Here's an idea for a knot-based Diffie–Hellman exchange:

• Public: random (oriented) knot $P$.
• Private: random (oriented) knots $A$ and $B$.
• Exchange: Alice sends (randomized or canonical representation of) $A'=A\oplus P$, Bob sends (randomized or canonical representation of) $B'=P\oplus B$. Here $\oplus$ is knot connected sum.
• Shared key: (invariant of) $A\oplus P\oplus B=A\oplus B'=A'\oplus B$.

Questions:

• Why is this a good/bad idea? References?
• What is the complexity of solving $P\oplus X=X'$ for $X$ given $P$ and $X'$? Is factoring knots difficult?
• What are good digital representations of knots? How to efficiently generate and randomize knot representations? (E.g., Gauss codes and Reidemeister moves?) "Diffusing" the sums $A'$, $B'$ (randomly or to some canonical form) would be important for security, assuming factoring is hard in the first place.
• What is a good choice of invariant in the last step? Should be efficient to compute for Alice and Bob but infeasible knowing $P$, $A'$, and $B'$.
• If this makes sense, can it be done in reasonable space/time (e.g. $n\log n$ diagram storage for $n$ crossings, obfuscation/diffusion, final invariant computation)?

TL;DR: Could the knot monoid be useful for public key cryptography? Is connected sum of knots one-way, even knowing one of the factors? Do random planar projections obfuscate knot factorizations?

EDIT: To make this question more pointed, given a "random" planar projection of a "random" knot with $n$ crossings, I would like to know:

• First, what are good notions of random above?
• What is the distribution of factors of such a random knot, e.g. is it mostly filled with trefoils and other small knots?
• What computational methods are used to factor knots?
• What are some bounds for space/time complexity of such methods (as a function of $n$)?
• What is the "average" complexity (at least intuitively or in practice)?
• For the envisioned cryptographic purposes, $n$ on the order of 1000 or 10000 might be of reasonable size. Where does this stand in regards to current computational capabilities?