BRCE: Braid-Ring Convolution Encryption – A Post-Abelian Cryptosystem Without Periodicity
Résumé
The advent of quantum computing poses a serious threat to conventional cryptographic systems, particularly those reliant on the hardness of integer factorization and discrete logarithm problems-problems that are efficiently solvable by Shor's algorithm on a quantum computer. In light of this existential cryptographic vulnerability, we propose a robust alternative framework rooted in non-abelian algebraic structures, specifically braid groups. Unlike abelian groups such as Z n , which are susceptible to Fourierbased quantum attacks, braid groups possess intricate non-commutative properties that render quantum period-finding techniques ineffective.
This paper explores the cryptographic potential of braid groups through a rigorous examination of the Conjugacy Search Problem (CSP) and its computational hardness. We formulate a braid-based key exchange protocol and analyze its resilience against both classical and quantum adversaries. Central to our approach is the substitution of abelian periodicity with the combinatorial complexity inherent in the braid group's conjugacy classes, thereby obstructing Shor's algorithm from exploiting any underlying algebraic regularity.
Further, we present formal mathematical proofs substantiating the intractability of the CSP within braid groups under standard and worst-case assumptions. A suite of reductions to known NP-complete problems is also provided to reinforce our security claims. From an implementation standpoint, we develop and simulate the full protocol in .NET Core, including modules for braid word manipulation, normal form reduction, and conjugacy computation. We also introduce a Shor Emulator to demonstrate the empirical failure of quantum Fourier-based attacks on our cryptosystem.
By synthesizing group-theoretic hardness assumptions with practical cryptographic engineering, this work contributes to the post-quantum cryptography domain with a non-abelian framework that fundamentally subverts quantum decryption strategies. We conclude by outlining potential extensions to braid-based signature schemes, multi-party key agreement protocols, and zero-knowledge proofs, all designed to be natively quantum-resistant.
Mots clés
- Post-Quantum Security 2.2 Word Problem
- Braid Groups Conjugacy Search Problem Quantum Cryptography Shor's Algorithm Non-Abelian Groups Word Problem Algebraic Cryptanalysis Post-Quantum Security 2.2 Word Problem Normal Forms and Centralizer
- Braid Groups
- Conjugacy Search Problem
- Quantum Cryptography
- Shor's Algorithm
- Non-Abelian Groups
- Algebraic Cryptanalysis
- Normal Forms
- and Centralizer
- Word Problem
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