  We solve an open question in code-based cryptography by introducing two
provably secure group signature schemes from code-based assumptions. Our basic
scheme satisfies the CPA-anonymity and traceability requirements in the random
oracle model, assuming the hardness of the McEliece problem, the Learning
Parity with Noise problem, and a variant of the Syndrome Decoding problem. The
construction produces smaller key and signature sizes than the previous group
signature schemes from lattices, as long as the cardinality of the underlying
group does not exceed $2^{24}$, which is roughly comparable to the current
population of the Netherlands. We develop the basic scheme further to achieve
the strongest anonymity notion, i.e., CCA-anonymity, with a small overhead in
terms of efficiency. The feasibility of two proposed schemes is supported by
implementation results. Our two schemes are the first in their respective
classes of provably secure groups signature schemes. Additionally, the
techniques introduced in this work might be of independent interest. These are
a new verifiable encryption protocol for the randomized McEliece encryption and
a novel approach to design formal security reductions from the Syndrome
Decoding problem.
