Is there a way to use Shamir Secret Sharing with updatable data? I want to divide a system that maintains these properties, based on Shamir's Secret Sharing:


A secret key is split up to N pieces, where T of them are enough to reconstruct the key.
The original key is then destroyed and doesn't exist anywhere except the N separate pieces.
The information unlocked by the T signatures is updatable - I want to retain some other information that will allow me to update the secret that is accessible by the secret holders.


The use case is a will that contains secret information. I have N people I trust, and I want any group of T of them to be able to read the secret information (hopefully, only after I'm dead). I want to be able to update this secret information from time to time, without the bother of recreating and redistributing another set of keys.

Is there something that answers these requirements?
I think there's a better way to do this, and I'm not sure the existing answers check all your boxes. I suggest using secret splitting together with asymmetric keys so that:


Nobody but you can write data.
Shareholders can come together to read data.
Each shareholder can individually verify, but not read data.
You can read and write data at any time.


Here's how I'd do it:


Create asymmetric (e.g. RSA) key pair "A".
Split the private half of A into $s$ shares, where $s=nx+t$.
Keep $t$ shares for yourself.
Create asymmetric key pair "B".
Give $x$ shares in A and the public half of B to each trusted person.
Secure the public half of A and the private half of B and keep them to yourself.
Create your data, encrypt it using the public half of A, then sign it using the private half of B. Distribute the signed, encrypted data.


Each shareholder can use the public half of B to verify the encrypted data; they may do this alone and at any time. Shareholders may combine their shares to recover the private half of A, which can then decrypt the data.

You, may at any time, use your shares to recover the private half of A and read the data. Also at any time, and as often as you wish, you may create new and/or additional data, encrypt and sign it, and distribute to your trusted people.