# Secret reconstruction¶

When at least $$t$$ users re-encrypted their shares with the receiver’s public key, the receiver can reconstruct the secret:

• Decrypt each re-encrypted share:
$$S_i = b_i \cdot a_i^{\frac{1}{x_r}}$$
• Reconstruct the secret:
$$S = \prod\limits_i S_i^{\lambda_i},~ \lambda_i = \prod\limits_{i', i' \ne i} \frac{i'}{i' - i}$$
where $$i, i'$$ are the user indices for all re-encrypted shares.