# Linear {VSS} and Distributed Commitments Based on Secret Sharing and Pairwise Checks

## Serge Fehr and Ueli Maurer

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```We present a general treatment of all non-cryptographic (i.e.,
information-theoretically secure) linear verifiable-secret-sharing
(VSS) and distributed-commitment (DC) schemes, based on an underlying
secret sharing scheme, pairwise checks between players, complaints, and
accusations of the dealer. VSS and DC are main building blocks for
unconditional secure multi-party computation protocols. This general
approach covers all known linear VSS and DC schemes. The main theorem
states that the security of a scheme is equivalent to a pure
linear-algebra condition on the linear mappings (e.g. described as
matrices and vectors) describing the scheme. The security of all known
schemes follows as corollaries whose proofs are pure linear-algebra
arguments, in contrast to some hybrid arguments used in the
literature. Our approach is demonstrated for the CDM DC scheme, which
we generalize to be secure against mixed adversary settings (some
curious and some dishonest players), and for the classical BGW VSS
scheme, for which we show that some of the checks between players are
superfluous, i.e., the scheme is not optimal. More generally, our
approach, establishing the minimal conditions for security (and hence
the common denominator of the known schemes), can lead to the design of
more efficient VSS and DC schemes for general adversary structures.