PQC: Triple Decomposition Problem Applied To GL(d, Fp) - A Secure Framework For Canonical Non-Commutative Cryptography

TL;DR

This paper introduces a new algebraic cryptographic framework based on triple decomposition and conjugacy problems within the general linear group, aiming for security against classical and quantum attacks across various cryptographic protocols.

Contribution

It develops a novel non-commutative cryptographic framework using triple decomposition and conjugacy problems in GL(d,Fp), enhancing security against quantum and classical threats.

Findings

Framework is secure against classical attacks like linear algebra and side-channel attacks.

The proposed cryptosystem is resistant to quantum algorithms Grover and Shor with proper parameters.

Supports multiple cryptographic protocols including key exchange, encryption, and digital signatures.

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 algebraic framework who could be applied to different asymmetric protocols like D-H KE (Diffie-Hellman key exchange), Public Key Encryption, Digital Signature, ZKP (zero-knowledge proof) authentication, Oblivious Transfer, Multi-Party Computing, and so on. The trapdoor one-way functions selected are (a) Triple decomposition Problem (TDP) developed by Kurt, where a known element is factored into a product of three unknown factors and (b) a new version of conjugacy search that we refer from now on as Blind Conjugacy Search Problem (BCSP). Our platform structure is the general linear group GL(d,F_p) d-square non-singular matrices of prime field values. We give support to the fact that this framework is cryptographically secure against classical attacks like linear algebra attacks, length-based attacks, side-channel attacks against square (or duplicate) and multiply (or sum) algorithm, high sensitivity to pseudo random deterministic generators, etc. At same time it is immune against quantum attacks (using Grover and Shor), if the size parameters are carefully selected. Semantic security and IND-CCA2 compliance for this framework is discussed.