Once this subset of valid tokens is determined, Lagrange interpolation formula
is executed to test each remaining token one at a time to identify whether it is an invalid token or not.
When the number of the signature shares got by any one participant or signature receiver reaches 2t + 1, he can compute the signature s according to the interpolation formula
. Thus the signature (r, s) of the message m is got.
In [21, Section 1.5], Phillips has studied two new interpolation formulas
3 Secret Reconstruction Algorithm: In the secret reconstruction algorithm, any authorized subset of n participants with order has 4, S can be obtained using Lagrange's interpolation formula
as in equations (2, 3) to reconstruct the polynomial f (x), and then find f (16) = S.
Using the Lagrange interpolation formula
through five points, [V.sup.I.sub.[tau]i] and [V.sup.I.sub.ni] are calculated.
These formulas are independent of the true value of the spectrum, and then the interpolation formulas
are obtained by Fourier series developments.
Computational procedures for the two types of data are generally different because interpolation formulas
are often necessary when dealing with frequency distributions.
Yet another method for completing step 4 is to use Lagrange's interpolation formula
As an application of the previous results, in Section 3, we obtain Hermite interpolation formulas
for nodal systems on [-1,1].
Meanwhile, with the knowledge of any t master shares with respect to the threshold structure, matrix S* can be recovered by Lagrange interpolation formula
(denoted by [F.sub.X](x)).
Stirling's interpolation formula
is based on a finite number of evaluations of the function and does not require derivatives, with the first-order approximation yielding
When the conditions [C.sub.1], [C.sub.2], [C.sub.3], [C.sub.4] are verified, the following interpolation formula