Verifiable secret sharing

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In topology and in calculus, a round function is a scalar function M, over a manifold M, whose critical points form one or several connected components, each homeomorphic to the circle S1, also called critical loops. They are special cases of Morse-Bott functions.

The black circle in one of this critical loops.

For instance

For example, let M be the torus. Let

K=(0,2π)×(0,2π).

Then we know that a map

X:K3

given by

X(θ,ϕ)=((2+cosθ)cosϕ,(2+cosθ)sinϕ,sinθ)

is a parametrization for almost all of M. Now, via the projection π3:3 we get the restriction

G=π3|M:M,(θ,ϕ)sinθ

G=G(θ,ϕ)=sinθ is a function whose critical sets are determined by

G(θ,ϕ)=(Gθ,Gϕ)(θ,ϕ)=(0,0),

this is if and only if θ=π2,3π2.

These two values for θ give the critical sets

X(π/2,ϕ)=(2cosϕ,2sinϕ,1)
X(3π/2,ϕ)=(2cosϕ,2sinϕ,1)

which represent two extremal circles over the torus M.

Observe that the Hessian for this function is

Hess(G)=[sinθ000]

which clearly it reveals itself as of rankHess(G)=1 at the tagged circles, making the critical point degenerate, that is, showing that the critical points are not isolated.

Round complexity

Mimicking the L-S category theory one can define the round complexity asking for whether or not exist round functions on manifolds and/or for the minimum number of critical loops.

References

  • Siersma and Khimshiasvili, On minimal round functions, Preprint 1118, Department of Mathematics, Utrecht University, 1999, pp. 18.[1]. An update at [2]