Possible to split a secret to generate a re-encryption key (PRE)?

Question

I am wondering, if it is possible to split a secret (s.th. like shamir's secret sharing) which can be used to generate a re-encryption key in proxy re-encryption and only serve the splitted portions for the sake of generating a re-encryption key which can only be used for a specific public key?

So something like:

sec = secret to decrypt message
sec1 = portion of sec
sec2 = portion of sec

pkr = public key of receiver of the message
pkm = public key of bad actor who wants to re-encrypt for him
sec1_pkr = getKey(sec1, pkr)
sec2_pkr = getKey(sec2, pkr)

reKeyPkr = getReKey(sec1_pkr, sec2_pkr, [secn_pkr...], pkr) = re-encryption key for receiver pkr
reKeyPkm != getReKey(sec1_pkr, sec2_pkr, [secn_pkr...], pkm) = re-encryption key for receiver pkm
...

Answer 1

Something similar to what you ask, although not exactly, is the PRE scheme called Umbral. It has a reference implementation called pyUmbral. (Disclaimer: I'm the author of Umbral).

Umbral is a threshold PRE scheme, which means that re-encryption is divided among several proxies and at least certain threshold of them has to cooperate. With Umbral, the delegator can generate M-out-of-N re-encryption keys, in a way similar to Shamir's Secret Sharing.

I think you probably can reuse some of the techniques used in Umbral to split the delegator's private key and generate re-encryption key shares from the shares of this private key, rather than Umbral's approach which consists in generating directly re-encryption key shares. Anyway, both approaches should be very similar from the perspective of the application of Shamir's Secret Sharing.

I also think these techniques can be easily adapted to pairing-based PRE schemes, such as AFGH and others.