  In rank-metric cryptography, a vector from a finite dimensional linear space
over a finite field is viewed as the linear space spanned by its entries. The
rank decoding problem which is the analogue of the problem of decoding a random
linear code consists in recovering a basis of a random noise vector that was
used to perturb a set of random linear equations sharing a secret solution.
Assuming the intractability of this problem, we introduce a new construction of
injective one-way trapdoor functions. Our solution departs from the frequent
way of building public key primitives from error-correcting codes where, to
establish the security, ad hoc assumptions about a hidden structure are made.
Our method produces a hard-to-distinguish linear code together with low weight
vectors which constitute the secret that helps recover the inputs.The key idea
is to focus on trapdoor functions that take sufficiently enough input vectors
sharing the same support. Applying then the error correcting algorithm designed
for Low Rank Parity Check (LRPC) codes, we obtain an inverting algorithm that
recovers the inputs with overwhelming probability.
