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\\ From: "How to obtain full privacy in auctions" (2006) by Felix Brandt pages 19-20


\\\\\\\\\\\\
\\ Adapt the following values to your needs
\\\\\\\\\\\\

\\ amount of bidders
\\n = 3
\\ amount of possible prices
\\k = 2^2
\\ randomize bids (change to something static, if you like)
\\bid = vector(n,i,random(k)+1)
\\bid = vector(n,i,n-i+1)     \\ first bidder wins
\\bid = vector(n,i,i)         \\ last bidder wins
\\bid = vector(n,i,(i+1)%2)   \\ second bidder wins (with ties)

\\\\\\\\\\\\
\\ SETUP
\\\\\\\\\\\\

read(group);
read(zkp);

fp_priv(bids:vec, k:int) =
{
	local(n:int = length(bids));

	\\\\\\\\\\\\
	\\ PROLOG
	\\\\\\\\\\\\

	\\ private keys of agents
	x = vector(n,i,random(q));
	\\ first index level = owning agent id (additive share)
	\\ second index level = agent id, price id
	m = vector(n,i,matrix(n,k,a,b,random(q)));

	\\ zkp
	proofs1 = vector(n,i,zkp1_proof(G, x[i]));

	\\ public keyshares of agents
	yshares = vector(n,i,proofs1[i][4]);
	\\yshares = vector(n,i,G^x[i])

	\\ for performance evaluations we need to check the proofs for every bidder
	\\ i := checking bidder (0 == seller)
	\\ h := bidder to check
	for(i=0,n,
		for(h=1,n,
			if(1 != zkp1_check(proofs1[h]),
				error("zkp1 failure in round0")
			)
		)
	);

	\\ shared public key
	y = prod(X=1,n,yshares[X]);

	\\\\\\\\\\\\
	\\ ROUND1
	\\\\\\\\\\\\

	\\ bid matrix
	b = matrix(n,k,i,j,G^(bids[i]==j));

	\\ zkp
	proofs3 = matrix(n,k,i,j, zkp3_proof(G,y,G^(bids[i]==j)));

	\\ index = owning agent id, price id
	r = matrix(n,k,i,j,proofs3[i,j][13]);
	\\r = matrix(n,k,i,j,random(q))

	\\ encrypted bids
	Alpha = matrix(n,k,i,j, proofs3[i,j][3]);
	Beta  = matrix(n,k,i,j, proofs3[i,j][4]);
	\\Alpha = matrix(n,k,i,j, b[i,j]*y^r[i,j])
	\\Beta  = matrix(n,k,i,j,        G^r[i,j])

	proofs2 = vector(n,i, zkp2_proof(y,G,sum(j=1,k, r[i,j])));
	\\ i := checking bidder (0 == seller)
	\\ h := bidder to check
	\\ j := price index to check
	for(i=0,n,
		for(h=1,n,
			for(j=1,k,
				if(1 != zkp3_check(proofs3[h,j]),
					error("zkp3 failure in round1")
				)
			);
			if((prod(j=1,k,Alpha[h,j])/G) != proofs2[h][6],
				error("alpha product doesn't match")
			);
			if(prod(j=1,k,Beta[h,j]) != proofs2[h][7],
				error("beta product doesn't match")
			);
			if(1 != zkp2_check(proofs2[h]),
				error("zkp2 failure in round1")
			)
		)
	);

	\\\\\\\\\\\\
	\\ ROUND2
	\\\\\\\\\\\\

	\\ multiplicative shares
	\\ first index level = owning agent id (multiplicative share)
	\\ second index level = agent id, price id
	Gamma = vector(n,a,matrix(n,k,i,j, prod(h=1,n,prod(d=j+1,k,Alpha[h,d])) * prod(d=1,j-1,Alpha[i,d]) * prod(h=1,i-1,Alpha[h,j]) ));
	Delta = vector(n,a,matrix(n,k,i,j, prod(h=1,n,prod(d=j+1,k, Beta[h,d])) * prod(d=1,j-1, Beta[i,d]) * prod(h=1,i-1, Beta[h,j]) ));
	\\Gamma = vector(n,a,matrix(n,k,i,j, ( prod(h=1,n,prod(d=j+1,k,Alpha[h,d])) * prod(d=1,j-1,Alpha[i,d]) * prod(h=1,i-1,Alpha[h,j]) )^m[a][i,j] ))
	\\Delta = vector(n,a,matrix(n,k,i,j, ( prod(h=1,n,prod(d=j+1,k, Beta[h,d])) * prod(d=1,j-1, Beta[i,d]) * prod(h=1,i-1, Beta[h,j]) )^m[a][i,j] ))

	\\ random masking and zkp
	proofs2 = vector(n,a,matrix(n,k,i,j, zkp2_proof(Gamma[a][i,j], Delta[a][i,j], random(q)) ));

	\\ for performance evaluations we need to check the proofs for every bidder
	\\ i := checking bidder (0 == seller)
	\\ h := bidder to check
	\\ t := target bidder (creator of the proof)
	\\ j := price
	for(t=1,n,
		for(h=1,n,
			for(j=1,k,
				for(i=0,n,
					if(1 != zkp2_check(proofs2[t][h,j]),
						error("zkp2 failure in round2")
					)
				);
				\\ use masked values generated during the zkp
				Gamma[t][h,j] = proofs2[t][h,j][6];
				Delta[t][h,j] = proofs2[t][h,j][7];
			)
		)
	);


	\\\\\\\\\\\\
	\\ ROUND3
	\\\\\\\\\\\\

	\\ multiplicative shares (decryption)
	\\ first index level = owning agent id (multiplicative share)
	\\ second index level = agent id, price id
	Phi = vector(n,a,matrix(n,k,i,j, prod(h=1,n,Delta[h][i,j]) ));
	\\Phi = vector(n,a,matrix(n,k,i,j, prod(h=1,n,Delta[h][i,j])^x[a] ))

	proofs2 = vector(n,a,matrix(n,k,i,j, zkp2_proof(Phi[a][i,j], G, x[a]) ));

	\\ for performance evaluations we need to check the proofs for every bidder
	\\ i := checking bidder (0 == seller)
	\\ h := bidder to check
	\\ t := target bidder (creator of the proof)
	\\ j := price
	for(t=1,n,
		for(h=1,n,
			for(j=1,k,
				for(i=0,n,
					if(1 != zkp2_check(proofs2[t][h,j]),
						error("zkp2 failure in round2")
					)
				);
				\\ use masked values generated during the zkp
				Phi[t][h,j] = proofs2[t][h,j][6];
			)
		)
	);


	\\\\\\\\\\\\
	\\ EPILOG
	\\\\\\\\\\\\

	\\ winner matrix
	v = matrix(n,k,a,j, prod(i=1,n,Gamma[i][a,j]) / prod(i=1,n,Phi[i][a,j]) );
	vi = lift(v);

	print("bids are: ", bids);
	for(X=1,n,
		if(vecmin(vi[X,])==1,
			print("And the winner is ", X)
		)
	);

}

;