Algebraic cryptography: new constructions and their security against provable break

Very few known cryptographic primitives are based on noncommutative algebra. Each new scheme is of substantial interest, because noncommutative constructions are secure against many standard cryptographic attacks. On the other hand, cryptography does not provide security proofs that might allow the security of a cryptographic primitive to rely upon structural complexity assumptions. Thus, it is important to investigate weaker notions of security.
In this paper, new constructions of cryptographic primitives based on group invariants are proposed, together with new ways to strengthen them for practical use. Also, the notion of a provable break is introduced, which is a weaker version of the regular cryptographic break. In this new version, an adversary should have a proof that he has correctly decyphered the message. It is proved that the cryptosystems based on matrix group invariants and a version of the Anshel–Anshel–Goldfeld key agreement protocol for modular groups are secure against provable break unless $\mathrm{NP}=\mathrm{RP}$.