1. 1 4.1 Hash Functions and Data Integrity A cryptographic hash function can provide assurance of data integrity. ex: Bob can verify if y = h K (x) h is a hash function x is a message y is the authentication tag (message digest) K is key AliceBob (x, y)

2. 2 Hash Functions and Data Integrity Definition 4.1: A hash family is a four-tuple ( X, Y, K,H ), where the following condition are satisfied: 1 X is a set of possible messages 2 Y is a finite set of possible message digests or authentication tags 3 K, the keyspace, is a finite set of possible keys 4 For each K  K, there is a hash function h K  H. Each h k : X  Y

3. 3 Hash Functions and Data Integrity h is compress functions X is a finite set Y is a finite set | X |  | Y | or stronger, | X |  2| Y | A pair (x,y)  X  Y is said to be valid under the key K h K (x) = y. Let F X,Y denote the set of all function from X to Y. | X | = N and | Y | = M. | F X,Y | = M N. F  F X,Y is termed an (N,M)-hash family. An unkeyed hash function is a function h: X  Y

4. 4 4.2 Security of Hash Functions If a hash function is to be considered secure, these three problems are difficult to solve Problem 4.1: Preimage Instance: A hash function h: X  Y and an element y  Y. Find: x  X such that f(x) = y Problem 4.2: Second Preimage Instance: A hash function h: X  Y and an element x  X Find: x’  X such that x’ ≠ x and h(x’) = h(x) Problem 4.3: Collision Instance: A hash function h: X  Y. Find: x, x’  X such that x’ ≠ x and h(x’) = h(x)

5. 5 Security of Hash Functions A hash function for which Preimage cannot be efficiently solved is often said to be one-way or preimage resistant. A hash function for which Second Preimage cannot be efficiently solved is often said to be second preimage resistant. A hash function for which Collision cannot be efficiently solved is often said to be collision resistant.

6. 6 Security of Hash Functions 4.2.1 The Random Oracle Model The random oracle model provides a mathematical model of an “ ideal ” hash function. In this model, a hash function h: X  Y is chosen randomly from F X,Y The only way to compute a value h(x) is to query the oracle. THEOREM 4.1 Suppose that h  F X,Y is chosen randomly, and let X 0  X. Suppose that the values h(x) have been determined (by querying an oracle for h) if and only if x  X 0. Then Pr[h(x)=y] = 1/M for all x  X \ X 0 and all y  Y.

7. 7 Security of Hash Functions 4.2.2 Algorithms in the Random Oracle Model Randomized algorithms make random choices during their execution. A Las Vegas algorithm is a randomized algorithm may fail to give an answer if the algorithm does return an answer, then the answer must be correct. A randomized algorithm has average-case success probability ε if the probability that the algorithm returns a correct answer, averaged over all problem instances of a specified size, is at least ε (0≤ε<1).

8. 8 Security of Hash Functions We use the terminology (ε,q)-algorithm to denote a Las Vegas algorithm with average-case success probability ε the number of oracle queries made by algorithms is at most q. Algorithm 4.1: FIND PREIMAGE (h, y, q) choose any X 0  X,| X 0 | = q for each x  X 0 do if h(x) = y then return (x) return (failure)

9. 9 Security of Hash Functions THEOREM 4.2 For any X 0  X with | X 0 | = q, the average-case success probability of Algorithm 4.1 is ε=1 - (1-1/M) q. proofLet y  Y be fixed. Let Χ 0 = {x 1,x..,x q }. For 1 ≤ i ≤ q, let E i denote the event “h(x i ) = y”. From Theorem 4.1 that the E i ’ s are independent events, and Pr[E i ] = 1/M for all 1 ≤ i ≤ q. Therefore The success probability of Algorithm 4.1, for any fixed y, is constant. Therefore, the success probability averaged over all y  Y is identical, too.

10. 10 Security of Hash Functions Algorithm 4.3: FIND COLLISION (h,q) choose X 0  X, | X 0 | = q for each x  X 0 do y x  h(x) if y x = y x’ for some x ’ ≠ x then return (x, x ’ ) else return (failure)

11. 11 Security of Hash Functions Birthday paradox In a group of 23 randomly chosen people, at least two will share a birthday with probability at least ½. Finding two people with the same birthday is the same thing as finding a collision for this particular hash function. ex: Algorithm 4.3 has success probability at least ½ when q = 23 and M = 365 Algorithm 4.3 is analogous to throwing q balls randomly into M bins and then checking to see if some bin contains at least two balls.

12. 12 Security of Hash Functions THEOREM 4.4 For any X 0  X with | X 0 | = q, the success probability of Algorithm 4.3 is proofLet X 0 = {x 1,..,x q }. E i : the event “h(x i )  {h(x 1 ),..,h(x i-1 )}.”, 2  i  q Using induction, from Theorem 4.1 that Pr[E 1 ] = 1 and for 2 ≤ i ≤ q.

13. 13 Security of Hash Functions The probability of finding no collision is ε denotes the probability of finding at least one collision Ignore – q, ε= 0.5, q ≈ 1.17 Take M = 365, we get q ≈ 22.3 x is small 1-x  e -x

14. 14 Security of Hash Functions This says that hashing just over random elements of X yields a collision with a prob. of 50%. A different choice of ε leads to a different constant factor, but q will still be proportional to. So this algorithm is a (1/2, O( ))- algorithm.

15. 15 Security of Hash Functions The birthday attack imposes a lower bound on the size of secure message digests. A 40-bit message digest would be very in secure, since a collision could be found with prob. ½ with just over 2^20 (about a million) random hashes. It is usually suggested that the minimum acceptable size of a message digest is 128 bits (the birthday attack will require over 2^64 hashes in this case). In fact, a 160-bit message digest (or larger) is usually recommended.