Digital Signatures Based on Public-Key Systems

Digital Signatures

• Alice wants to send Bob a signed document x.

• The signature must unmistakably identifies the sender.

• Both Alice and Bob have public and private keys e_Alice, e_Bob, d_Alice, d_Bob.

• Every cryptosystem guarantees D(d, E(e, x)) = x.

• Assume the cryptosystem also satisfies the commutative property

E(e, D(d, x)) = D(d, E(e, x)). (15) – E.g., the RSA system satisfies it as (x^d)^e = (x^e)^d.

aDiffie and Hellman (1976).

Digital Signatures Based on Public-Key Systems

• Alice signs x as

(x, D(d_Alice, x)).

• Bob receives (x, y) and verifies the signature by checking E(e_Alice, y) = E(e_Alice, D(d_Alice, x)) = x

based on Eq. (15).

• The claim of authenticity is founded on the difficulty of inverting E_Alice without knowing the key d_Alice.

Probabilistic Encryption

• A deterministic cryptosystem can be broken if the

plaintext has a distribution that favors the “easy” cases.

• The ability to forge signatures on even a vanishingly small fraction of strings of some length is a security weakness if those strings were the probable ones!

• A scheme may also “leak” partial information.

– Parity of the plaintext, e.g.

• The first solution to the problems of skewed distribution and partial information was based on the QRA.

aGoldwasser and Micali (1982). This paper “laid the framework for modern cryptography” (2013).

Shafi Goldwasser

(1958–)

aTuring Award (2013).

Silvio Micali

(1954–)

aTuring Award (2013).

Goldwasser and Micali

A Useful Lemma

Lemma 75 Let n = pq be a product of two distinct primes.

Then a number y ∈ Zn^∗ is a quadratic residue modulo n if and only if (y | p) = (y | q) = 1.

• The “only if” part:

– Let x be a solution to x² = y mod pq.

– Then x² = y mod p and x² = y mod q also hold.

– Hence y is a quadratic modulo p and a quadratic residue modulo q.

The Proof (concluded)

• The “if” part:

– Let a²₁ = y mod p and a²₂ = y mod q.

– Solve

x = a₁ mod p, x = a₂ mod q,

for x with the Chinese remainder theorem.

– As x² = y mod p, x² = y mod q, and gcd(p, q) = 1, we must have x² = y mod pq.

The Jacobi Symbol and Quadratic Residuacity Test

• The Legendre symbol can be used as a test for quadratic residuacity by Lemma 62 (p. 525).

• Lemma 75 (p. 636) says this is not the case with the Jacobi symbol in general.

• Suppose n = pq is a product of two distinct primes.

• A number y ∈ Zn^∗ with Jacobi symbol (y | pq) = 1 may be a quadratic nonresidue modulo n when

(y | p) = (y | q) = −1, because (y | pq) = (y | p)(y | q).

The Setup

• Bob publishes n = pq, a product of two distinct primes, and a quadratic nonresidue y with Jacobi symbol 1.

• Bob keeps secret the factorization of n.

• Alice wants to send bit string b¹b₂ · · · b^k to Bob.

• Alice encrypts the bits by choosing a random quadratic residue modulo n if b_i is 1 and a random quadratic

nonresidue (with Jacobi symbol 1) otherwise.

• So a sequence of residues and nonresidues are sent.

• Knowing the factorization of n, Bob can efficiently test quadratic residuacity and thus read the message.

The Protocol for Alice

1: for i = 1, 2, . . . , k do

2: Pick r ∈ Zn^∗ randomly;

3: if b_i = 1 then

4: Send r² mod n; {Jacobi symbol is 1.}

5: else

6: Send r²y mod n; {Jacobi symbol is still 1.}

7: end if

8: end for

The Protocol for Bob

1: for i = 1, 2, . . . , k do

2: Receive r;

3: if (r| p) = 1 and (r | q) = 1 then

4: b_i := 1;

5: else

6: b_i := 0;

7: end if

8: end for

Semantic Security

• This encryption scheme is probabilistic.

• There are a large number of different encryptions of a given message.

• One is chosen at random by the sender to represent the message.

• This scheme is both polynomially secure and semantically secure.

What Is a Proof?

• A proof convinces a party of a certain claim.

– “xⁿ + yⁿ ̸= zⁿ for all x, y, z ∈ Z⁺ and n > 2.”

– “Graph G is Hamiltonian.”

– “x^p = x mod p for prime p and p ̸ |x.”

• In mathematics, a proof is a fixed sequence of theorems.

– Think of it as a written examination.

• We will extend a proof to cover a proof process by which the validity of the assertion is established.

– Recall a job interview or an oral examination.

Prover and Verifier

• There are two parties to a proof.

– The prover (Peggy).

– The verifier (Victor).

• Given an assertion, the prover’s goal is to convince the verifier of its validity (completeness).

• The verifier’s objective is to accept only correct assertions (soundness).

• The verifier usually has an easier job than the prover.

• The setup is very much like the Turing test.^a

aTuring (1950).

Interactive Proof Systems

• An interactive proof for a language L is a sequence of questions and answers between the two parties.

• At the end of the interaction, the verifier decides whether the claim is true or false.

• The verifier must be a probabilistic polynomial-time algorithm.

• The prover runs an exponential-time algorithm.^a

– If the prover is not more powerful than the verifier, no interaction is needed!

aSee the problem to Note 12.3.7 on p. 296 and Proposition 19.1 on p. 475, both of the textbook, about alternative complexity assumptions without affecting the definition. Contributed by Mr. Young-San Lin

Interactive Proof Systems (concluded)

• The system decides L if the following two conditions hold for any common input x.

– If x ∈ L, then the probability that x is accepted by the verifier is at least 1 − 2^{−| x |}.

– If x ̸∈ L, then the probability that x is accepted by the verifier with any prover replacing the original prover is at most 2^{−| x |}.

• Neither the number of rounds nor the lengths of the messages can be more than a polynomial of | x |.

An Interactive Proof

3

3

3

3

3

9

9

9

9

9

IP

• IP is the class of all languages decided by an interactive proof system.

• When x ∈ L, the completeness condition can be modified to require that the verifier accepts with certainty without affecting IP.^b

• Similar things cannot be said of the soundness condition when x ̸∈ L.

• Verifier’s coin flips can be public.^c

aGoldwasser, Micali, and Rackoff (1985).

bGoldreich, Mansour, and Sipser (1987).

cGoldwasser and Sipser (1989).

The Relations of IP with Other Classes

• NP ⊆ IP.

– IP becomes NP when the verifier is deterministic and there is only one round of interaction.^a

• BPP ⊆ IP.

– IP becomes BPP when the verifier ignores the prover’s messages.

• IP actually coincides with PSPACE.^b

aRecall Proposition 35 on p. 306.

bShamir (1990).

Graph Isomorphism

• V¹ = V₂ = {1, 2, . . . , n}.

• Graphs G¹ = (V₁, E₁) and G₂ = (V₂, E₂) are isomorphic if there exists a permutation π on

{1, 2, . . . , n} so that (u, v) ∈ E¹ ⇔ (π(u), π(v)) ∈ E².

• The task is to answer if G1 ∼= G2.

• No known polynomial-time algorithms.

• The problem is in NP (hence IP).

• It is not likely to be NP-complete.^a

aSch¨oning (1987).

graph nonisomorphism

• V¹ = V₂ = {1, 2, . . . , n}.

• Graphs G¹ = (V₁, E₁) and G₂ = (V₂, E₂) are

nonisomorphic if there exist no permutations π on {1, 2, . . . , n} so that (u, v) ∈ E¹ ⇔ (π(u), π(v)) ∈ E².

• The task is to answer if G¹ ̸∼= G2.

• Again, no known polynomial-time algorithms.

– It is in coNP, but how about NP or BPP?

– It is not likely to be coNP-complete.

• Surprisingly, graph nonisomorphism ∈ IP.^a

aGoldreich, Micali, and Wigderson (1986).

A 2-Round Algorithm

1: Victor selects a random i ∈ { 1, 2 };

2: Victor selects a random permutation π on { 1, 2, . . . , n };

3: Victor applies π on graph Gi to obtain graph H;

4: Victor sends (G1, H) to Peggy;

5: if G₁ ∼= H then

6: Peggy sends j = 1 to Victor;

7: else

8: Peggy sends j = 2 to Victor;

9: end if

10: if j = i then

11: Victor accepts;

12: else

13: Victor rejects;

14: end if

Analysis

• Victor runs in probabilistic polynomial time.

• Suppose G¹ ̸∼= G2.

– Peggy is able to tell which Gi is isomorphic to H, so j = i.

– So Victor always accepts.

• Suppose G¹ ∼= G2.

– No matter which i is picked by Victor, Peggy or any prover sees 2 identical graphs.

– Peggy or any prover with exponential power has only probability one half of guessing i correctly.

– So Victor erroneously accepts with probability 1/2.

• Repeat the algorithm to obtain the desired probabilities.

Knowledge in Proofs

• Suppose I know a satisfying assignment to a satisfiable boolean expression.

• I can convince Alice of this by giving her the assignment.

• But then I give her more knowledge than is necessary.

– Alice can claim that she found the assignment!

– Login authentication faces essentially the same issue.

– See

www.wired.com/wired/archive/1.05/atm pr.html for a famous ATM fraud in the U.S.

Knowledge in Proofs (concluded)

• Suppose I always give Alice random bits.

• Alice extracts no knowledge from me by any measure, but I prove nothing.

• Question 1: Can we design a protocol to convince Alice (the knowledge) of a secret without revealing anything extra?

• Question 2: How to define this idea rigorously?

Zero Knowledge Proofs

An interactive proof protocol (P, V ) for language L has the perfect zero-knowledge property if:

• For every verifier V ^′, there is an algorithm M with expected polynomial running time.

• M on any input x ∈ L generates the same probability distribution as the one that can be observed on the communication channel of (P, V ^′) on input x.

aGoldwasser, Micali, and Rackoff (1985).

Comments

• Zero knowledge is a property of the prover.

– It is the robustness of the prover against attempts of the verifier to extract knowledge via interaction.

– The verifier may deviate arbitrarily (but in

polynomial time) from the predetermined program.

– A verifier cannot use the transcript of the interaction to convince a third-party of the validity of the claim.

– The proof is hence not transferable.

Comments (continued)

• Whatever a verifier can “learn” from the specified prover P via the communication channel could as well be

computed from the verifier alone.

• The verifier does not learn anything except “x ∈ L.”

• Zero-knowledge proofs yield no knowledge in the sense that they can be constructed by the verifier who believes the statement, and yet these proofs do convince him.

Comments (continued)

• The “paradox” is resolved by noting that it is not the transcript of the conversation that convinces the verifier.

• But the fact that this conversation was held “on line.”

• Computational zero-knowledge proofs are based on complexity assumptions.

– M only needs to generate a distribution that is

computationally indistinguishable from the verifier’s view of the interaction.

Comments (concluded)

• It is known that if one-way functions exist, then

zero-knowledge proofs exist for every problem in NP.^a

• The verifier can be restricted to the honest one (i.e., it follows the protocol).^b

• The coins can be public.^c

aGoldreich, Micali, and Wigderson (1986).

bVadhan (2006).

cVadhan (2006).

Are You Convinced?

• A newspaper commercial for hair-growing products for men.

– A (for all practical purposes) bald man has a full head of hair after 3 months.

• A TV commercial for weight-loss products.

– A (by any reasonable measure) overweight woman loses 10 kilograms in 10 weeks.

Quadratic Residuacity

• Let n be a product of two distinct primes.

• Assume extracting the square root of a quadratic residue modulo n is hard without knowing the factors.

• We next present a zero-knowledge proof for the input x ∈ Zn^∗

being a quadratic residue.

Zero-Knowledge Proof of Quadratic Residuacity

1: for m = 1, 2, . . . , log₂ n do

2: Peggy chooses a random v ∈ Zn^∗ and sends y = v² mod n to Victor;

3: Victor chooses a random bit i and sends it to Peggy;

4: Peggy sends z = uⁱv mod n, where u is a square root of x; {u² ≡ x mod n.}

5: Victor checks if z² ≡ xⁱy mod n;

6: end for

7: Victor accepts x if Line 5 is confirmed every time;

A Useful Corollary

Corollary 76 Let n = pq be a product of two distinct

primes. (1) If x and y are both quadratic residues modulo n, then xy ∈ Zn^∗ is a quadratic residue modulo n. (2) If x is a quadratic residue modulo n and y is a quadratic nonresidue modulo n, then xy ∈ Zn^∗ is a quadratic nonresidue modulo n.

• Suppose x and y are both quadratic residues modulo n.

• Let x ≡ a² mod n and y ≡ b² mod n.

• Now xy is a quadratic residue as xy ≡ (ab)² mod n.

The Proof (concluded)

• Suppose x is a quadratic residue modulo n and y is a quadratic nonresidue modulo n.

• By Lemma 75 (p. 636), (x | p) = (x | q) = 1 but, say, (y | p) = −1.

• Now xy is a quadratic nonresidue as (xy | p) = −1, again by Lemma 75 (p. 636).

Analysis

• Suppose x is a quadratic residue.

– Then x’s square root u can be computed by Peggy.

– Peggy can answer all challenges.

– So Victor will accept x.

• Suppose x is a quadratic nonresidue.

– Corollary 76 (p. 664) says if a is a quadratic residue, then xa is a quadratic nonresidue.

– As y is a quadratic residue, xⁱy can be a quadratic residue (see Line 5) only when i = 0.

Analysis (continued)

• (continued)

– Peggy can answer only one of the two possible challenges, when when i = 0.^a

– So Peggy will be caught in any given round with probability one half.

• How about the claim of zero knowledge?

• The transcript between Peggy and Victor when x is a quadratic residue can be generated without Peggy!

• Here is how.

aLine 5 (z² ≡ xⁱy mod n) cannot equate a quadratic residue z² with a quadratic nonresidue xⁱy when i = 1.

Analysis (continued)

• Suppose x is a quadratic residue.^a

• In each round of interaction with Peggy, the transcript is a triplet (y, i, z).

• We present an efficient Bob that generates (y, i, z) with the same probability without accessing Peggy’s power.

aThere is no zero-knowledge requirement when x ̸∈ L.

Analysis (concluded)

1: Bob chooses a random z ∈ Zn^∗;

2: Bob chooses a random bit i;

3: Bob calculates y = z²x⁻ⁱ mod n;^a

4: Bob writes (y, i, z) into the transcript;

aRecall Line 5 on p. 663.

Comments

• Assume x is a quadratic residue.

• For (y, i, z), y is a random quadratic residue, i is a random bit, and z is a random number.

• Bob cheats because (y, i, z) is not generated in the same order as in the original transcript.

– Bob picks Peggy’s answer z first.

– Bob then picks Victor’s challenge i.

– Bob finally patches the transcript.

Comments (concluded)

• So it is not the transcript that convinces Victor, but that conversation with Peggy is held “on line.”

• The same holds even if the transcript was generated by a cheating Victor’s interaction with (honest) Peggy.

• But we skip the details.