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4

Shamir's $(t,n)$ secret sharing scheme involves picking a random polynomial $p$ (over a finite field) of degree $t-1$, such that $p(0) = s$ is the secret value to be shared (this is easy to do, since $p(0)$ is just the constant term of the polynomial), and then evaluating the polynomial at $n$ distinct non-zero points $x_1, \dotsc, x_n$ to construct $n$ ...

3

This is because $t$ shares uniquely defines the polynomial of degree $t-1$. $t-1$ shares still leaves $k$ possible and equally likely polynomials, for $k$ the size of the field, so the secret is information theoretically hidden. Think of a degree 1 polynomial, essentially a line. If you know just one point on the line, you cannot say anything about the ...

5

Shamir Secret Sharing (SSS) is based on constructing a polynomial of degree $t-1$, whose independent term is the secret $S$. Each share is actually a point of the polynomial. The security of SSS is based on the fact that, when one wants to interpolate a polynomial of degree $t-1$, one needs at least $t$ points of the polynomial. It can be seen graphically ...

4

Take a linear polynomial: $y=mx+b$. If I tell you that the point $(1,5)$ is on the line, can you tell me $m$ and $b$? No, because in fact there are infinitely many lines that pass through the point $(1,5)$. It takes 2 points to uniquely identify a line. In general it takes $t$ points to uniquely identify a degree $t-1$ polynomial. Furthermore, given $t-1$ ...

5

You could view it as a form of One-Time-Pad encryption. In this view one of the two "messages" is the key and the other is the ciphertext (it does not matter which you think of as which as their roles are symmetric). It may, however, be more natural to think of as a simple secret sharing scheme. Roughly speaking a secret sharing scheme is a scheme where a ...

1

Since Bob already knows $K_{AB}$ he can run $D_{K_{AB}}(E_{K_{AB}}(E_{K_{AC}}(K)))$. Which will leave him with $E_{K_{AC}}(K)$. And to decrypt that any further (in order to get $K$) he would need either Alice or Charlie's help.

4

Is the Kurihara algorithm really what it purports to be (dramatically faster but equally secure replacement for Shamir Secret Sharing)? The algorithm being referred to is in this paper, and I believe that the speed benefits are at best marginal, if not nonexistent. As for the speed benefits being marginal, well, normally we use secret sharing as a part ...

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