  We introduce an information-theoretic framework, named Coded State Machine
(CSM), to securely and efficiently execute multiple state machines on untrusted
network nodes, some of which are Byzantine. The standard method of solving this
problem is using State Machine Replication, which achieves high security at the
cost of low efficiency. We propose CSM, which achieves the optimal linear
scaling in storage efficiency, throughput, and security simultaneously with the
size of the network. The storage efficiency is scaled via the design of
Lagrange coded states and coded input commands that require the same storage
size as their origins. The computational efficiency is scaled using a novel
delegation algorithm, called INTERMIX, which is an information-theoretically
verifiable matrix-vector multiplication algorithm of independent interest.
Using INTERMIX, the network nodes securely delegate their coding operations to
a single worker node, and a small group of randomly selected auditor nodes
verify its correctness, so that computational efficiency can scale almost
linearly with the network size, without compromising on security.
