For all valid assertions, Verifier says "yes"
For all invalid assertions, Verifier says "no"
Verifier is convinced, but gains "zero knowledge"
For all valid assertions, Verifier says "yes"
For all invalid assertions, Verifier says "no"
( with a very high probability )
Verifier is convinced, but gains "zero knowledge"
\(\color{blue}g\) is the generator of cyclic group \(\color{blue}G\) of prime order \(\color{blue}p\)
\(\color{blue}A \in \color{blue}G \) is an arbitrary group element having discrete logarithm $$ \color{red}a = \log_\color{blue}g(\color{blue}A) $$
testNonInteractiveProof :: IO Bool
testNonInteractiveProof = do
let secret = 42
let a' = pointBaseMul secp256k1 secret
-- Prover
(r, k) <- commmitment
-- Prover
let e = oracle (show r)
-- Prover
let s = response k e secret
-- Verifer
pure (verify a' r e s)
testInteractiveProof :: IO Bool
testInteractiveProof = do
let secret = 42
let a' = pointBaseMul secp256k1 secret
-- Prover
(r, k) <- commmitment
-- Verifier
e <- challange
-- Prover
let s = response k e secret
-- Verifer
pure (verify a' r e s)
commmitment :: MonadRandom m => m (Point, Integer)
commmitment = do
k <- generateBetween 0 p
pure (pointBaseMul secp256k1 k, k)challange :: MonadRandom m => m Integer
challange = generateBetween 0 presponse :: Integer -> Integer -> Integer -> Integer
response k chal secret = (k + chal*secret)
verify :: Point -> Point -> Integer -> Integer -> Bool
verify a' r e s = lhs == rhs
where
lhs = pointBaseMul secp256k1 s
rhs = pointAdd secp256k1 r (pointMul secp256k1 e a')
testAndProof :: IO (Bool, Bool)
testAndProof= do
a <- testNonInteractiveProof
b <- testNonInteractiveProof
pure (a, b)
testAndProof' :: IO Bool
testAndProof' = do
let secretA = 42
let a' = pointBaseMul secp256k1 secretA
let secretB = 43
let b' = pointBaseMul secp256k1 secretB
-- Prover
(r, k) <- commmitment
-- Prover
let e = oracle (show r)
-- Prover
let s = (response k e secretA) + e * secretB
-- Verifer
pure (verifyAnd a' b' r e s)
verifyAnd a' b' r e s = lhs == rhs
where
lhs = pointBaseMul secp256k1 s
rhs = pointAdd secp256k1 r (pointAdd secp256k1 a b)
a = pointMul secp256k1 e a'
b = pointMul secp256k1 e b'
testEqProof :: IO Bool
testEqProof = do
let secret = 42
let a' = pointBaseMul secp256k1 secret
let b' = pointBaseMul secp192r1 secret
-- Prover
k <- generateBetween 0 p
let rA = pointBaseMul secp256k1 k
let rB = pointBaseMul secp192r1 k
-- Prover
let e = oracle (show [rA,rB])
-- Prover
let s = (response k e secret)
-- Verifer
pure (verifyEq secp256k1 a' rA e s && verifyEq secp192r1 b' rB e s)
verifyEq :: Curve -> Point -> Point -> Integer -> Integer -> Bool
verifyEq curve a' r e s = lhs == rhs
where
lhs = pointBaseMul curve s
rhs = pointAdd curve r (pointMul curve e a')