In this paper we investigate the operator H=??(??) in L2 (R2), where> 0 and? is a closed C4 Jordan curve in R2. We obtain the asymptotic form of each eigenvalue of H as tends to in nity. We also get the asymptotic form of the number of negative eigenvalues of H in the strong coupling asymptotic regime. MSC: 35J10, 35P15

We show that the Masur–Veech volumes and area Siegel–Veech constants can be obtained using intersection theory on strata of Abelian differ- entials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel–Veech constants for all strata. We also show that the same results hold for the spin and hyperelliptic components of the strata.

Sflash had recently been broken by Dubois, Stern, Shamir, etc., using a differential attack on the public key. The C^{*-} signature schemes are hence no longer practical. In this paper, we will study the new attack from the point view of symmetry, then (1) present a simple concept (projection) to modify several multivariate schemes to resist the new attacks; (2) demonstrate with practical examples that this simple method could work well; and (3) show that the same discussion of attack-and-defence applies to other big-field multivariates. The speed of encryption schemes is not affected, and we can still have a big-field multivariate signatures resisting the new differential attacks with speeds comparable to Sflash.

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