# Multi-Linear Secret Sharing vs. Shamir Secret Sharing

I hope someone can help me. I read several papers including the original paper, "Multi-linear Secret-sharing Schemes," by Beimel et al., and watched the following video https://www.youtube.com/watch?v=3GFdp7V_yog but still struggling with the concept of multi-linear secret sharing scheme. I know that in LSSS, the secret is an element of a field, and a dealer distributes shares to parties. Consequently, the parties can independently use linear mapping to recover the secret. I will use the same example in the video; a secret is shared in field, F_2, and the matrix is as follows:

My questions are:

1. The first and the second rows of the matrix are labeled to the same party, ie P_2, but how can you give different shares to the same party?
2. Only P_2 and P_4 can recover the secret because when they combine their shares, the r_2, r_3, and r_4 gets canceled, But P_2 already has r_2 in two different rows. What is the difference between row 1 and row 2?
3. what is the difference between Shamir's secret sharing and multi-linear secret sharing.

Thank you

• I think you’re more likely to get help if you make your question as self-contained as possible by, for instance, defining the construction you’re considering explicitly – Daniel Feb 15 at 3:38
• @Daniel. Thank you very much for your feedback. I edited my question. – Mona Feb 15 at 13:31

Shamir Secret Sharing, on the other hand, rely on polynomial interpolation to reconstruct the hidden secret. Using the fact that d+1 points can interpolate degree d polynomial with Lagrange's Coefficient, we can assure that for d+1 shares can reconstruct the hidden secret. This is a good explanation video on SSS.