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CIS 5371 Cryptography 8. Data Integrity Techniques 1 Asymmetric techniques, I Digital signatures With PK encryption, Alice can use her private key to decrypt a message and the resultant “ciphertext’’ can be “encrypted’’ to recover the message. This ciphertext can serve as a Manipulation Detection Code (MDC). The verification of a MDC can be performed by anyone since the public key is available to anyone. Example of an MDC based on RSA • Let p = 101, q = 113. Then n = 11413. (n) = 100 112 = 11200 = 26527 • Alice takes e = 3533, d = 6597 • Alice publishes: n = 11413, e = 3533. • Let the message be m = 5761 • Alice computes the MDC: • 57616597 (mod 11413) = 9726 • Suppose Bob wants to verify that 9726 is the MDC of Alice • Bob computes 97263533 mod 11413 = 5761 Digital signature schemes • M, message space • S, signature space • K, signing key space • K’, verifying key space • Gen: 1 KK’, an efficient key generating algorithm • Sign: MK S, an efficient signing algorithm • Verify: MSK’ {true,false} an efficient verifying algorithm. The RSA signature scheme Signature setup: n = pq, where p and q are primes. M = S = Zn , with keyspace K = {(n,e,d) : ed = 1 mod (n) }. Public key = (n,e), Private key (n,d). Signature generation: for m Zn, = = Signature Verification , , = if and only if = Security issues for Digital Signatures Active attacks digital signatures • Adaptive Chosen-Message Attack (CMA): – The attacker chooses adaptively a number of messages and obtains the corresponding signatures: the task of the attacker is successful if he can sign a (new) target message. • Existential forgery under CMA: – The algorithms (Sign,Verify) form a one-way trapdoor pair. This means that it is easy to compute valid “message-signature” pairs (by first selecting a signature and then finding the corresponding message). – However, computing message-signature pairs should be hard. – A usual way to control this is add redundancy to the message. Rabin signatures Signature setup: Same as RSA Public key = (n,b), Private key = (p,q). Signature generation: Exercise Signature Verification: Exercise The ElGamal signature scheme Signature setup: Same as ElGamal encryption scheme, with: M = Zp* , S = Zp* Zp-1, and keyspace K = Zp* Zp-1. Public key = (p, g, y) Private key = (p, g, x). The ElGamal signature scheme • Signing Let m Zp* be a message. For public key (p,g,y), with y = gx modp, and a secret random number k Zp-1, define: sigx,k (m,k) = (s,t), where • • • s = gk modp t = (m-xs) k-1 mod (p-1) Verification Verify(p,,g,y)(m,(s,t)) = true st·ys = gm mod p. Toy example Let p = 467, g = 2, x = 127, y = 132 message m = 100, Choose k = 213. Then k -1mod 466 = 431. The signature is: s = 2213 mod 467 = 29 t = (m-xs) k -1 mod (p-1) = (100-127×29) 431 mod 466 = 51 Verification: 2100 ? 2951×13229 mod 467 The security of ElGamal signatures • • • If the DL problem is feasible then ElGamal signatures can be forged. The converse may not be true. The exponent k must be • • • private cannot be used twice best: chosen at random. The Digital Signature Algorithm Let p be a an L-bit prime prime, 512 L 1024 and L 0 mod 64 , let q be a 160-bit prime that divides p-1 and Let e Zp* be a q-th root of 1 modulo p. Let M = Zp-1, S = Zq x Zq and K = {(p,q,,x,y): y = x modp }. • • The public key is (p,q,,y). The private key is (p,q,,x). The Digital Signature scheme • Signing Let m Z be a message. For public key (p,g,,y), with y = x mod p, and secret random number k Zp-1, define: sigx,k(m) = (s,t), where • s = ( k mod p) mod q • t = (SHA(m) + xs) k -1 mod q • Verification Let • e1 = (SHA(m)) t -1 mod q • e2 = s t -1 mod q Verify(p,,y),(m,(s,t)) = true ( e1 y e2 mod p) mod q = s. Provable security Forging signatures • • • • We must how that given a message it is hard to forge a signature. Is this enough? There are several attacks we already discussed: • Existential forgery • Adaptive Chosen-Message Attacks What is really needed is a formal security model for digital signatures, that allows for all possible threat scenarios and all protocol aspects. One such model is the Random Oracle model. Asymmetric techniques, II Data Integrity without source Identification Optimal Asymmetric Encryption Padding RSA-OAEP RSA with OAEP Key Parameters Let (N,e,d,G,H,n,k0,k1) U Gen (1x) satisfy: (N,e,d) are RSA parameters |N| = k = n+k0+k1, with 2k0, 2k1 negligible quantities G, H hash functions with: G: {0,1}k0 {0,1}k-k0 , H: {0,1}k-k0 {0,1}k0 n is the length of the plaintext (n, k0,k1,G,H,e) is Alice’s RSA public key, (n, k0,k1,G,H,d) is Alice’s RSA private key. RSA with OAEP Encryption Let m {0,1}n be the message to be sent to Alice. Bob (the adversary?) performs the following: 1. .r U {0,1}k0 ; s (m || 0k1) G(r) ; t r H(s) 2. .c (s || t) e mod N RSA with OAEP Decryption Upon receipt of the ciphertext c Alice performs: 1. .s || t c d mod N satisfying |s| = n+k1 , |t| = k0 2. .u t H(s); v s G(u) 3. Output m if v = m || 0k0, else reject. RSA with OAEP Security RSA with OAEP provides data-integrity, but not origin integrity. It can be shown that RSA-OAEP is secure against adaptive chosen ciphertext (CCA2) attacks in the Random Oracle Model. The Random Oracle Model (ROM) • • • • • Security is defined in terms of a game involving two parties: the system and the adversary. All authorized parties of the system are represented by random oracles (Alice, Bob, …) Access to any party is via its oracle. Access to an oracle G is by a query a, to get the response G(a). The system of oracles is managed by a Simulator (who arranges that the oracles simulate the behavior of the real parties). The Random Oracle Model • There are two phases: • • • A training phase in which adversary is allowed to make queries (adaptively) and get responses. A test phase in which adversary must answer 0 or 1 as his educated guess to a challenge. The adversary wins if at the test phase he can distinguish with probability better than 0.5 + negl between two strings. e.g. if a public-key encryption system is analyzed, the adversary must distinguish between the ciphertexts c1,c2 of two new messages m1, m2. The Random Oracle Model • • The system is secure if the adversary cannot win. The type of queries the adversary can make is determined by the threat model used. • in CCA2 the adversary can adaptively chose ciphertexts an get the corresponding plaintexts. One-time signatures Lamport signature scheme Let k be an integer, P = {0,1}k. Suppose that f : Y Z is a one-way function, and A = Y k. Let yi,j Y be chosen at random, 1 ≤ i ≤ k, j =0,1, and zi,j = f (yi,j), The key K consists of the 2k pairs : (yi,j, zi,j). The y’s are the private key, the z’s are the pubic key. Lamport signature scheme Signing Let x = (x1,x2, … , xk) P be a message. For K = (yi,j, zi,j) define sigK (x1,x2, … , xk) = (y1,x1,y2,x2, … , yk,xk ) . Verification verK((x1,x2, … xk),(y1,x1,y2,x2, … , yk,xk )) = true f(yi) = zixi , 1 ≤ i ≤ k The security of the Lamport signature scheme The security of the Lamport signature scheme can be proven if we assume that: • • The one-way function is bijective, and that The public key consists of distinct elements. Certificates and Public-Key Infrastructures We have not shown how to distribute securely public keys. We shall now show how this can be done by using public-key cryptography Sounds circular, and is! but works. 26 Certificates and Public-Key Infrastructures A first approach: bootstrap all public-keys to a single one using certificates. To be concrete suppose that Charlie has generated a key pair ( , ). Consider the certificate → = ′ ′ ′ Give this to Bob. Bob publishes: ( , → ) This authenticates the public key of Bob to anyone who trusts Charlie. 27