We consider the task of using one or more Unmanned Aerial Vehicles (UAVs)

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We analyze how fast we can solve general systems of multivariate equations of various low degrees over F2; this is a well known hard problem which is important both in itself and as part of many types of algebraic cryptanalysis. Compared to the standard exhaustive search technique, our improved approach is more efficient both asymptotically and practically. We implemented several optimized versions of our techniques on CPUs and GPUs. Our technique runs more than 10 times faster on modern graphic cards than on the most powerful CPU available. Today, we can solve 48+ quadratic equations in 48 binary variables on a 500-dollar NVIDIA GTX 295 graphics card in 21 minutes. With this level of performance, solving systems of equations supposed to ensure a security level of 64 bits turns out to be feasible in practice with a modest budget. This is a clear demonstration of the computational power of GPUs in solving many types of combinatorial and cryptanalytic problems.

Abstract. This paper reports record-setting performance for the ellipticcurve method of integer factorization: for example, 926.11 curves/second for ECM stage 1 with B1 = 8192 for 280-bit integers on a single PC. The state-of-the-art GMP-ECM software handles 124.71 curves/second for ECM stage 1 with B1 = 8192 for 280-bit integers using all four cores of a 2.4 GHz Core 2 Quad Q6600.The extra speed takes advantage of extra hardware, specifically two NVIDIA GTX 295 graphics cards, using a new ECM implementation introduced in this paper. Our implementation uses Edwards curves, relies on new parallel addition formulas, and is carefully tuned for the highly parallel GPU architecture. On a single GTX 295 the implementation performs 41.88 million modular multiplications per second for a general 280-bit modulus. GMP-ECM, using all four cores of a Q6600, performs 13.03 million modular multiplications per second. This paper also reports speeds on other graphics processors: for example, 2414 280-bit elliptic-curve scalar multiplications per second on an older NVIDIA 8800 GTS (G80), again for a general 280-bit modulus. For comparison, the CHES 2008 paper "Exploiting the Power of GPUs for Asymmetric Cryptography" reported 1412 elliptic-curve scalar multiplications per second on the same graphics processor despite having fewer bits in the scalar (224 instead of 280), fewer bits in the modulus (224 instead of 280), and a special modulus (2 224 − 2 96 + 1).

Abstract. A recently proposed class of multivariate Public-Key Cryptosystems, the Rainbow-Like Digital Signature Schemes, in which successive sets of central variables are obtained from previous ones by solving linear equations, seem to lead to e cient schemes (TTS, TRMS, and Rainbow) that perform well on systems of low computational resources. Recently SFLASH (C * − ) was broken by Dubois, Fouque, Shamir, and Stern via a di erential attack. In this paper, we exhibit similar algebraic and di ential attacks, that will reduce published Rainbow-like schemes below their security levels. We will also discuss how parameters for Rainbow and TTS schemes should be chosen for practical applications. Keywords: rank, di erential attack, algebraic attack, oil-and-vinegar Note: This is an update to the paper to appear at ACNS 2008, New York 1 Outline Multivariate Public-Key Cryptosystems (MPKCs, or trapdoor MQ schemes) are cryptosystems for which the public key is a set of polynomials P = (p 1 , . . . , p m ) in variables x = (x 1 , . . . , x n ) where all variables and coe cients are in K = GF(q). In practice this is always accomplished viaIn any given scheme, the central map Q belongs to a certain class of quadratic maps whose inverse can be computed relatively easily. The maps S, T are a ne. The polynomials giving y i in x are called the central polynomials, and the x j are called the central variables.In 1999, the Unbalanced Oil-and-Vinegar multivariate structure is proposed by Patarin et al [16]. Lately the Rainbow class of signatures [7,20,25], based on repeated applications of the Unbalanced Oil-and-Vinegar principle, shows some promise on systems of low computational resources.Given that the well-known C * − class of signature schemes including SFLASH was broken by di erential attacks [8], we examine similar attacks on Rainbow, with the following conclusions:Di erentials improve on the High-Rank attacks on Rainbow-like systems.