On proper secrets, \((t,k)\)-bases and linear codes (Q735016)

The authors introduce the concept of \textit{proper secret} in Shamir's \((k, n)\) threshold scheme. In this scheme the secret and the shares are elements of \(\mathbb{F}_q ,\) where \(q>n.\) They prove, that all values of the secret generating polynomial are proper secrets, but not all linear combinations of coefficients of this polynomial serve as a secret. They also study \((t,k)\) spanning sets, independent sets, bases (which are generalizations of the usual concepts). and the relations between them. It is shown that \((t,k)\)-bases over \(\mathbb{F}_q\) may have different sizes, lower and upper bounds are derived for these sizes. Finally the notions, introduced here are compared with some well-known projective geometric objects.