  Post-Quantum Cryptography (PQC) attempts to find cryptographic protocols
resistant to attacks using Shor polynomial time algorithm for numerical field
problems or Grover search algorithm. A mostly overlooked but valuable line of
solutions is provided by non-commutative algebraic structures, specifically
canonical protocols that rely on one-way trapdoor functions (OWTF). Here we
develop an evolved algebraic framework who could be applied to different
asymmetric protocols. The (canonic) trapdoor one-way function here selected is
a fortified version of the Triple decomposition Problem (TDP) developed by
Kurt. The original protocol relies on two linear and one quadratic algebraic
public equation. As quadratic equations are much more difficult to
cryptanalyze, an Algebraic Span Attack (ASA) developed by Boaz Tsaban, focus on
the linear ones. This seems to break our previous work. As a countermeasure, we
present here an Extended TDP (cited as XTDP in this work). The main point is
that the original public linear equations are transformed into quadratic ones
and the same is accomplished for exchanged tokens between the entities. All
details not presented here could be found at the cited references.
