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It appears, braid groups are indeed a good bet, and you can do Diffie-Hellman on them, see Braid Group Cryptography.


Let's assume that everyone agreed on some elliptic curve and a public base point $g$ somewhere on the curve. When two parties Alice and Bob want to agree on a shared secret, they proceed as follows: Alice chooses some random number $a$ and applies the curve operation to $g$, the public base point, $a$ times. She obtains some result ...

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