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TL;DR

How can the shares of the PRZS protocol proposed in [CDI05] reconstruct the "secret" $0$?

Is the number of shares required to reconstruct $0$ equal to $2t + 1$?


Long question

The paper "Share conversion, pseudorandom secret-sharing and applications to secure computation." by Cramer, Damgård, and Ishai shows a protocol for pseudorandom zero-sharing that enables every player $P_j$ to locally compute his share $s_j$ of the polynomial $f_0$ of degree $2t$ from initially distributed keys $r_A^i$, where $t = k - 1$ in a $(k, n)$ secret sharing scheme.

$s_j$ is computed as follows: $$s_j = \sum_{A \subseteq [n] : |A| = n - t, j \in A} \sum_{i = 1}^{t} \psi_{r_A^i}(a) \cdot f_A^i(j)$$

The authors say that: "it is straightforward to verify that this results in shares consistent with polynomial $f_0$" and that: "$\mathit{deg}(f_0) \leq 2t$ and $f_0(0) = 0$".

If I've correctly understood, if we make $f_A^i(j) = 1, j \in A$ and we consider $\psi$ as an HMAC function, then every player $P_j$ simply sums the HMAC of $a$ with all the keys $r_A$ he has received from all the sets $A$ he belongs to.

How can this result in shares (polynomial $f_0$) that can reconstruct the secret 0? It still remains quite obscure to me.

Furthermore, since $\mathit{deg}(f_0) \leq 2t$, does this imply that to reconstruct $0$ we need $2t + 1$ shares at most?

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To your question:

  1. About the HMAC example, maybe the hypothesis that $f^i_A(j)$ is not feasible. Because $f_A(x)$ servers as a role somelike base polynomial while $\psi$ servers as a random scalors of these base polynomials.

$f_A(x)$ is constructed in this way: first, we know $t$ roots of $f_A(x)$, then we assign $f_A(0)=1$. Therefore, $f_A(x)$ is a $t$ degree polynomial whose constant term is 1. Besides, other points like $(j, f_A(j))$ are computed by interpolating. We cannot simply treat $f_A(j)$ as 1.

  1. About reconstructing secret 0. Now that we know the secret is 0, why we need to concern about how to reconstruct the secret. I think maybe the real secret, things we want to hide from is coefficients of non-constant terms in $f_A(x)$, because we usually use PRZS protocol to randomize an existing polynomial whose constant term is a secret.

BTW, there is a simpler version of PRZS:

\begin{equation*} f(X) = \sum_{A\subset X\text{, }|A|=n-t}\sum_{j=1}^t \psi_{r_A}(a, j) \cdot X^j \cdot f_A(X) \end{equation*}

Three observations here:

  1. $f_A(X)$ is a polynomial whose degree is at most $t$ with random constant term.

  2. $X^t \cdot f_A(X)$ is a polynomial whose degree is at most $2t$.

  3. $f(X)$ is divisible by $X$ which means its constant term is 0. Beacause $X^1 \cdot f_A(X)$ contributes $f(X)$'s lowest degree.

Therefore, we come to the conclusion that $f(X)$ is a polynomial that $deg(f) \leq 2t$, and $f(0) = 0$.

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