# Digital Signatures. Murat Kantarcioglu. Based on Prof. Li s Slides. Digital Signatures: The Problem

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1 Digital Signatures Murat Kantarcioglu Based on Prof. Li s Slides Digital Signatures: The Problem Consider the real-life example where a person pays by credit card and signs a bill; the seller verifies that the signature on the bill is the same with the signature on the card Contracts, they are valid if they are signed. Can we have a similar service in the electronic world? 2 1

2 Digital Signatures Digital Signature: a data string which associates a message with some originating entity. Digital Signature Scheme: for each key, there is a SECRET signature generation algorithm and a PUBLIC verification algorithm. Services provided: Authentication Data integrity Non-Repudiation (MAC does not provide this.) 3 Adversarial Goals Total break: adversary is able to find the secret for signing, so he can forge then any signature on any message. Selective forgery: adversary is able to create valid signatures on a message chosen by someone else, with a significant probability. Existential forgery: adversary can create a pair (message, signature), s.t. the signature of the message is valid. A signature scheme can not be perfectly secure; it can only be computationally secure. Given enough time and adversary can always forge Alice s signature on any message. 4 2

3 Attack Models for Digital Signatures Key-only attack: Adversary knows only the verification function (which is supposed to be public). Known message attack: Adversary knows a list of messages previously signed by Alice. Chosen message attack: Adversary can choose what messages wants Alice to sign, and he knows both the messages and the corresponding signatures. 5 Digital Signatures and Hash Very often digital signatures are used with hash functions, hash of a message is signed, instead of the message. Hash function must be: Pre-image resistant Weak collision resistant Strong collision resistant 6 3

4 RSA Signature Key generation (as in RSA encryption): Select 2 large prime numbers of about the same size, p and q n = pq, and Φ = (q - 1)(p - 1) Select a random integer e, 1 < e < Φ, s.t. gcd(e, Φ) = 1 d, 1 < d < Φ s.t. ed 1 mod Φ Public key: (e, n) Secret key: d, p and q must also remain secret 7 Signing message M M must verify 0 < M < n Use private key (d) compute S = M d mod n RSA Signature (cont.) Verifying signature S Use public key (e, n) S e mod n = (M d mod n) e mod n = M Note: in practice, a hash of the message is signed and not the message itself. 8 4

5 RSA Signature (cont.) Example of forging Attack based on the multiplicative property of property of RSA. y 1 = sig K (x 1 ) y 2 = sig K (x 2 ), then ver K (x 1 x 2 mod n, y 1 y 2 mod n) = true So adversary can create the valid signature y 1 y 2 mod n on the message x 1 x 2 mod n This is an existential forgery using a known message attack. 9 El Gamal Signature Key Generation (as in ElGamal encryption) Generate a large random prime p such that DLP is infeasible in and a generator α of the multiplicative group of the integers modulo p Select a random integer a, 1a p-2, and compute β = α a mod p Public key is (p, α, β) Private key is a Recommended sizes: 1024 bits for p and 160 bits for a. 10 5

6 ElGamal Signature (cont.) Signing message M Select random k, 1 k p-1, k Z p-1 * r = α k mod p s = k -1 (M - ar) mod (p-1) Signature is: (r,s) Size of signature is double size of p NOTE: In practice, instead of M, h(m) is used where h is a hash function 11 ElGamal Signature (cont.) Signature is: (r, s) r = α k mod p s = k -1 ( M - ar) mod (p-1) Verification Verify that r is in Z p-1* : 1 r p-1 v 1 = β r r s mod p v 2 = α M mod p Accept iff v 1 =v

7 ElGamal Signature (cont.) Security of ElGamal signature Weaker than DLP k must be unique for each message signed Hash function h must be used, otherwise easy for an existential forgery attack without h, a signature on M Z p, is (r,s) s.t. β r r s = α M mod p choose u,v s.t. gcd(v,p-1)=1, then let r= α u β v mod p= α u+av mod p, and let s=-rv -1 mod (p-1) then β r r s = α ar (α u+av ) s = α ar g avs g us = α ar α av(-rv^{-1}) α us = α ar α -ar α us = g us i.e., (r,s) is a signature of the message u.s 13 ElGamal Signature (Continued) 0 < r < p must be checked, otherwise easy to forge a signature on any message if an valid signature is available. given M, and r= α k, s=k -1 (M- ar) mod (p-1) for any message M, let u=m /M mod (p-1) computes s =su mod (p-1) and r s.t. r ru (mod (p-1)) AND r r (mod p), then β r r s = β ru r su = (β r r s ) u = (α M ) u = α M 14 7

8 Digital Signature Algorithm (DSA) Specified as FIPS 186 Key generation Select a prime q of 160-bits Choose 0 t 8 Select t < p < t with q p-1 Let α be a generator of Z p*, and set g = α (p-1)/q mod p Select 1 a q-1 β = g a mod p Public key: (p, q, g, β) Private key: a 15 DSA Signing message M: Select a random integer k, 0 < k < q k -1 mod q r = (g k mod p) mod q s = k -1 ( h(m) + ar) mod q Signature: (r, s) Note: FIPS recommends the use of SHA-1 as hash function. 16 8

9 DSA Signature: (r, s) r = (g k mod p) mod q s = k -1 ( h(m) + ar) mod q Verification Verify 0 < r < q and 0 < s < q, if not, invalid w = s -1 mod q u 1 = w h(m) mod q, u 2 = r w mod q v = (g u 1 β u 2 mod p) mod q Valid iff v = r 17 Schnorr Signature Key generation (as in DSA, h:{0,1} * Z q ) Select a prime q Select 1 a q-1 β= g a mod p Public key: (p,q,g,β) Private key: a 18 9

10 Schnorr Signature Signing message M Select random secret k, 1 k q-1 r = g k mod p, e = h(m r) s = ae + k mod q Signature is: (s, e) 19 Schnorr Signature Verification v = g s β -e mod p, Valid iff e = e e = h(m v) 20 10

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