  Post-Quantum Cryptography (PQC) attempts to find cryptographic protocols
resistant to attacks by means of for instance Shor's polynomial time algorithm
for numerical field problems like integer factorization (IFP) or the discrete
logarithm (DLP). Other aspects are the backdoors discovered in deterministic
random generators or recent advances in solving some instances of DLP. The use
of alternative algebraic structures like non-commutative or non-associative
partial groupoids, magmas, monoids, semigroups, quasigroups or groups, are
valid choices for these new kinds of protocols. In this paper, we focus in an
asymmetric cipher based on a generalized ElGamal non-arbitrated protocol using
a non-commutative general linear group. The developed protocol forces a hard
subgroup membership search problem into a non-commutative structure. The
protocol involves at first a generalized Diffie-Hellman key interchange and
further on the private and public parameters are recursively updated each time
a new cipher session is launched. Security is based on a hard variation of the
Generalized Symmetric Decomposition Problem (GSDP). Working with GF(251^8) a
64-bits security is achieved, and if GF(251^16) is chosen, the security rises
to 127-bits. An appealing feature is that there is no need for big number
libraries as all arithmetic if performed in Z_251 and therefore the new
protocol is particularly useful for computational platforms with very limited
capabilities like smartphones or smartcards.
