Computing functions of a shared secret

In this work we introduce and study threshold (t-out-of-n) secret sharing s chemes for families of functions F. Such schemes allow any set of at least t parties to compute privately the value f(s) of a (previously distributed) secret s, for any f is an element of F. Smaller sets of players get no mor e information about the secret than what follows from the value f(s). The g oal is to make the shares as short as possible. Results are obtained for tw o different settings: we study the case when the evaluation is done on a br oadcast channel without interaction, and we examine what can be gained by a llowing evaluations to be done interactively via private channels.