Zero Knowledge Proofs

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Zero Knowledge Proofs

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Interactive proof

• An Interactive Proof System for a language L is
  a two-party game between a verifier and a prover
  that interact on a common input in a way
  satisfying the following properties

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Interactive proof

• The verifiers strategy is a probabilistic
  polynomial-time procedure.
• Correctness requirements
• Completeness There exists a prover strategy P,
  such that for every x?L, when interacting on a
  common input x, the prover P convinces the
  verifier with probability at least 2/3.
• Soundness For every x?L, when interacting on the
  common input x, any prover strategy P convinces
  the verifier with probability at most 1/3.

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Zero Knowledge Proof

• Let (P,V) be an interactive proof system for some
  language L. We say that (P,V), actually P, is
  zero-knowledge if for every probabilistic
  polynomial-time ITM V there exists a
  probabilistic polynomial-time machine M s.t. for
  every x?L holds
• ltP,Vgt(x)x?L ? M(x)x?L
• Machine M is called the simulator for the
  interaction of V with P.

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Perfect Zero Knowledge

• Definition
• Let (P,V) be an interactive proof system for
  some language L. We say that (P,V), actually P,
  is perfect zero-knowledge (PZK) if for every
  probabilistic polynomial time ITM V there exists
  a probabilistic polynomial-time machine M s.t.
  for every x?L the distributions ltP,Vgt(x)x?L
  and M(x)x?L are identical, i.e.,
  ltP,Vgt(x)x?L ? M(x)x?L

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Statistical Zero Knowledge

• Definition
• Let (P,V) be an interactive proof system for some
  language L. We say that (P,V), actually P, is
  statistical zero knowledge (SZK) if for every
  probabilistic polynomial time verifier V there
  exists a probabilistic polynomial-time machine M
  s.t. the ensembles ltP,Vgt(x)x?L and M(x)x?L
  are statistically close.

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Statistical Zero Knowledge

• Definition-cont.
• The distribution ensembles Axx?L and Bxx?L
  are statistically close or have negligible
  variation distance if for every polynomial p()
  there exits integer N such that for every x?L
  with x ? N holds?? Pr Ax ? Pr Bx
  ? ? p(x)-1

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Computational Zero Knowledge

• Definition
• Let (P,V) be an interactive proof system for some
  language L. (P,V), actually P, is computational
  zero knowledge (CZK) if for every probabilistic
  polynomial-time verifier V there exists a
  probabilistic polynomial-time machine M s.t. the
  ensembles ltP,Vgt(x)x?L and M(x)x?L are
  computationally indistinguishable.

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Computational Zero Knowledge

• Definition
• Two ensembles Axx?L and Bxx?L are
• computationally indistinguishable if for
• every probabilistic polynomial time
• distinguisher D and for every polynomial p()
• there exists an integer N such that for every
• x?L with x ? N holds
• Pr D(x,Ax) 1 Pr D(x,Bx) 1 ? p(x)-1

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Graph Isomorphism problem

• Definition
• Graph Isomorphism two graphs G0 (V0,E0) and G1
  (V1, G1) are isomorphic ? ? permutation ?
• s.t
• ? (u,v) ? E0 ?(? (u), ?(v)) ? E1
• if G0 and G1 are isomorphic and ? is an
  isomorphism between G0 to G1 we write G1 ?(G0)
  .

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Graph Isomorphism problem

• Graph Isomorphism problem Given Two Graphs G1
  and G2 Are They Isomorphic ?
• Lemma GI ?ZK
• Proof Zero Knowledge Interactive Proof for GI.

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Zero Knowledge Interactive proof for Graph
Isomorphism

• 1. Repeat the following n times
• 2. The Prover chooses a random permutation ? of
  (1n) and computes H ?(G1) and send it to the
  verifier.
• 3. The verifier chooses randomly i1 or 2 and
  sends it to the prover.

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Zero Knowledge Interactive proof for Graph
Isomorphism-cont.

• 4. The prover chooses permutation ? s.t H
  ?(Gi).
• If i1 the prover sends ? to the verifier
  otherwise the prover will send ? ?-1 .(? is the
  isomorphism between G1 and G2.
• 5. The verifier checks if H is the image of Gi
  under ?.
• 6. The verifier accepts if H is the image of Gi
  in all n rounds.

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Zero Knowledge Interactive proof for Graph
Isomorphism-cont.

Prover
Verifier
? H ?(G1)
i1,2
R
? or ? ?-1
Checks if H is the image of Gi
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Building simulator M for graph isomorphism
problem

• We will define simulator M as follows
• Input(G0, G1) ? ISO
• 1.Randomly chooses a random string RANDOM and
  puts it on the Random tape of Verifier V.
• 2. Randomly chooses a ?0,1 and permutation ?
  and construct H ?(Ga) send H to V .

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Building simulator M for graph isomorphism
problem

• 3. Receive b from V .
• If b ?0,1 then outputs RANDOM,H,b and
  STOP.
• If a b then outputs RANDOM,H,b, ? and
  STOPelse GOTO 1 .

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Zero-Knowledge Password Proofs

• 1. The prover finds two large primal numbers - p
  and q and sends npq to the verifier
• 2. r is a random number belongs to n, n4. The
  prover sends x2 modn and r2 modn to the verifier.
• 3. The verifier then randomly asks for r or xr
  and checks the prover.

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Zero-Knowledge Password Proofs
Prover
Verifier

npq x2 modn r2 modn
Asks for xr or r
xr or r
Checks the Prover
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NP and Zero Knowledge proofs

• Lemma NP?ZK
• Proof 3col?ZK .

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Zero Knowledge proof for 3col problem

• 1. The prover randomly chooses a permutation ?.
  Computes ?(c(v)), puts in envelopes and sends to
  the verifier.
• 2. The verifier chooses randomly
• (u,v) ?E and opens the envelope.
• If the colors are different and legal he answers
  yes.

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Zero Knowledge proof for 3col problem
Prover
Verifier

permutation ?. ?(c(v))
Chooses (u,v) ?E
envelope
Checks that colors are different
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ZK protocol for Co-SAT

• Transform the CNF to a polynom by these
  transformation rules
• 1. T ? positive value
• 2. F ? 0
• 3. Xi ? Xi
• 3. ? Xi ? (1-Xi)
• 4. OR ?
• 5. AND ?

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ZK protocol for Co-SAT

• The protocol
• 1. The prover selects a prime number q gt 2n 3m
  and sends to the verifier.
• 2. The verifier checks that q is prime. If q
  isnt prime halts and rejects.

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ZK protocol for Co-SAT

• 3. V0 is at the initialized at value zero. The
  prover does the following for i1n. The prover
  computes polynom Pi that its rank is at most m .
  
• The construction of Pi
• P1(x) ? xn 0,1. ? xn0,1 p(x1 xn)
• P2(x) ? xn 0,1. ? xn0,1 p(r1,x, x3 xn)
• Pn(x) p(r1,... Rn-1, xn ) the prover puts
  polynom Pi in envelopes and send to the verifier.

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ZK protocol for Co-SAT

• 4. The prover moves to the next stage(ii1).
• 5. We know that the verifier will accept
• if ? r1 ri rn s.t Pi(0) Pi(1) vi -1modq.
  
• Since checking each assignment is polynomial this
  problem is in NP .
• We can now do a reduction from any NP problem to
  3col ? ZK .

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ZK protocol for Graph non isomorphism

• Definition
• Graph non Isomorphism given two graphs G0
  (V0,E0) and G1 (V1, G1) .
• (G0, G1 )?GNI ?
• there is no permutation ?
• s.t
• ? (u,v) ? E0 ?(? (u), ?(v)) ? E1

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ZK protocol for Graph non isomorphism

• 1. The verifier chooses randomly a number i
  ?(0,1) . The verifier chooses a random
  permutation ? and computes H ? (Gi). Then the
  verifier chooses randomly j ?(0,1) . The verifier
  creates the pair of graphs (H0, H1) such that
• if j0
• H0 is a permutation of G0
• H1 is a permutation of G1

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ZK protocol for Graph non isomorphism

• if j1
• H0 is a permutation of G1
• H1 is apermutation of G0
• the verifier sends H and the pair (H0, H1).

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ZK protocol for Graph non isomorphism

• 2. The prover chooses randomly
• b ?(0,1) . The prover sends b to the verifier .
  
• If b0 then the verifier sends the prover the
  isomorphism between (G0, G1) and (H0, H1).
• If b1 the verifier sends the prover the
  isomorphism between H and (H0, H1) .

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ZK protocol for Graph non isomorphism

• 3. The prover checks that the right isomorphism
  is sent otherwise it stops. the prover computes b
  such that Gb is isomorphic to H and sends b to V
  . If there is no such b , the prover sends a
  random b.
• 4. The verifier accepts if jb.

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ZK protocol for Graph non isomorphism
Prover
Verifier

1. i ?(0,1) 2.H ? (Gi) 3. H and the pair (H0,
H1)
1.Isomorphism between (G0, G1) and (H0, H1).
OR 2.Isomorphism between (H0, H1) and H.
Check isomorphism computes b
checks that jb
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ZK protocol for Graph non isomorphism

• Lemma GNI ? PZK
• Proof building M
• s.t ltP,Vgt(x)x?L ? M(x)x?L
• 1. The machine M takes random string of bits and
  puts ot on a Random tape.

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ZK protocol for Graph non isomorphism

• Mv does the following n times
• 2. Mv waits to get H and the pair (H0, H1) from
  V .
• 3. Mv chooses a random b .
• 4. Mv gets from V the isomorphism between H
  and (H0, H1) and (G0, G1). Mv checks if it is
  not the right isomorphism it stops.

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ZK protocol for Graph non isomorphism

• Otherwise1. Returns V to the point after H and
• (H0, H1) were received.
• 2. choose b again and sends to V
• 3. Waits to get I from V
• I- isomorphism received from V.

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ZK protocol for Graph non isomorphism

• If b?b then the Mv finds isomorphism from I and
  I, from G0,G1 to (H0, H1) and from (H0, H1) to
  H. The machine uses this information to find
  Isomorphism from H to G0 , G1.
• 4. The machine Mv uses this information to
  compute V and sends it to V.