Invariant-based Cryptosystems and Their Security Against Provable Worst-Case Break?

Cryptography based on noncommutative algebra still suffers from lack of schemes and lack of interest. In this work, we show new constructions of cryptosystems based on group invariants and suggest methods to make such cryptosystems secure in practice. Cryptographers still cannot prove security in its cryptographic sense or even reduce it to some statement about regular complexity classes. In this paper we introduce a new notion of cryptographic security, a provable break, and prove that cryptosystems based on matrix group invariants and also a variation of the Anshel-Anshel-Goldfeld key agreement protocol for modular groups are secure against provable worst-case break unless NP ⊆ RP.