Cryptanalysis of block ciphers with overdefined systems of equations

Several recently proposed ciphers, for example Rijndael and Serpent, are built with layers of small S-boxes interconnected by linear key-dependent layers. Their security relies on the fact, that the classical methods of cryptanalysis (e.g. linear or differential attacks) are based on probabilistic characteristics, which makes their security grow exponentially with the number of rounds N r r. In this paper we study the security of such ciphers under an additional hypothesis: the S-box can be described by an overdefined system of algebraic equations (true with probability 1). We show that this is true for both Serpent (due to a small size of S-boxes) and Rijndael (due to unexpected algebraic properties). We study general methods known for solving overdefined systems of equations, such as XL from Eurocrypt’00, and show their inefficiency. Then we introduce a new method called XSL that uses the sparsity of the equations and their specific structure. The XSL attack uses only relations true with probability 1, and thus the security does not have to grow exponentially in the number of rounds. XSL has a parameter P, and from our estimations is seems that P should be a constant or grow very slowly with the number of rounds. The XSL attack would then be polynomial (or subexponential) in N r> , with a huge constant that is double-exponential in the size of the S-box. The exact complexity of such attacks is not known due to the redundant equations. Though the presented version of the XSL attack always gives always more than the exhaustive search for Rijndael, it seems to (marginally) break 256-bit Serpent. We suggest a new criterion for design of S-boxes in block ciphers: they should not be describable by a system of polynomial equations that is too small or too overdefined.

Amplitude equations and asymptotic expansions for multi-scale problems

In this paper we introduce a new technique to obtain the slow-motion dynamics in nonequilibrium and singularly perturbed problems characterized by multiple scales. Our method is based on a straightforward asymptotic reduction of the order of the governing differential equation and leads to amplitude equations that describe the slowly-varying envelope variation of a uniformly valid asymptotic expansion. This may constitute a simpler and in certain cases a more general approach toward the derivation of asymptotic expansions, compared to other mainstream methods such as the method of Multiple Scales or Matched Asymptotic expansions because of its relation with the Renormalization Group. We illustrate our method with a number of singularly perturbed problems for ordinary and partial differential equations and recover certain results from the literature as special cases. © 2010 - IOS Press and the authors. All rights reserved.

Examples illustrating the use of renormalization techniques for singularly perturbed differential equations

With nine examples, we seek to illustrate the utility of the Renormalization Group approach as a unification of other asymptotic and perturbation methods.

Reduction of amplitude equations by the renormalization group approach

This article elucidates and analyzes the fundamental underlying structure of the renormalization group (RG) approach as it applies to the solution of any differential equation involving multiple scales. The amplitude equation derived through the elimination of secular terms arising from a naive perturbation expansion of the solution to these equations by the RG approach is reduced to an algebraic equation which is expressed in terms of the Thiele semi-invariants or cumulants of the eliminant sequence { Zi } i=1 . Its use is illustrated through the solution of both linear and nonlinear perturbation problems and certain results from the literature are recovered as special cases. The fundamental structure that emerges from the application of the RG approach is not the amplitude equation but the aforementioned algebraic equation. © 2008 The American Physical Society.

The renormalization group and the implicit function theorem for amplitude equations

This article lays down the foundations of the renormalization group (RG) approach for differential equations characterized by multiple scales. The renormalization of constants through an elimination process and the subsequent derivation of the amplitude equation [Chen, Phys. Rev. E 54, 376 (1996)] are given a rigorous but not abstract mathematical form whose justification is based on the implicit function theorem. Developing the theoretical framework that underlies the RG approach leads to a systematization of the renormalization process and to the derivation of explicit closed-form expressions for the amplitude equations that can be carried out with symbolic computation for both linear and nonlinear scalar differential equations and first order systems but independently of their particular forms. Certain nonlinear singular perturbation problems are considered that illustrate the formalism and recover well-known results from the literature as special cases. © 2008 American Institute of Physics.

Secular series and renormalization group for amplitude equations

We have developed a technique that circumvents the process of elimination of secular terms and reproduces the uniformly valid approximations, amplitude equations, and first integrals. The technique is based on a rearrangement of secular terms and their grouping into the secular series that multiplies the constants of the asymptotic expansion. We illustrate the technique by deriving amplitude equations for standard nonlinear oscillator and boundary-layer problems. © 2008 The American Physical Society.

Renormalization group interpretation of the Born and Rytov approximations

In this paper the method of renormalization group (RG) [Phys. Rev. E 54, 376 (1996)] is related to the well-known approximations of Rytov and Born used in wave propagation in deterministic and random media. Certain problems in linear and nonlinear media are examined from the viewpoint of RG and compared with the literature on Born and Rytov approximations. It is found that the Rytov approximation forms a special case of the asymptotic expansion generated by the RG, and as such it gives a superior approximation to the exact solution compared with its Born counterpart. Analogous conclusions are reached for nonlinear equations with an intensity-dependent index of refraction where the RG recovers the exact solution. © 2008 Optical Society of America.

Effect of hydrophilicity of carbon nanotube arrays on the release rate and activity of recombinant human bone morphogenetic protein-2

Novel nanostructures such as vertically aligned carbon nanotube (CNT) arrays have received increasing interest as drug delivery carriers. In the present study, two CNT arrays with extreme surface wettabilities are fabricated and their effects on the release of recombinant human bone morphogenetic protein-2 (rhBMP-2) are investigated. It is found that the superhydrophilic arrays retained a larger amount of rhBMP-2 than the superhydrophobic ones. Further use of a poloxamer diffusion layer delayed the initial burst and resulted in a greater total amount of rhBMP-2 released from both surfaces. In addition, rhBMP-2 bound to the superhydrophilic CNT arrays remained bioactive while they denatured on the superhydrophobic surfaces. These results are related to the combined effects of rhBMP-2 molecules interacting with poloxamer and the surface, which could be essential in the development of advanced carriers with tailored surface functionalities.

Length control of He atmospheric plasma jet plumes: Effects of discharge parameters and ambient air

The effects of various discharge parameters and ambient gas on the length of He atmospheric plasma jet plumes expanding into the open air are studied. It is found that the voltage and width of the discharge-sustaining pulses exert significantly stronger effects on the plume length than the pulse frequency, gas flow rate, and nozzle diameter. This result is explained through detailed analysis of the I-V characteristics of the primary and secondary discharges which reveals the major role of the integrated total charges of the primary discharge in the plasma dynamics. The length of the jet plume can be significantly increased by guiding the propagating plume into a glass tube attached to the nozzle. This increase is attributed to elimination of the diffusion of surrounding air into the plasma plume, an absence which facilitates the propagation of the ionization front. These results are important for establishing a good level of understanding of the expansion dynamics and for enabling a high degree of control of atmospheric pressure plasmas in biomedical, materials synthesis and processing, environmental and other existing and emerging industrial applications. © 2009 American Institute of Physics.

High-rate, room temperature plasma-enhanced deposition of aluminum-doped zinc oxide nanofilms for solar cell applications

A custom-designed inductively coupled plasma (ICP)-assisted radio-frequency magnetron sputtering deposition system has been employed to synthesize aluminium-doped zinc oxide (ZnO:Al) nanofilms on glass substrates at room temperature. The effects of film thickness and ZnO target (partially covered by Al chips) power on the structural, electrical and optical properties of the ZnO:Al nanofilms are studied. A high growth rate (∼41 nm/min), low electrical sheet resistance (as low as 30 Ω/□) and high optical transparency (>80%) over the visible spectrum has been achieved at a film thickness of ∼615 nm and ZnO target power of 150 W. The synthesis of ZnO:Al nanofilms at room temperature and with high growth rates is attributed to the unique features of the ICP-assisted radio-frequency magnetron sputtering deposition approach. The results are relevant to the development of photovoltaic thin-film solar cells and flat panel displays.

A model of a large-area planar plasma producer based on surface wave propagation in a plasma-metal structure with a dielectric sheath

A theoretical model of a large-area planar plasma producer based on surface wave (SW) propagation in a plasma-metal structure with a dielectric sheath is presented. The SW which produces and sustains the microwave gas discharge in the planar structure propagates along an external magnetic field and possesses an eigenfrequency within the range between electron cyclotron and electron plasma frequencies. The spatial distributions of the produced plasma density, electromagnetic fields, energy flow density, phase velocity and reverse skin depth of the SW are obtained analytically and numerically.

High-rate, low-temperature synthesis of composition controlled hydrogenated amorphous silicon carbide films in low-frequency inductively coupled plasmas

It is commonly believed that in order to synthesize high-quality hydrogenated amorphous silicon carbide (a-Si1-xCx : H) films at competitive deposition rates it is necessary to operate plasma discharges at high power regimes and with heavy hydrogen dilution. Here we report on the fabrication of hydrogenated amorphous silicon carbide films with different carbon contents x (ranging from 0.09 to 0.71) at high deposition rates using inductively coupled plasma (ICP) chemical vapour deposition with no hydrogen dilution and at relatively low power densities (∼0.025 W cm -3) as compared with existing reports. The film growth rate R d peaks at x = 0.09 and x = 0.71, and equals 18 nm min-1 and 17 nm min-1, respectively, which is higher than other existing reports on the fabrication of a-Si1-xCx : H films. The extra carbon atoms for carbon-rich a-Si1-xCx : H samples are incorporated via diamond-like sp3 C-C bonding as deduced by Fourier transform infrared absorption and Raman spectroscopy analyses. The specimens feature a large optical band gap, with the maximum of 3.74 eV obtained at x = 0.71. All the a-Si1-xCx : H samples exhibit low-temperature (77 K) photoluminescence (PL), whereas only the carbon-rich a-Si1-xCx : H samples (x ≥ 0.55) exhibit room-temperature (300 K) PL. Such behaviour is explained by the static disorder model. High film quality in our work can be attributed to the high efficiency of the custom-designed ICP reactor to create reactive radical species required for the film growth. This technique can be used for a broader range of material systems where precise compositional control is required. © 2008 IOP Publishing Ltd.

Effect of soil properties on the response of pile to underground explosion

This paper develops and presents a fully coupled non-linear finite element procedure to treat the response of piles to ground shocks induced by underground explosions. The Arbitrary Lagrange Euler coupling formulation with proper state material parameters and equations are used in the study. Pile responses in four different soil types, viz, saturated soil, partially saturated soil and loose and dense dry soils are investigated and the results compared. Numerical results are validated by comparing with those from a standard design manual. Blast wave propagation in soils, horizontal pile deformations and damages in the pile are presented. The pile damage presented through plastic strain diagrams will enable the vulnerability assessment of the piles under the blast scenarios considered. The numerical results indicate that the blast performance of the piles embedded in saturated soil and loose dry soil are more severe than those in piles embedded in partially saturated soil and dense dry soil. Present findings should serve as a benchmark reference for future analysis and design.

Effects of wall transition regions on surface-wave propagation in a plasma layer bounded by a metal

Effect of near-wall transition regions on the surface wave propagation in a magnetoactive plasma layer bounded by a metal. It is shown that the account for inhomogeneities of plasma density or magnetic field causes an appearance of coupling between surface waves, propagating across magnetic field and localized near difference boundaries of the structure. The resonance damping of surface waves is analyzed too.

Effect of heating nonlinearity on propagation of high-frequency surface waves at the semiconductor-metal interface

In approximation of weak heating influence of electron heating in the high-frequency surface wave field on propagation of surface wave (heating nonlinearity) is considered. It is shown that high-frequency surface wave propagates in direction perpendicular to the external magnetic field at the semiconductor-metal interface. A nonlinear dispersion equation is obtained and studied that allows to make conclusions about the contribution of heating nonlinearity to nonlinear process of considered interaction.