Isaac Lam, HW25, April 21st 2024

#C25.txt: Zero-Knowledge Proofs. April 18, 2024
Help:=proc(): print(`LD(p), RG(n,p), RHG(n,p), ZKP1(n,G,pi,Opt)`): end:
with(combinat):
#LD(p): inputs a pos. rational number from 0 to 1 and  outputs true with prob. p
LD:=proc(p) local i:i:=rand(1..denom(p))(): if i<=numer(p) then true:else false:fi:end:

#RG(n,p): inputs a pos. integer n and outputs a simple graph on n vertices where the
#prob of an edge is p (independetly)
RG:=proc(n,p) local i,j,G:
G:={}:
for i from 1 to n do
 for j from i+1 to n do
  if LD(p) then
    G:=G union {{i,j}}:
  fi:
 od:
od:
G:  
end:


#RHG(n,p): inputs a pos. integer n and outputs a simple  HAMILTONIAN graph on n vertices 
#where the
#prob of an edge is p (independetly) together with the Hamiltonian path
#The output is a pair [G,permutation] where the permutation of {1,...,n}
#is the Hamiltonian cycle that you know and you don't want anyone else to know
#BUT you do want to convince them that you DO know
RHG:=proc(n,p) local i,j,G,pi:
pi:=randperm(n):
G:={seq({pi[i],pi[i+1]},i=1..n-1), {pi[n],pi[1]}}:

for i from 1 to n do
 for j from i+1 to n do
  if LD(p) then
    G:=G union {{i,j}}:
  fi:
 od:
od:
[G, pi]:  
end:

#ZKP1(n,G,pi,Opt): Does ONE ROUND of the Blum protocol. Inputs a graph n,G,
#a Hamiltonian pi, and Opt=1 or 2 Opt=1 you show all the n+binomial(n,2) boxes
#Opt2, you show the contents of the n boxes correponsing to the mapping of
#your Hamiltonian
ZKP1:=proc(n,G,pi,Opt) local B1,B2,i,j,sig:
sig:=randperm(n):
for i from 1 to n do
 B1[i]:=sig[i]:
od:

for i from 1 to n do
 for j from i+1 to n do
   if member ({i,j},G) then
     B2[{sig[i],sig[j]}]:=1:
   else
    B2[{sig[i],sig[j]}]:=0:
   fi:
od:
od:

if Opt=1 then
 [op(B1),op(B2)]:
else
 {seq(B2[{sig[pi[i]],sig[pi[i+1]]}],i=1..n-1),B2[{sig[pi[n]],sig[pi[1]]}]}:
fi:
end:








VerifyOpt1 := proc(n, G, B1, B2) local edge, faithful; 
faithful := true; 
for edge in G do 
if B2[B1[edge[1]], B1[edge[2]]] = false then faithful := false; end if; 
end do; 

faithful; 

end proc;

ZKP1 := proc(n, G, pi, Opt) local B1, B2, i, j, sig; 

sig := randperm(n); 

for i to n do B1[i] := sig[i]; end do; 
for i to n do for j from i + 1 to n do 
if member*({i, j}, G) then B2[{sig[i], sig[j]}] := 1; 
else B2[{sig[i], sig[j]}] := 0; 
end if; 
end do; 
end do; 

if Opt = 1 then VerifyOpt1(n, G, B1, B2); 
else if {B2[{sig[pi[1]], sig[pi[n]]}], seq(B2[{sig[pi[i + 1]], sig[pi[i]]}], i = 1 .. n - 1)} = {1} then return true; 
else return false; 
end if; 
end if; 
end proc;

ZKP := proc(n, G, pi, K) local i; 
for i to K do 
if LD(1/2) then if ZKP1(n, G, pi, 1) = false then 
return false; 
end if; 

else if ZKP1(n, G, pi, 2) = false then return false; 
end if; 
end if; 
end do; 

true; 

end proc;