacks on reduced MD4 algorithm.