Higher order differentiation over finite fields with applications to generalising the cube attack

Higher order differentiation was introduced in a cryptographic context by Lai. Several attacks can be viewed in the context of higher order differentiations, amongst them the cube attack of Dinur and Shamir and the AIDA attack of Vielhaber. All of the above have been developed for the binary case. We examine differentiation in larger fields, starting with the field GF(p) of integers modulo a prime p, and apply these techniques to generalising the cube attack to GF(p). The crucial difference is that now the degree in each variable can be higher than one, and our proposed attack will differentiate several times with respect to each variable (unlike the classical cube attack and its larger field version described by Dinur and Shamir, both of which differentiate at most once with respect to each variable). Connections to the Moebius/Reed Muller Transform over GF(p) are also examined. Finally we describe differentiation over finite fields GF(ps) with ps elements and show that it can be reduced to differentiation over GF(p), so a cube attack over GF(ps) would be equivalent to cube attacks over GF(p)

Higher order differentiation over finite fields with applications to generalising the cube attack

Higher order differentiation was introduced in a cryptographic context by Lai. Several attacks can be viewed in the context of higher order differentiations, amongst them the cube attack of Dinur and Shamir and the AIDA attack of Vielhaber. All of the above have been developed for the binary case. We examine differentiation in larger fields, starting with the field GF(p) of integers modulo a prime p, and apply these techniques to generalising the cube attack to GF(p). The crucial difference is that now the degree in each variable can be higher than one, and our proposed attack will differentiate several times with respect to each variable (unlike the classical cube attack and its larger field version described by Dinur and Shamir, both of which differentiate at most once with respect to each variable). Connections to the Moebius/Reed Muller Transform over GF(p) are also examined. Finally we describe differentiation over finite fields GF(ps) with ps elements and show that it can be reduced to differentiation over GF(p), so a cube attack over GF(ps) would be equivalent to cube attacks over GF(p)

Point compression for the trace zero subgroup over a small degree extension field

Using Semaev's summation polynomials, we derive a new equation for the $\mathbb{F}_q$-rational points of the trace zero variety of an elliptic curve defined over $\mathbb{F}_q$. Using this equation, we produce an optimal-size representation for such points. Our representation is compatible with scalar multiplication. We give a point compression algorithm to compute the representation and a decompression algorithm to recover the original point (up to some small ambiguity). The algorithms are efficient for trace zero varieties coming from small degree extension fields. We give explicit equations and discuss in detail the practically relevant cases of cubic and quintic field extensions.Comment: 23 pages, to appear in Designs, Codes and Cryptograph

Viscous vortex flows

Several computational studies are currently being pursued that focus on various aspects of representing the entire lifetime of the viscous trailing vortex wakes generated by an aircraft. The formulation and subsequent near-wing development of the leading-edge vortices formed by a delta wing are being calculated at modest Reynolds numbers using a three-dimensional, time-dependent Navier-Stokes code. Another computational code was developed to focus on the roll-up, trajectory, and mutual interaction of trailing vortices further downstream from the wing using a two-dimensional, time-dependent, Navier-Stokes algorithm. To investigate the effect of a cross-wind ground shear flow on the drift and decay of the far-field trailing vortices, a code was developed that employs Euler equations along with matched asymptotic solutions for the decaying vortex filaments. And finally, to simulate the conditions far down stream after the onset of the Crow instability in the vortex wake, a full three-dimensional, time-dependent Navier-Stokes code was developed to study the behavior of interacting vortex rings

An overlapped grid method for multigrid, finite volume/difference flow solvers: MaGGiE

The objective is to develop a domain decomposition method via overlapping/embedding the component grids, which is to be used by upwind, multi-grid, finite volume solution algorithms. A computer code, given the name MaGGiE (Multi-Geometry Grid Embedder) is developed to meet this objective. MaGGiE takes independently generated component grids as input, and automatically constructs the composite mesh and interpolation data, which can be used by the finite volume solution methods with or without multigrid convergence acceleration. Six demonstrative examples showing various aspects of the overlap technique are presented and discussed. These cases are used for developing the procedure for overlapping grids of different topologies, and to evaluate the grid connection and interpolation data for finite volume calculations on a composite mesh. Time fluxes are transferred between mesh interfaces using a trilinear interpolation procedure. Conservation losses are minimal at the interfaces using this method. The multi-grid solution algorithm, using the coaser grid connections, improves the convergence time history as compared to the solution on composite mesh without multi-gridding