We continue our series on polynomial secret sharing. In the previous post of this series we discussed secret sharing with a passive adversary. In this post we assume crash failures and in later posts we will extend to malicious failures. As before, we must assume parties have private channels: the adversary cannot see the content of messages sent between two non-faulty parties.

### What is a Secret Sharing scheme?

An secret sharing scheme is composed of two protocols: Share and Reconstruct. These protocols are run by the $n$ parties. The dealer has a secret $s$ in a commonly known finite field $\mathbb{F}_p$ with $p>n$, which is given as input to the Share protocol. The two properties are:

1. Validity: If the share protocol completes, the output of the Reconstruct protocol is the dealer’s input value $s$.
2. Hiding: If no honest party has begun the Reconstruct protocol, then the adversary can gain no information about $s$.

In later posts we will address the malicious dealer case which will introduce the third property: Binding.

We also need termination properties:

1. Weak Termination of Share: if the dealer is non-faulty then all non-faulty parties complete the Share protocol. In some cases a stronger property is needed: Strong Termination of Share: if non-faulty completes the Share protocol then all non-faulty parties complete the Share protocol.
2. Termination of Reconstruct: if all non-faulty parties complete the share protocol then all non-faulty parties complete the Reconstruct protocol.

See the last section of this post for a discussion on the strong termination property.

### The main idea

TL;DR: For an adversary controlling $f$ parties, use a degree $f$ polynomial and $n>2f$.

Parties are enumerated $N={1,2,3,\dots,n}$. Given input $s \in \mathbb{F}_p$, the dealer chooses a uniformly random polynomial $p(x)$, conditioned on $p(0)=s$. The dealer then gives party $i$ the value $p(i)$ which we call its share.

The adversary controls $f$ parties, hence gets to see $f$ shares and can also crash these parties. So what degree should $p(x)$ be? and how many parties do we need to tolerate $f$ crash failures?

1. For Hiding to hold, we need to share using a polynomial of degree $\geq f$ (otherwise the adversary can learn the secret during the Share protocol simply by interpolating its shares).
2. Since at least $f+1$ shares are required for unique reconstruction (for a polynomial of degree $\geq f$) and $f$ parties may crash, then we need $n \geq 2f+1$ for Validity.

Hence we choose $p(x)$ from all polynomials of degree at most $f$ and choose $n=2f+1$.

### The Secret Sharing protocol

Share protocol: Given a secret $s$ as input, the dealer randomly chooses $f$ values $p_1,\dots,p_{f} \in_R \mathbb{F}_p$ and defines a degree $\le f$ polynomial:

$p(X)=s+p_1 X + \dots + p_f X^f$

The dealer then sends $p(i)$ to party $i$, for each $i \in N$.

Reconstruct protocol: Each party $i$ sends its share $p(i)$ to all other parties. Each party receives at least $n-f$ shares. Let $I$ be the subset of the first $f+1$ parties whose shares are received. Note that at most $f$ parties may crash, so at least $n-f \geq f+1$ shares will arrive (here we use $n>2f$).

Each party outputs the dealer’s original secret $s$ as follows:

$s=p(0)=\sum_{i \in I} \lambda_i p(i)$

Here, $p(i)$ is the share sent by party $i \in I$ and $\lambda_i$ are the Lagrange interpolation coefficients:

$\lambda_i = \frac{\prod_{j \in I \setminus \{i\}} j}{\prod_{j \in I \setminus \{i\}} (j-i)}$

### Proof of Validity, Hiding, and Weak Termination

Validity follows from the one-to-one mapping of degree at most $f$ polynomials and any $f+1$ shares.

Similarly, Hiding follows since, for any $s$, conditioned on $p(0)=s$, the one-to-one mapping of degree at most $f$ polynomials to any $f$ shares (held by the adversary) implies that the uniform distribution on the coefficients $p_1,\dots,p_f$ induce a uniform distribution on the adversary view. The adversary view is therefore independently and uniformly random, for any value $s$. You can find a more detailed proof in our previous post.

Weak Termination follows since the share is non-blocking and the reconstruct only needs $n-f \geq f+1$ shares.

### Note on strong termination of the share protocol

As written, the dealer may crash in the middle of sending shares and parties have no way of knowing if all non-faulty parties received their phase. So parties need to reach agreement on whether the dealer completed or not. In particular all the lower bounds on agreement must hold.

One way to handle this is to abstract it away :-). Simply assume parties have access to broadcast channel. So the dealer, after sending all the shares, simply broadcasts OK. If parties dont hear an OK, they know the dealer crashed and use 0 as the reconstruct value.

In this model, the share protocol requires $O(n)$ words to be sent on private channels and $O(1)$ words of broadcast. The reconstruct protocol requires $O(n^2)$ words on private channels.