Fast Lagrange--Newton transformations

Authors:
Ryszard Smarzewski;Joanna Kapusta
Affiliations:
Department of Mathematics and Computer Sciences, The John Paul II Catholic University of Lublin, ul. Konstantynow 1H, 20708 Lublin, Poland;Department of Mathematics and Computer Sciences, The John Paul II Catholic University of Lublin, ul. Konstantynow 1H, 20708 Lublin, Poland
Venue:
Journal of Complexity
Year:
2007

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Abstract

In this paper we present explicit vector formulae for the Lagrange-Newton transformation L:K^n-K^n and its inverse L^-^1 with respect to interpolating knots x"i=@ax"i"-"1+@bi=1,2,...,n-1;x"0=@c, where @a0,@b,@c belong to a field K. These formulae depend on the wrapped convolution, Horner transformation, iterative product and coordinatewise vector operations. All these transformations and operations, except of Onlogn-wrapped convolution, have running time of On base operations from the field K. Moreover, we give an application of these fast interpolating transformations to threshold secret sharing schemes in cryptography.