PQC: Extended Triple Decomposition Problem (XTDP) Applied To GL(d, Fp)-An Evolved Framework For Canonical Non-Commutative Cryptography

TL;DR

This paper introduces an evolved algebraic framework using Extended Triple Decomposition Problem (XTDP) for non-commutative cryptography, enhancing security against algebraic attacks in post-quantum cryptographic protocols.

Contribution

It develops an extended version of the Triple Decomposition Problem that transforms linear equations into quadratic ones, strengthening cryptographic security in non-commutative algebraic protocols.

Findings

Transforming linear equations into quadratic ones increases cryptographic robustness.

The XTDP framework can be applied to various asymmetric cryptographic protocols.

Enhanced security against algebraic span attacks in non-commutative cryptography.

Abstract

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.