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Published bySydney Dixon Modified over 8 years ago
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Data Integrity / Data Authentication
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Definition Authentication (Signature) algorithm - A Verification algorithm - V Authentication key – k Verification key – k’ Message space (usually binary strings) Every message between Alice and Bob is a pair (m, A k (m))
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Definition (cont.) Requirement – V k’ (m,A k (m)) = “yes” In the symmetric case –k=k’ –The authentication algorithm is called MAC (Message Authentication Code) –A k (m) is frequently denoted MAC k (m) –Verification is by executing authentication and comparing with MAC k (m))
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Definition (cont.) In the a-symmetric case –k k’ –The authentication algorithm is called “Digital signature” –A k (m) is frequently denoted SIG k (m)
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Model Alice Eve Bob
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Properties of MAC Functions Security requirement – adversary can’t construct a legal pair (m, MAC k (m)) Output should be as short as possible The MAC function is not 1-to-1
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Adversarial Model Security Model: –Known message –Chosen message Note: chosen MAC is less realistic Goal: given n legal pairs (m 1, MAC k (m 1 )), …, (m n, MAC k (m n )) find a new legal pair (m, MAC k (m))
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One-Way Functions A function f: D R is called one-way if: –Computing f(x) is “easy” –Computing f -1 (y) for almost all the images is “hard” Given the “real-world” definition of “hard” a one- way function may have a constant-sized output (e.g. SHA-1) Given the theoretical definition, we require a function with variable sized output or a family of one-way functions of constant sized output
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Example The Domain is all the pairs of prime numbers. The function is f(p,q) = pq Multiplication is easy – naïve algorithm is O(n 2 ) Factoring is difficult – simple algorithm is O(2 n/2 ). NFS and ECM are better but not polynomial. The function f(p,q) = pq (almost) maintains length
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Hash Functions Map large domains to smaller ranges Example h: N {0,1,…,m-1} is defined by h(k) = k mod m Used extensively for searching in hash tables Collisions are resolved by several possible means – chaining, open hashing etc.
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Collision Resistance A hash function h: D R is called weakly collision resistant for x D if it is hard to find x’ x such that h(x’)=h(x) A function h: D R is called strongly collision resistant if it is hard to find x, x’ such that x’ x but h(x)=h(x’) Note: given certain constraints a strongly collision-resistant hash function is a one-way function
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The Birthday Paradox If 23 people are chosen at random the probability that two of them have the same birth-day is greater than 0.5 Let h:D R be a mapping. If 1.17|R| 1/2 elements of D are chosen at random, the probability that two of them are mapped to the same image is greater than 0.5 Leads to the birthday attack
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Cryptographic Hash Functions Cryptographic hash functions are hash functions that are strongly collision resistant. Usually defined by: –Compression function mapping n bits (e.g. 672) to m bits (e.g 160). –Extension via chaining to arbitrary strings. –Padding
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Merkle - Damgard HHH M1M1 M2M2 MkMk IV h(M)
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Merkle - Damgard (cont.) Unlike most encryption systems, the IV is usually constant Typically, there is padding (including text length before the final output) Claim: if the basic function H is collision resistant then so is its extension.
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Lengths Input message length should be arbitrary Block length is usually 512 bits Output length should be at least 160 bits
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Real-World Hash Functions MD family –MD-2 –MD-4 –MD-5 SHA and SHA-1 RIPE-MD SHA-256, 384 and 512
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Hash Function Status - 2005 Internet standards (IPSec, SSL, X.509, many others) use MD-5, SHA-1 almost exclusively Multi-block differential attacks: –First block get near collisions (small number of different bits) –Second block achieve full collision MD-5 fully broken, large number of collisions of Word, Postscript files and others. SHA-1 – collision attack requires 2 63 steps, no practical collisions found yet
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HMAC Attempts First goal: combine message and key, hash and produce MAC Second goal: work with any cryptographic hash function First attempt: MAC k (m)=h(k,m) Second attempt: MAC k (m)=h(m,k)
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HMAC Proposed in 1996 by [BCK] Receives as input a message m, a key k and a hash function h Outputs a MAC by: –HMAC k (m,h)= h(k opad, h(k ipad,m)) Claim [BCK]: HMAC can be broken if and only if the underlying hash function is broken.
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HMAC in Practice SSL / TLS BPI IPSec: –AH –ESP SRTP
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