Revisiting the Karnin, Greene and Hellman Bounds

The algebraic setting for threshold secret sharing scheme can vary, dependent on the application. This algebraic setting can limit the number of participants of an ideal secret sharing scheme. Thus it is important to know for which thresholds one could utilize an ideal threshold sharing scheme and for which thresholds one would have to use nonideal schemes. The implication is that more than one share may have to be dealt to some or all parties. Karnin, Greene and Hellman constructed several bounds concerning the maximal number of participants in threshold sharing scheme. There has been a number of researchers who have noted the relationship between k-arcs in projective spaces and ideal linear threshold secret schemes, as well as between MDS codes and ideal linear threshold secret sharing schemes. Further, researchers have constructed optimal bounds concerning the size of k-arcs in projective spaces, MDS codes, etc. for various finite fields. Unfortunately, the application of these results on the Karnin, Greene and Hellamn bounds has not been widely disseminated. Our contribution in this paper is revisiting and updating the Karnin, Greene, and Hellman bounds, providing optimal bounds on the number of participants in ideal linear threshold secret sharing schemes for various finite fields, and constructing these bounds using the same tools that Karnin, Greene, and Hellman introduced in their seminal paper. We provide optimal bounds for the maximal number of players for a t out of n ideal linear threshold scheme when t = 3, for all possible finite fields. We also provide bounds for infinitely many t and infinitely many fields and a unifying relationship between this problem and the MDS (maximum distance separable) codes that shows that any improvement on bounds for ideal linear threshold secret sharing scheme will impact bounds on MDS codes, for which there is a number of conjectured (but open) problems.