By Tal Malkin, Vladimir Kolesnikov, Allison Lewko, Michalis Polychronakis

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**Extra resources for Applied Cryptography and Network Security: 13th International Conference, ACNS 2015, New York, NY, USA, June 2-5, 2015, Revised Selected Papers**

**Sample text**

Y := Xoutgate · D−1 19. if ¬fsver(ΣCD ; Y, Y0 , v, v0 ; adec , cdec , rdec ; D) then return ⊥ 20. y := paillierdecode(Y0 ) // encryption of y + d = r 21. return (1 + N )y D In more detail, during the input phase of the protocol, the input parties provide their inputs (lines 4–8). As in the CDN protocol, each party encrypts its input and compiles a FS proof of knowledge (line 5). In the original CDN protocol, these encryptions and proofs would be broadcast directly; however, if a majority of computation parties are corrupted, then this allows corrupted parties to adapt their inputs based on the inputs of the honest parties.

Then each party acts as the trusted party in the steps above. The fact that the original vector of n − 3t sharings was padded with an extra t sharings prevents the adversary from learning any information on the secrets. 34 J. Baron et al. Once each party is done acting as the trusted party, she then sends the shares of the results to the other parties. Each party, upon receiving the n (or fewer) shares, can apply the Berlekamp-Welch algorithm to interpolate the vector of n shares in the presence of errors to reconstruct the pre-image under multiplication by the hyper-invertible matrix, which is a vector of n − 2t shares.

If |P \ F | < n/2 then break 20. Sc := Xa · Dc 21. parties i ∈ P \ F do Si := (Sc )2Δsi ; bcast(Si ) 22. (·, S0,c , ·, ·; πCDc ) := 23. MΣ(ΣCD , ΦCD , P \ F, {(Sc , Si , v, vi )}i∈P\F , {Δsi }i∈P\F ) 24. if |P \ F | < n/2 then break 25. s := paillierdecode(S0,c ); Xc := (Xb )s · Ec−1 26. if |P \ F | < n/2 then parties i ∈ I ∪ P ∪ {R} do return ⊥ 27. party R do d ∈R ZN ; s ∈R Z∗N ; D := (1 + N )d sN // output phase 28. party R do πPKd := fsprove(ΣPK ; D; d, s; R); bcast(D, πPKd ) 29. if ¬fsver(ΣPK ; D; πPKd ; R) then parties i ∈ I ∪ P ∪ {R} do return ⊥ 30.