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Additional resources for Applied Cryptography and Network Security: 13th International Conference, ACNS 2015, New York, NY, USA, June 2-5, 2015, Revised Selected Papers
Ua(n−2t,m) T . Each party in P locally computes their shares of these polynomials. 2 Each party in P sends all their shares of Ha party Pk for each a, i, and m. 3 Each Pk uses Berlekamp-Welch on the shares of each U (i,k,m) to interpolate U (i,k,m) (ζ j ) for each j ∈ [ ]. 4 Each Pk uses Berlekamp-Welch on the shares of each Ha . to (i,k,m) interpolate H (αi ) for each i ∈ [n]. 5 Each Pk checks if the shares of Ha are consistent with the interpolation of the polynomial U (i,k,m) . That is, Pk checks if (k,m) (αi ) for each j ∈ [ ].
A , ya ) lie on the polynomial of degree ≤ a − 1 which evaluates to xj at φj for each j ∈ [a]. (In other words, M interpolates the points with x-coordinates θ1 , . . , θa on a polynomial given the points with x-coordinates φ1 , . . ) Then any submatrix of M is hyper-invertible. For our protocol, we let M be some (publicly known) hyper-invertible matrix with n rows and n − 2t columns. Throughout the protocol, the Berlekamp-Welch algorithm is used to interpolate polynomials in the presence of corrupt shares introduced by the adversary.
Each party locally computes and similarly deﬁne Ra (j,m) and their shares of these polynomials and sends his share of each Ha (j,m) Ra to party Pj . (i,m) Each Pi uses Berlekamp-Welch to interpolate the shares of Ha and (i,m) Ra received in the previous step. 1 h2a−1 (ζ j ) = Ha (ζ ) for j ∈ [ ]. (i,m) (i,m) j +j ) = R2a−1 (ζ ) for j ∈ [d − + 1]. 2 h2a−1 (ζ (i,m) (i,m) (ζ +j ) for j ∈ [ ]. 4 h2a (ζ ) = R2a (ζ ) for j ∈ [d − + 1]. (i,m) Each Pi sends each ha (αj ) to each Pj . 4 h2a (ζ then it is clear that ha(1,m) , .