No abstract uploaded!

In this paper, we first prove an explicit formula which bounds the degree of regularity of the family of HFEv (“HFE with vinegar”) and HFEv- (“HFE with vinegar and minus”) multivariate public key cryptosystems over a finite field of size q. The degree of regularity of the polynomial system derived from an HFEv- system is less than or equal to \frac{(q−1)(r+v+a−1)}{2}+2 if q is even and r+a is odd, \frac{(q−1)(r+v+a−1)}{2}+2 otherwise, where the parameters v, D, q, and a are parameters of the cryptosystem denoting respectively the number of vinegar variables, the degree of the HFE polynomial, the base field size, and the number of removed equations, and r is the “rank” paramter which in the general case is determined by D and q as r=⌊log_q (D−1)⌋+1. In particular, setting a = 0 gives us the case of HFEv where the degree of regularity is bound by \frac{(q−1)(r+v−1)}{2}+2 if q is even and r is odd, \frac{(q−1)(r+v)}{2}+2 otherwise. This formula provides the first solid theoretical estimate of the complexity of algebraic cryptanalysis of the HFEv- signature scheme, and as a corollary bounds on the complexity of a direct attack against the QUARTZ digital signature scheme. Based on some experimental evidence, we evaluate the complexity of solving QUARTZ directly using F_4/F_5 or similar Gröbner methods to be around 2^{92}.

It is well known that del Pezzo surfaces of degree 9 − n one-to-one correspond to flat En bundles over an elliptic curve. In this paper, we construct ADE-bundles over a broader class of rational surfaces that we call ADE-surfaces, and extend the above correspondence to all flat G-bundles over an elliptic curve, where G is any simply laced, simple, compact and simply connected Lie group. In what follows, we will construct G-bundles for a non-simply laced Lie group G over these rational surfaces, and extend the above correspondence to non-simply laced cases.

We study the stability properties of explicit marching schemes for second-kind Volterra integral equations that arise when solving boundary value problems for the heat equation by means of potential theory. It is well known that explicit finite-difference or finite-element schemes for the heat equation are stable only if the time step 􏰃t is of the order O(􏰃x2), where 􏰃x is the finest spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all dimensions d ≥ 1, we show that the simplest Volterra marching scheme, i.e., the forward Euler scheme, is unconditionally stable. Our proof is based on an explicit spectral radius bound of the marching matrix, leading to an estimate that an L2-norm of the solution to the integral equation is bounded by cd T d /2 times the norm of the right-hand side. For the Robin problem on the half-space in any dimension, with constant Robin (heat transfer) coefficient κ, we exhibit a constant C such that the forward Euler scheme is stable if 􏰃t < C /κ 2 , independent of any spatial discretization. This relies on new lower bounds on the spectrum of real symmetric Toeplitz matrices defined by convex sequences. Finally, we show that the forward Euler scheme is unconditionally stable for the Dirichlet problem on any smooth convex domain in any dimension, in the L∞-norm.

No abstract uploaded!