Security Arguments for Digital Signatures and Blind Signatures Journal of Cryptology, (2000) 13: Authors: D. Pointcheval and J. Stern Presented by J. Liu

Outline Introduction Definitions 1.The random oracle model 2.Digital signature schemes Preliminaries 1. Complexity theory and “Oracle replay attack” 2. Distinguishability of distributions of probability Security arguments for digital signatures

Introduction Provable security has tried to provide proof in the asymptotic framework of complexity theory. That is, poly reductions the problem to well-established problems, such as factorization, DLP, NPC…. One way function  NP vs. P

The random oracle model Hash function (e.g. MD5, SHA1-2, …) long message  short digest. Nonrepudiation  it is impossible to find two different messages providing the same hash value (collision freeness) The hash function can be seen as an oracle which produces a truly random value for each “new” query.

Digital signature schemes 1.Key generation algo. G (probabilistic): input: k and w, output: (Kp, Ks) 2.Signing algo. Σ(may be probabilistic): input: message m, (Kp, Ks) output: signature σ 3.Verification algo. V (not probabilistic): input: m, Kp, σ output: accept or reject

Fig. 1. signature schemes

Example: RSA signature N = pq, ed = 1 mod φ(N) where e is p and d is s. The signature of a message m with respect to d is σ= m d mod N It is not secure under existential forgery. σ’ = σ 2 = (m d ) 2 = (m 2 ) d mod N Not intelligible or without the proper redundancy

Example: Schnorr signature p, q two large prime and q ｜ p-1 with q ≧ 2 k. g  (Z/pZ)* of order q, y = g -x mod p σ= (r, e, s), where r = g K mod p with random K, e = H(m, r) mod q and s =K+ex mod q Verify by e = H(m, g s y e mod p) [g s y e = g K+ex (g -x ) e = g K+ex-ex = g K =r mod p]

No-message attack vs. known- message attack NMA: Attacker only knows public key of the signer. KMA: Attacker can access a list of (m, σ) pairs. 1)Plan known-message attack 2)Generic chosen-message attack 3)Oriented chosen-message attack 4)Adaptively chosen-message attack

Plan known-message attack Attacker has access to a list of signed messages, but he has not chosen them.

Generic chosen-message attack Attacker can choose the list of messages to be signed. This choice must be made before accessing the public key of the signer. That is the choice is independent of the signer.

Oriented chosen-message attack Choose the message for specific signer.

Adaptively chosen-message attack Having knowledge of the public key of the signer, the attacker can ask the signer to sign any message that he wants. He can then adapt his queries according to previous message-signature pairs.

Forgeries Total break: Disclose the secret key of the signer. Universal forgery: Constructing an efficient algorithm which can sign any message. Existential forgery: providing a new message-signature pair. (not dangerous ∵ meaningless)

Secure signature scheme A signature scheme is secure if an existential forgery is computationally impossible, even under an adaptively chosen-message attack.

Preliminaries Complexity theorem and “Oracle replay attack” Distinguishability of distributions of probability

Complexity theorem and “Oracle replay attack” All participants are modeled by probabilistic polynomial time Turing machine. Generic reduction technique. Oracle replay attack: by a polynomial replay of the attacker with different random oracle.Oracle replay attack

 : random tape A query the random oracle Q times,  i is the answer of the i-th query.  +1: the index of Q(m,  1)

Lemmas Splitting lemma Lemma 2 Forking lemma Theorem 2

Splitting lemma

Lemma 2

Forking lemma

Theorem 2

Proof By forking lemma, we obtain 2 valid signatures (m, r, e, s) and (m, r, e’, s’) with e  e’.