This paper presents a time-area efficient hardware architecture for the multivariate signature scheme Rainbow. As a part of this architecture, a high-performance hardware optimized variant of the well-known Gaussian elimination over GF(2^l) and its efficient implementation are presented. The resulting signature generation core of Rainbow requires 63,593 gate equivalents and signs a message in just 804 clock cycles at 67 MHz using AMI 0.35μm CMOS technology. Thus, Rainbow provides significant performance improvements compared to RSA and ECDSA.

We study the asymptotic behaviour of$N$(α)—the number of negative eigenvalues of the operator (-τ)^{$l$}-α$V$in$L$_{2}($R$^{$d$}) for an even$d$and$2l≥d$. This is the only case where the previously known results were far from being complete. In order to describe our results we introduce an auxiliary ordinary differential operator (system) on the semiaxis. Depending on the spectral properties of this operator we can distinguish between three cases where$N$(α) is of the Weyl-type,$N$(α) is of the Weyl-order but not the Weyl-type coefficient and finally where$N$(α)=$O$(α^{q}) with$q>d/2l$.

In this note, we characterize maximal invariant subspaces for a class of operators. Let$T$be a Fredholm operator and $1-TT^{*}\in\mathcal{S}_{p}$ for some$p$≥1. It is shown that if$M$is an invariant subspace for$T$such that dim$M$$⊖$$TM$<∞, then every maximal invariant subspace of$M$is of codimension 1 in$M$. As an immediate consequence, we obtain that if$M$is a shift invariant subspace of the Bergman space and dim$M$$⊖$$zM$<∞, then every maximal invariant subspace of$M$is of codimension 1 in$M$. We also apply the result to translation operators and their invariant subspaces.

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