Since this question hast just been updated I am wondering what the benefits of this scheme are. The only motivation I could find with a quick search are speed and power efficiency.
Am I missing anything?
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Beside speed and power efficiency, there's also claimed resistance to attacks by quantum computers; quoting Iris Anshel, Derek Atkins, Dorian Goldfeld, Paul E. Gunnells: Post Quantum Group Theoretic Cryptography (2016-12-05), at SecureRF website:
(..) unlike the classic protocols, these group theoretic protocols are not based on finite cyclic abelian groups. Instead, the underlying algebraic object is the braid group, an infinite non-abelian group. The security of these group theoretic protocols is not based on any problem known to be susceptible to a quantum attack, which makes them viable candidates for post-quantum asymmetric cryptography. In addition, since the data transmitted over public channels is contained in the Braid group, linear algebra/permutation based attack are not applicable to the protocols (..)
Also, there is novelty, which allows a patent, which serves some interests; obviously those of the patent holder, but occasionally those of the customer: when one purchases a door cylinder or lock with a patented physical key feature (which abound), one might gain some small level of protection against unauthorized key reproduction by someone who temporarily gets hold of the key, and better insurance that extra keys ordered from a shop will actually work, if the patent holder actively uses it as a legal leverage to enforce a key reproduction policy including proving possession of the key reproduction card sold with the cylinder/lock, and quality control. Similarly, the holder of a patent for a cryptographic technique can try to use the patent as leverage to force implementations to pass some security and/or interoperability tests, both of which potentially beneficial to customers.
Note: I'm making no statement about the claims of resistance to attacks by quantum computers, the validity of the patents, or their licensing policy.