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 Feb 16 comment $(k, n)$ threshold secret sharing to $(k+k', n + n')$ without reissuing all shares Precisely. $(k,n)$ means that any subset of size $k$ can have the secret. Let's say $n\ge k + 1$. Then your method doesn't give a $(k+1, n+1)$ because if you take a subset of size $k+1$ consisting of only people who already have shares, you can't find the secret. You give more importance to new shares, somehow. Feb 16 comment $(k, n)$ threshold secret sharing to $(k+k', n + n')$ without reissuing all shares Let's say we call $Q$ the original degree $k-1$ polynomial so that the secret is $Q(0)$ and the shares are $Q(i)$ for $1\le i \le n$. Then we generate your random polynomial $P$. The new shares are $Q(i)+P(i)$ and $P(0)$. If I have $k$ values of $Q+P$ and the value of $P(0)$, I can indeed get $Q(0)$ (assuming that $P$ is of degree $\le k-1$). But what if I have $k+1$ values of $Q+P$? All I can compute is $Q(0)+P(0)$. To get the secret, you need to have the new share: $Q(0)$. Feb 16 comment $(k, n)$ threshold secret sharing to $(k+k', n + n')$ without reissuing all shares The problem is that decoding then takes 2 steps: Getting the old secret shifted by a constant and then unshifting by the constant. So that the person holding the constant is necessary, unless I'm mistaken. Feb 13 awarded Yearling Aug 12 awarded Commentator Aug 12 comment Is there an encryption that is only reversible with a key? There is the one time pad. Aug 10 revised $(k, n)$ threshold secret sharing to $(k+k', n + n')$ without reissuing all shares added 1105 characters in body Aug 10 comment $(k, n)$ threshold secret sharing to $(k+k', n + n')$ without reissuing all shares You second point about $x$ having to be private is really interesting (and really tricky). I never would've though about it (O_O) but I think the trustworthy group fixes that too. I'm not sure though. Anyway, I'm not going to make the $x$s public. But since I was thinking of only updating $y$ if a share was compromised, I'll probably have to rework my recovery protocol. I'll think about it and ask another question if I'm not sure :) Aug 10 comment $(k, n)$ threshold secret sharing to $(k+k', n + n')$ without reissuing all shares I edited and added a lot of detail. I think that the trustworthy group of shareholders that accept to delete their shares makes it possible. Aug 10 revised $(k, n)$ threshold secret sharing to $(k+k', n + n')$ without reissuing all shares added 2317 characters in body Aug 10 revised $(k, n)$ threshold secret sharing to $(k+k', n + n')$ without reissuing all shares added 2317 characters in body Aug 10 comment $(k, n)$ threshold secret sharing to $(k+k', n + n')$ without reissuing all shares The parties that I see often can be trusted to delete their own shares (and not compromising them). We can also, if necessary, assume that whenever a share is compromised, I know it. Aug 10 comment $(k, n)$ threshold secret sharing to $(k+k', n + n')$ without reissuing all shares Not really. But the adversary can only figure out who the people are and then try to find where they hid the key. Why? Aug 10 asked $(k, n)$ threshold secret sharing to $(k+k', n + n')$ without reissuing all shares Aug 9 comment Recover from compromised shares with Shamir Secret Sharing Since I use secret sharing to back up a key that I use daily, I totally missed the fact that you need the secret to issue new shares. Thanks for pointing that out :-) Aug 9 accepted Recover from compromised shares with Shamir Secret Sharing Aug 9 asked Recover from compromised shares with Shamir Secret Sharing Jun 11 accepted Encrypt HDD with keyfile and password and allow backing up the keyfile with secret sharing May 10 comment Encrypt HDD with keyfile and password and allow backing up the keyfile with secret sharing @e-sushi : The compiled default in Debian: aes-xts-plain64. May 10 comment Encrypt HDD with keyfile and password and allow backing up the keyfile with secret sharing @RickyDemer : No idea. That's what the tutorials I read did.