In this paper we propose a new approach to code-based signatures that
makes use in particular of rank metric codes. When the classical approach consists in finding the unique
preimage of a syndrome through a decoding algorithm, we propose to introduce the notion
of mixed decoding of erasures and errors for building signature schemes.
In that case the difficult problem becomes, as is the case in lattice-based cryptography,
finding a preimage of weight above the Gilbert-Varshamov bound (case where
many solutions occur) rather than finding a unique preimage of weight below
the Gilbert-Varshamov bound. The paper describes RankSign: a
new signature algorithm for the rank metric
based on a new mixed algorithm for decoding erasures and errors for
the recently introduced Low Rank Parity Check (LRPC) codes.
We explain how it is possible (depending on choices
of parameters) to obtain a full decoding algorithm which is able
to find a preimage of reasonable rank weight for any random syndrome
with a very strong probability. We study the semantic security
of our signature algorithm and show how it is possible to reduce
the unforgeability to direct attacks on the public matrix, so that
no information leaks through signatures. Finally, we give several examples of parameters
for our scheme, some of which with public key of size $5760$ bits and signature of size $1728$ bits.
Moreover the scheme can be very fast for small base fields.