3.5.2 Shuffling the Deck

Every player must perform a shuffle of the deck, because only such a procedure guarantees that no coalition has influence on the outcome. Thus we build a shuffle chain (e.g. using the strict total order P_i < P_j, if and only if i < j) such that each player shuffles the deck once.

First the regular stack s is initialized with open cards from deck. Then each player shuffles the stack (see SchindelhauerTMCG, i.e, TMCG_MixStack) using randomness (see TMCG_CreateStackSecret) and proves the correctness of this operation (see TMCG_ProveStackEquality). Consequently, every player should verify these proofs (see TMCG_VerifyStackEquality) and complain deviations immediately. Finally, the stack s contains the shuffled result. Consider the following code fragment for the player P_j.

for (size_t i = 0; i < 5; i++)
{
  TMCG_Stack<VTMF_Card> s2;
  if (i == j)
  {
    TMCG_StackSecret<VTMF_CardSecret> ss;
    tmcg->TMCG_CreateStackSecret(ss, false, s.size(), vtmf);
    tmcg->TMCG_MixStack(s, s2, ss, vtmf);
    for (size_t i2 = 0; i2 < 5; i2++)
    {
      if (i2 == j)
        continue;
      out_stream[i2] << s2 << std::endl;
      tmcg->TMCG_ProveStackEquality(s, s2, ss, false, vtmf,
        in_stream[i2], out_stream[i2]);
    }
  }
  else
  {
    in_stream[i] >> s2;
    if (!in_stream[i].good())
      std::cerr << "Read or parse error!" << std::endl;
    if (!tmcg->TMCG_VerifyStackEquality(s, s2, false, vtmf,
      in_stream[i], out_stream[i]))
        std::cerr << "Verification failed!" << std::endl;
  }
  s = s2;
}

If you want to use the more efficient shuffle verification protocol of Groth, then you have to replace TMCG_ProveStackEquality and TMCG_VerifyStackEquality by TMCG_ProveStackEquality_Groth and TMCG_VerifyStackEquality_Groth, respectively.¹⁶