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Round-Optimal and Efficient Verifiable Secret Sharing

Round-Optimal and Efficient Verifiable Secret Sharing. Matthias Fitzi (Aarhus University) Juan Garay (Bell Labs) Shyamnath Gollakota (IIT Madras) C. Pandu Rangan (IIT Madras) Kannan Srinathan (IIIT Hyderabad). Secret Sharing Protocols [Sha79,Bla79]. Two phases Sharing phase

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Round-Optimal and Efficient Verifiable Secret Sharing

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  1. Round-Optimal and EfficientVerifiable Secret Sharing Matthias Fitzi (Aarhus University) Juan Garay (Bell Labs) Shyamnath Gollakota (IIT Madras) C. Pandu Rangan (IIT Madras) Kannan Srinathan (IIIT Hyderabad)

  2. Secret Sharing Protocols [Sha79,Bla79] • Two phases • Sharing phase • Reconstruction phase • Sharing Phase • D initially holds s and each player Pi finally holds some private information vi. • Reconstruction Phase • Each player Pi reveals (some of) his private information v’i on which a reconstruction function is applied to obtain s = Rec(v’1, v’2, …, v’n). • Set of players P = {P1 , P2, … ,Pn}, dealer D (e.g., D = P1). Round-Optimal and Efficient VSS —TCC’06

  3. Sharing Phase … vn v1 v3 v2 Reconstruction Phase Less than t +1 players have no info’ about the secret Secret Sharing (cont’d) Secret s Dealer Round-Optimal and Efficient VSS —TCC’06

  4. Sharing Phase vn v1 v3 v2  t +1 players can reconstruct the secret Secret s Secret Sharing (cont’d) Secret s Dealer … Reconstruction Phase Players are assumed to give their shares honestly Round-Optimal and Efficient VSS —TCC’06

  5. Verifiable Secret Sharing(VSS)[CGMA85] • Extends secret sharing to the case of active corruptions • (corrupted players, incl. Dealer, may not follow the protocol) • Up to t corrupted players • Adaptive adversary • Reconstruction Phase • Each player Pi reveals (some of) his private information v’i • on which a reconstruction function is applied to obtain • s’ = Rec(v’1, v’2, …, v’n). Round-Optimal and Efficient VSS —TCC’06

  6. VSS Requirements • Privacy • If D is honest, adversary has no Shannon information about s during the Sharing phase. • Correctness • If D is honest, the reconstructed value s’ = s. • Commitment • After Sharing phase, s’ is uniquely determined. Round-Optimal and Efficient VSS —TCC’06

  7. Weak VSS (WSS) [RB89] • Privacy • If D is honest, adversary has no Shannon information about s during the Sharing phase. • Correctness • If D is honest, the reconstructed value s’ = s. • Weak Commitment • After Sharing phase, s’ is uniquely determined such that • Rec(v’1, v’2, …, v’n)  {, s’}. Round-Optimal and Efficient VSS —TCC’06

  8. Communication Model and Round Complexity • Synchronous, fully connected network of pair-wisesecure channels + broadcast channel. • Round complexity:Number of communication rounds in the Sharing phase. • Efficiency:Total computation and communication polynomial in n and size of the secret. Round-Optimal and Efficient VSS —TCC’06

  9. Prior (Relevant) Work • Perfect VSS possible iff n > 3t [BGW88, DDWY90] • Round complexity of VSS [GIKR01] • n > 4t: Efficient 2-round protocol • n > 3t: No 2-round protocol exists Efficient 4-round protocol Inefficient3-round protocol Round-Optimal and Efficient VSS —TCC’06

  10. Our Contributions • VSS:Efficient3-round protocol for n > 3t • WSS: • Efficient 3-round protocol for n > 3t — round optimal • Efficient 1-round protocol for n > 4t • (1+) amortized-round VSS protocol for n > 3t Round-Optimal and Efficient VSS —TCC’06

  11. Our Contributions • VSS:Efficient3-round protocol for n > 3t • WSS: • Efficient 3-round protocol for n > 3t — round optimal • Efficient 1-round protocol for n > 4t • (1+ ) amortized-round VSS protocol for n > 3t Round-Optimal and Efficient VSS —TCC’06

  12. 3-Round (n/3)-WSS Secret s Dealer Sharing Phase … vn v1 v3 v2 Reconstruction Phase Round-Optimal and Efficient VSS —TCC’06

  13. Secret s’ 3-Round (n/3)-WSS Secret s … vn v1 v3 v2 Reconstruction Phase Round-Optimal and Efficient VSS —TCC’06

  14. F(j,i) + r 3-Round (n/3)-WSS — Sharing Phase • Round 1: • D selects a random bivariate polynomial F(x,y) of degree t in each variable, s.t. F(0,0) = s; sends F(x,i) = fi(x) and F(i,y) = gi(y) to Pi. • Player Pi sends to Pj a random pad rij. • Round 2:Pi broadcasts • aij = fi(j) + rij • bij = gi(j) + rji Pj broadcasts • aiji = fj(i) + rji • bji = gj(i) + rij Round-Optimal and Efficient VSS —TCC’06

  15. 3-Round (n/3)-WSS — Sharing Phase • Round 1: • D selects a random bivariate polynomial F(x,y) of degree t in each variable, s.t. F(0,0) = s; sends F(x,i) = fi(x) and F(i,y) = gi(y) to Pi. • Player Pi sends to Pj a random pad rij. • Round 2:Pi broadcasts • aij = fi(j) + rij • bij = gi(j) + rji • Round 3:For each aij≠bji • Pi broadcasts fi(j) • Pj broadcasts gj(i) • D broadcasts F(j,i) • A player is said to be unhappy if his value does not match D’s value. If no. unhappy players > t, disqualify D. Pj broadcasts • aij = fj(i) + rji • bji = gj(i) + rij Round-Optimal and Efficient VSS —TCC’06

  16. 3-Round (n/3)-WSS — Reconstruction Phase • Every happy player Pi broadcasts fi(x) and gi(y). • Local computation: • Every player constructs a consistency graph G over the set of happy players: there exists an edge between Pi,Pj G iff fi(j) = gj(i) and gi(j) =fj(i). • Every player constructs a set CORE as follows: • Initially all nodes with degree at least n–t in G are in CORE. • Players in CORE consistent with less than n–t players in CORE are removed. • Repeat until no more players can be removed from CORE. • Secret determined by the polynomial defined by any t+1 players from CORE. If |CORE| < n–t, the secret is . Round-Optimal and Efficient VSS —TCC’06

  17. 3-Round (n/3)-WSS — Proof Sketch • Privacy: (D is honest) • D distributes consistent information any pair of honest players publish same mutual padded values. • Randomness of pads leads to indistinguishability of adversary’s view under different secrets. • Correctness: (D is honest) • All honest players (at least n–t) are happy  no disqualification of D in Sharing Phase. • They all end up in CORE, thus the secret reconstructed is s. Round-Optimal and Efficient VSS —TCC’06

  18. 3-Round (n/3)-WSS — Proof Sketch • Weak Commitment: • |CORE| < n – t: All honest players output . • |CORE|  n – t: All players in CORE are consistent with a polynomial fixed at the end of the Sharing Phase: • The n–2thonest happy players define a unique polynomial F’(x,y) (at the end of Sharing Phase). • Every dishonest happy player in CORE is consistent with at least n–t players in CORE, of which n–2tt+1 are honest  every dishonest happy player in CORE is also consistent with F’(x,y). Round-Optimal and Efficient VSS —TCC’06

  19. Recall: 3-Round (n/3)-WSS — Sharing Phase • Round 1: • D selects a random bivariate polynomial F(x,y) of degree t in each variable, s.t. F(0,0) = s; sends F(x,i) = fi(x) and F(i,y) = gi(y) to Pi. • Player Pi sends to Pj a random pad rij. • Round 2:Pibroadcasts • aij = fi(j) + rij • bij = gi(j) + rji • Round 3:For each aij≠bji • Pi broadcasts fi(j) • Pj broadcasts gj(i) • D broadcasts F(j,i) • A player is said to be unhappy if his value does not match D’s value. If no. unhappy players > t, disqualify D. Round-Optimal and Efficient VSS —TCC’06

  20. 3-Round (n/3)-VSS — Sharing Phase • Round 1: • D selects a random bivariate polynomial F(x,y) of degree t in each variable, s.t. F(0,0) = s; sends F(x,i) = fi(x) and F(i,y) = gi(y) to Pi. • Player Pi selects randomriand starts (n/3)-WSS onriusingFiW(x,y). Round-Optimal and Efficient VSS —TCC’06

  21. 3-Round (n/3)-VSS — Sharing Phase • Round 1: • D selects a random bivariate polynomial F(x,y) of degree t in each variable, s.t. F(0,0) = s; sends F(x,i) = fi(x) and F(i,y) = gi(y) to Pi. • Player Pi selects randomriand starts (n/3)-WSSi onriusingFiW(x,y). • Round 2:Pi broadcasts • aij = fi(j) + FiW(0,j) • bij = gi(j) + FjW(0,i) • Concurrently, round 2 of (n/3)- WSSi • takes place. Round-Optimal and Efficient VSS —TCC’06

  22. 3-Round (n/3)-VSS — Sharing Phase • Round 1: • D selects a random bivariate polynomial F(x,y) of degree t in each variable, s.t. F(0,0) = s; sends F(x,i) = fi(x) and F(i,y) = gi(y) to Pi. • Player Pi selects randomriand starts (n/3)-WSSi onriusingFiW(x,y). • Round 2:Pi broadcasts • aij = fi(j) + FiW(0,j) • bij = gi(j) + FjW(0,i) • Round 3:For each aij≠bji • Pi broadcasts fi(j) • Pj broadcasts gj(i) • D broadcasts F(j,i) • Concurrently, round 2 of (n/3)-WSSi • takes place. • Concurrently, round 3 of (n/3)-WSSi • takes place. Round-Optimal and Efficient VSS —TCC’06

  23. 3-Round (n/3)-VSS — Sharing Phase • Round 1: • D selects a random bivariate polynomial F(x,y) of degree t in each variable, s.t. F(0,0) = s; sends F(x,i) = fi(x) and F(i,y) = gi(y) to Pi. • Player Pi selects randomriand starts (n/3)-WSSi onriusingFiW(x,y). • Round 2:Pi broadcasts • aij = fi(j) + FiW(0,j) • bij = gi(j) + FjW(0,i) • Round 3:For each aij≠bji • Pi broadcasts fi(j) • Pj broadcasts gj(i) • D broadcasts F(j,i) • A player is said to be unhappy if his value does not match D’s value. If no. unhappy players > t, disqualify D. • Concurrently, round 2 of (n/3)-WSSi • takes place. • Concurrently, round 3 of (n/3)-WSSi • takes place. Round-Optimal and Efficient VSS —TCC’06

  24. 3-Round (n/3)-VSS — Sharing Phase • Local Computation: • H = {happy players} – {players disqualified as WSS dealers} • If |H| < n–t, disqualify D and stop. • For Pi H, if |H ∩ HiW| < n–t, remove Pi from H. • Call the final set COREsh. If |COREsh| < n–t disqualify D and stop. • Properties of COREsh: • If D is honest, then COREsh contains all honest players  D is not disqualified during the Sharing phase. • Every player in COREsh is consistent with n–t players in COREsh At least t+1 honest players in COREsh (defining a unique polynomial FH(x,y)). Round-Optimal and Efficient VSS —TCC’06

  25. 3-Round (n/3)-VSS — Reconstruction Phase • For each Pi COREsh, run Rec. phase of (n/3)-WSSi, concurrently. • Local computation: • CORErec := COREsh • CORErec := CORErec – {Pi : (n/3)-WSSi } • For each Pi COREreccompute fi(j) = aij – FiW(0,j),1≤ j ≤ n If fi(x) not a t-degree polynomial, remove Pi from CORErec. • ObtainF’(x,y) by taking any t+1 polynomials fi(x)from CORErec; s’ := F’(0,0). Round-Optimal and Efficient VSS —TCC’06

  26. 3-Round (n/3)-VSS — Reconstruction Phase • Properties of CORErec: • At least n–2t ( t+1) honest players in COREsh  unique t-degree polynomial FH(x,y). • Dishonest Pi in CORErec: WSSi succeeded; fi(j) lie on at-degree polynomial f’i(x) ; F’iW(x,y)is … consistent with t+1 honest players in CORErec  f’i(x) is consistent with FH(x,y). • Privacy: • The only difference with WSS protocol is the pads. • Prove that aij = fi(j) + FiW(0,j)does not reveal any info’ about fi(j). Round-Optimal and Efficient VSS —TCC’06

  27. Amortized VSS Round Complexity • Say,m k-round sequential VSS protocols (e.g., MPC) • Using “deferred commitment,”m+2 total rounds  • 1+ O(1/m) amortized-round VSS protocol • Initial phase: Dealer(s) share random values r1, r2,…, rm using the given VSS protocol. • Sharing Phaseof jth VSS protocol: • Broadcast correction term cj = sj – rj • Correction:(two ways) • In Reconstruction Phase each player computes sj = cj + rj. • At the end of Sharing Phase every player Pi computes F*j(x,i) = Fj(x,i) + cj and F*j(i,y) = Fj(i,y) + cj Round-Optimal and Efficient VSS —TCC’06

  28. Summary • VSS:Efficient3-round protocol for n > 3t • WSS: • Efficient 3-round protocol for n > 3t — round optimal • Efficient 1-round protocol for n > 4t • (1+) amortized-round VSS Round-Optimal and Efficient VSS —TCC’06

  29. Round-Optimal and EfficientVerifiable Secret Sharing Matthias Fitzi (Aarhus University) Juan Garay (Bell Labs) Shyamnath Gollakota (IIT Madras) C. Pandu Rangan (IIT Madras) Kannan Srinathan (IIIT Hyderabad)

  30. (n/3)-WSS Round Optimality • Based on impossibility of 3-round Weak Secure Multicast: • P = {P1 , P2, … ,Pn}; D P holds input m; multicast setM P. • Privacy: If all players in M are honest, then adversary learns no information about m. • Correctness: If D is honest, then all honest players in M output m. • Weak Agreement:Even if D is dishonest, all honest players in M output a value in {m’, }. • r-round WSS  r-round WSM Round-Optimal and Efficient VSS —TCC’06

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