To prevent people from switching pseudonym parties after they have been assigned to them (which makes a certain attack possible), people commit short selfie-videos, just 15 seconds or so, and share a hash of the video with relayers, who perform “proof-of-commitment” and are rewarded for doing so.

The idea to use relayers instead of just uploading the hash on-state, is that all subjective data should be off-state, so that there is plausible deniability.

So that the pseudonym party participants can know that a certain selfie-video hash came from a specific person, a separate set of public-private keys are used, off-state keys, making it possible to have “subjective proofs”. These off-state keys are then used to sign data in ways that each participant can know that which on-state key the data originates from, without having “consensus proof”.

Using asymmetric cryptography to generate a shared symmetric key

Asymmetric cryptography, for example elliptic curve cryptography, can be used to generate a shared symmetric key, with what is often called a “Diffie-Hellman” exchange.

Wikipedia explains it,

"ECDH is used for the purposes of key agreement. Suppose two people, Alice and Bob, wish to exchange a secret key with each other. Alice will generate a private key dA and a public key QA=dAG (where G is the generator for the curve). Similarly Bob has his private key dB and a public key QB=dBG. If Bob sends his public key to Alice then she can calculate dAQB=dAdBG. Similarly if Alice sends her public key to Bob, then he can calculate dbQA=dAdBG. The shared secret is the x co-ordinate of the calculated point dAdBG. Any eavesdropper would only know QA and QB, and would be unable to calculate the shared secret. "

In online pseudonym parties, the on-state keys would be used to generate a symmetric key, in other words, an end-to-end encrypted channel where any data is also authenticated to the people who hold a specific on-state key. That channel would then be used to exchange the off-state public keys, which are then used to sign/encrypt all data in the pseudonym party.