# MTAT Cryptology II. Digital Signatures. Sven Laur University of Tartu

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1 MTAT Cryptology II Digital Signatures Sven Laur University of Tartu

2 Formal Syntax

3 Digital signature scheme pk (sk, pk) Gen (m, s) (m,s) m M 0 s Sign sk (m) Ver pk (m, s)? = 1 To establish electronic identity, Charlie must generate (pk, sk) Gen and convinces others that the public information pk represents him. A keyed hash function Sign sk : M S takes in a plaintext and outputs a corresponding digital signature. A public verification algorithm Ver pk : M S {0,1} tries to distinguish between altered and original message pairs. The signature scheme is functional if for all (pk, sk) Gen and m M: Ver pk (m, Sign sk (m)) = 1. MTAT Cryptology II, Digital Signatures, 16 April,

4 Key generation Gen: Example. RSA-1024 signature scheme 1. Choose uniformly 512-bit prime numbers p and q. 2. Compute N = p q and φ(n) = (p 1)(q 1). 3. Choose uniformly e Z φ(n) and set d = e 1 mod φ(n). 4. Output sk = (p,q, e, d) and pk = (N, e). Signing and verification: M = Z N, S = Z N, R = Sign sk (m) = m d mod N Ver pk (m,s) = 1 m = s e mod N. MTAT Cryptology II, Digital Signatures, 16 April,

5 Security definitions Attack scenarios Key only attack. The adversary has access only to the public key pk. Chosen message attack. Besides the public key pk, the adversary can adaptively query a list of valid signatures (m 1,s 1 ),...,(m n,s n ). Attack types Universal forgery. The adversary must create a valid signature for a prescribed message m that is chosen from a distribution M 0. Existential forgery. The adversary must create a valid signature for some message m, i.e., there are no limitations on the choice of message. One more signature attack = Chosen message attack + Existential forgery MTAT Cryptology II, Digital Signatures, 16 April,

6 Security against one more signature attack A signature scheme is (t, q, ε)-secure against one more signature attack if for any t-time adversary A where G A Adv forge (A) = Pr[G A = 1] ε 2 (pk, sk) Gen For i {1,..., q} do " mi A(s i 1 ) s i Sign sk (m i ) (m, s) A 6 if (m, s) {(m 4 1, s 1 ),..., (m q, s q )} return 0 return Ver pk (m, s) MTAT Cryptology II, Digital Signatures, 16 April,

7 A conceptual description of digital signatures A digital signature is a non-interactive and transferable counterpart of the following extended identification protocol: 1. Verifier sends a message m to a prover. 2. The prover accepts the message m. 3. The prover authenticates him or herself by using sk. Transferability means that signature must be verifiable by other parties that did not participate in the creation of the signature. As a result, digital signature is either a non-interactive proof of possession that is linked to the message m. a non-interactive proof of knowledge that is linked to the message m. MTAT Cryptology II, Digital Signatures, 16 April,

8 Digital Signatures Based on Proofs of Possession

9 Lamport one-time signatures Public parameters: Let {0,1} n be the message space and f : X Y a one-way function. Key Generation: Generate 2 n random elements x ij u X and compute y ij f(x ij ). The secret key is sk = (x ij ) and the public key is pk = (y ij ). Signing: To sign a message m = m n...m 1 release σ = (x 1m1,...,x nmn ). Verification: A signature (m,σ) is valid if f(σ i ) = y imi for i = 1,...,n. MTAT Cryptology II, Digital Signatures, 16 April,

10 Security of one-time signatures Theorem. If f is (t, ε)-one-way function, then Lamport signature scheme is (t,1, 2nε)-secure against one more signature attack. Proof. A successful forgery must reveal an inverse of y i mi for some i. B(y) Choose 2n 1 elements of (x ij ) u X. Compute 2n 1 entries y ij f(x ij ). Put y to the missing place and send pk to A. Reveal n elements of (x ij ) to A if possible. Return x if A reveals x such that f(x) = y. MTAT Cryptology II, Digital Signatures, 16 April,

11 Merkle signature scheme c c 0 c 1 y 00 y 01 y 10 y 11 x 00 x 01 x 10 x 11 General description Public key pk is a root hash c Secret key sk consists of all leafs x ij. A signature σ is a minimal amount of information needed to recompute c. A secret key can be compressed by a pseudoramndom function family. Detailed description All intermediate values c u are computed by hashing c u h(c u0, c u1 ). The second level values y u are computed as before y u f(x u ). The secret key can be further compacted by computing x u g(u,k) where g : I K X induces a pseudorandom function family G = {g k }. MTAT Cryptology II, Digital Signatures, 16 April,

12 Trapdoor one-way permutations A collection of trapdoor permutations F tp is determined by three algorithms (Gen, Map, Inv) such that (pk, sk) Gen, m M pk : Inv sk (Map pk (m)) = m and both algorithms Map pk ( ) and Inv sk ( ) are deterministic. MTAT Cryptology II, Digital Signatures, 16 April,

13 OW-CPA security A collection of trapdoor permutations F tp is (t, ε)-secure if for any t-time adversary A where Adv inv-cpa (A) = Pr [ G A = 1 ] ε G A (pk,sk) Gen m u M pk y Map pk (m) return [A(pk,y) =? m] MTAT Cryptology II, Digital Signatures, 16 April,

14 Full Domain Hash Setup Run (pk, sk) Gen to create an instance of trapdoor permutation. Choose h : M M pk from a collision resistant function family H. Signing To sign m M, compute x h(m) and output s Inv sk (x). Verification A pair (m, s) is a valid signature if h(m) = Map pk (s). MTAT Cryptology II, Digital Signatures, 16 April,

15 Random oracle model Let us model the hash function by an oracle O pk ( ) that evaluates h u H all where H all = {h : M M pk } is a set of all functions. Thus, the advantage is now computed as average Adv forge H all (A) = 1 H all Adv forge h (A). h H all In reality, we substitute H all with a suitable hash family H and hope Adv forge H (A) = 1 H h H Adv forge h (A) Adv forge H all (A) for all relevant adversaries A A humans devise. MTAT Cryptology II, Digital Signatures, 16 April,

16 Security in the random oracle model Theorem. Let F tp is (t, ε)-secure collection of trapdoor permutations such that Map pk ( ) is always τ-time computable. Then the FDH signature scheme is (t (q h + q s + 1) τ, q s,(q h + q s + 1) ε)-secure against one more signature attack provided that the adversary can do up to q h hash queries. Drawbacks The result holds only in random oracle model. The security bound increases linearly with q h and d s. PSS signature scheme achieves a sublinear bound. The proof cannot be generalised to the standard model [Pallier2007]. MTAT Cryptology II, Digital Signatures, 16 April,

17 The corresponding proof We can always assume that A queries hash for the forgery. The following evaluation strategies lead to identical results O pk (m) if H[m] = then [ H[m] u M pk return H[m] O pk (m) if H[m] = then [ S[m] u M pk H[m] Map pk (S[m]) return H[m] Let ε i be the probability that forgery was successful and the ith hash query corresponded to the message. Then ε i ε, since otherwise we would have a too efficient inverter. MTAT Cryptology II, Digital Signatures, 16 April,

18 Digital Signatures Based on Proofs of Knowledge

19 Sigma protocols α R α β γ = γ sk (α, β) β B sk V pk (α, β, γ) A sigma protocol for an efficiently computable relation R {0, 1} {0,1} is a three move protocol that satisfies the following properties. Σ-structure. A prover first sends a commitment, next a verifier sends varying challenge, and then the prover must give a consistent response. Functionality. The protocol run between an honest prover P(sk) and verifier V(pk) is always accepting if (sk,pk) R. MTAT Cryptology II, Digital Signatures, 16 April,

20 Security properties of sigma protocols α R α β γ = γ sk (α, β) β B sk V pk (α, β, γ) Perfect simulatability. There exists an efficient non-rewinding simulator S such that the output distribution of a semi-honest verifier V in the real world and the output distribution of S V in the ideal world coincide. Special soundness. There exists an efficient extraction algorithm Extr that, given two accepting protocol runs (α, β 0,γ 0 ) and (α, β 1,γ 1 ) with β 0 β 1 that correspond to pk, outputs sk such that (sk,pk) R MTAT Cryptology II, Digital Signatures, 16 April,

21 Fiat-Shamir heuristics Fiat-Shamir heuristics converts any sigma-protocol to a signature scheme. α R α β γ β B Sigma protocols are interactive non-transferrable sk V pk (α,β,γ) and cannot be linked to particular messages α R sk,m α If β h(m, α) then β h(m, α) the signer cannot cheat m the protocol is non-interactive the protocol is transferable s = (α,β,γ) V pk (α, β, γ) h(m, α) =? β MTAT Cryptology II, Digital Signatures, 16 April,

22 Generic signature schemes Let π be a sigma protocol specified by the prover and verifier algorithms P sk and V pk. Then the corresponding generic signature scheme is following. Setup Signing Use output (pk,sk) Gen of the sigma protocol setup. To sign a message m, output (α,β, γ) as a signature where Verification α P sk, β h(m, α), γ P sk (β). A tuple (m, α,β, γ) is a valid signature if β = h(m, α) Ver pk (α, β, γ) = 1. MTAT Cryptology II, Digital Signatures, 16 April,

23 Famous examples Schnorr authentication protocol Schnorr signature scheme DSA algorithm ElGamal signature scheme Nyberg-Rueppel signature scheme Fiat-Shamir identification protocol Feige-Fiat-Shamir signature scheme Guillou-Quisquater identification protocol Guillou-Quisquater signature scheme MTAT Cryptology II, Digital Signatures, 16 April,

24 Security in the random oracle model Basic Forking Lemma. Let κ be the knowledge error of the sigma protocol. If for a particular public key pk, a t-time adversary A manages to output valid signature by making at most l queries to the random oracle and no queries to the signing oracle with probability ε, then there exist an extraction algorithm K for that runs in expected time ( ) (l + 2) t E[τ] = Θ ε κ and returns the corresponding secret key sk. Simulatability Lemma. Oracle calls to the signing oracle can be replaced with the runs of the sigma protocol simulator. The probability of simulation failures due to the contradicting assignments for O( ) are negligible. MTAT Cryptology II, Digital Signatures, 16 April,

25 An encoding of the extraction task Assume that A never queries the same value h(m i, α i ) twice and that A itself verifies the validity of the candidate signature (m n+1, s n+1 ). Let ω 0 denote the randomness used by A and let ω 1,...ω l+1 be the replies for the hash queries h(m i,α i ). Now define A(ω 0, ω 1,...,ω l+1 ) = { i, if the i th reply ω i is used in forgery, 0, if A fails. For any ω = (ω 0,...,ω i 1,ω i,...,ω l+1 ), A behaves identically up to the i th query as with the randomness ω. To extract the secret key sk, we must find ω and ω such that A(ω) = i and A(ω) = i and ω i ω i. MTAT Cryptology II, Digital Signatures, 16 April,

26 Classical algorithm Rewind: 1. Probe random entries A(ω) until A(r, c) Store the matrix location ω and the rewinding point i A(ω). 3. Probe random entries A(ω) until A(ω) = i. 4. Output the location tuple (ω,ω). Rewind-Exp: 1. Repeat the procedure Rewind until ω i ω i. 2. Use the knowledge extraction lemma to extract sk. MTAT Cryptology II, Digital Signatures, 16 April,

27 Average-case running time Theorem. If a array A(ω) with entries in {0,...,l} contains ε-fraction of nonzero entries, then Rewind and Rewind-Exp make on average probes where the knowledge error E[probes Rewind] = 2 ε E[probes Rewind-Exp] = l + 1 ε κ κ = l Pr [ω i = ω i ]. i=1 Proof. We prove this theorem in another lecture. MTAT Cryptology II, Digital Signatures, 16 April,

28 Interpretation of the Results

29 Objective and subjective security All forms of the Forking Lemma are stated with respect to a fixed key and thus give only soundness guarantees. If we believe that no human can devise an algorithm that for a particular public key pk computes the corresponding secret key, then we also get the corresponding subjective security guarantee. For objective security guarantees, we have to consider average success probability over all public keys. In this situation, the knowledge extraction with high probability is overkill. Fixed number of probes together with Jensen s inequality give more tight bounds. All these security guarantees hold in the random oracle model! MTAT Cryptology II, Digital Signatures, 16 April,

30 Average case nature of advantages Adv forge (A h) 1 Trivial ε General purpose Weakly specialised h The limit on the average advantage over all functions means: An attack algorithm A can be successful on few functions For randomly chosen function family H the corresponding average advantage is comparable with high probability over the choice of H. Such argumentation does not rule out possibility that one can choose adaptively a specialised attack algorithm A based on the description of h. MTAT Cryptology II, Digital Signatures, 16 April,

31 Security against generic attacks An adaptive choice of a specialised attack algorithm implies that the attack depends on the description of the hash function and not the family H. Often, it is advantageous to consider only generic attacks that depend on the description of function family H and use only black-box access to the function h. Therefore, we can consider two oracles O Hall and O H. If H is pseudorandom function family then for any generic attack, we can substitute H with the H all and the success decreases marginally. Theorem. Security in the random oracle model implies security against generic attacks if H is a pseudorandom function family. The assumption that attackers use only generic attacks is subjective. Such an assumption are not universal, i.e., there are settings where this assumption is clearly irrational (various non-instantiability results). MTAT Cryptology II, Digital Signatures, 16 April,

32 Why the random oracle argument truly fails? All currently used hash functions are iterative h(m 1,...,m k ) = f( f(f(iv, m 1 ),m 2 ),...,m k ). Consequently, the class of generic attack A bb that treats h( ) as a black box without internal structure is sub-optimal. Instead, we should consider a wider class of attacks that treat f as a black box, but still try to exploit iterative structure of the hash function. However, the corresponding black box adversaries A f( ),h( ) can distinguish H form H all with high probability. As a result, there might be a successful attack strategy A f( ),h( ) that works for all possible iterative hash functions, although the signature scheme is secure in the random oracle model. MTAT Cryptology II, Digital Signatures, 16 April,

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