BRCE: Braid-Ring Convolution Encryption – A Post-Abelian Cryptosystem Without Periodicity

We propose BRCE (Braid-Ring Convolutional Encryption), a novel quantum-resilient group ring-based encryption scheme built over the non-abelian structure of braid groups and their integral group rings. The encryption mechanism applies convolution-based masking with randomized braid elements to achieve semantic obfuscation without relying on periodic structures. Our design embeds high-entropy messages into non-commutative algebraic carriers , resulting in ciphertexts that resist both classical and quantum algebraic attacks. We provide a provable security reduction from the Conjugacy Search Problem (CSP) in braid groups and show that the scheme satisfies IND-CPA and IND-CCA security notions under reasonable pseudorandomness assumptions. We anchor security in a new hardness assump-tion—the Group Ring Convolution Inversion Problem (GRCIP)—tailored to non-abelian group rings. Further, we demonstrate that the structure resists reductions to the Hidden Subgroup Problem (HSP), positioning BRCE as a competitive post-quantum candidate rooted in topological and algebraic hardness. MSC 2020: 94A60, 20F36, 68Q12, 11T71