Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 11.5, Problem 1E
Program Plan Intro
To show that the probability of no collision is
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Computer Science
(a) Let h be a collision resistant hash function. Now consider the following function H that is defined for binary strings of even length. Let x = x1||x2 with x1 ∈ F2n and x2 ∈ Fn2 . Then
H(x) = h(x1 ⊕ x2).
Prove that this function is not collision resistant.
The hash function
h(k)= k mod m
and linear probing are used to insert the keys
37, 38, 72, 48, 98, 11, 56
into a hash table of size = 7 having indices 0...6. The order of the hash codes in the
array are given by
a. 72, 11, 37, 38, 56, 98, 47
b. 98, 56, 37, 38, 72, 11, 48
c. 98, 11, 37, 38, 72, 56, 48
d. 11, 37, 48, 38, 72, 98, 56
e. 11, 38, 98, 37, 72, 48, 56
Given:• a hash function: h(x) = | 3x + 1 | mod M• bucket array of capacity 'N'• set of objects with the folloeing keys: 12, 44, 13, 88, 23, 94, 11, 39, 20, 16, 5 (to input from left to right)
1. What would be the hash table where M=N=11 and collisions are taken care of using linear probing?
2. What would be the hash table where M=N=11 and collisions are taken care of using separate chaining?
3. Would a size N for the bucket array be able to exist, so that no collisions happen with thehash function h(x) = | 2x + 5 | mod 11 and the keys above?
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- code in python attached please Define a hash table with an associated hash function ℎ(?)h(k) mapping keys ?k to their associated hash value. b) In simple uniform hashing, each key is assumed to have equal probability to map to any of the hashes in a given table of size m. Given an open-address table of size 500500 and 22 random keys, what is the probability that they hash to the same value? What is the probability that they hash to different values?arrow_forwardLet S be the set consisting of all 10 digits and all upper- and lowercase letters. Let U be the set of all (ordered) strings consisting of exactly four elements from S. Construct an injective hash function h: U -> N with the smallest possible value of maxUh(u). Modify your hash function to give a simply uniform map h' to the set {0, ..., 30}. In other words, if U is a probability space with P(u) = P(u') for all u, u' in U, then the random variable h': U -> {0, ..., 30} should have a uniform distribution..arrow_forward1 2 3 4 6 7 8 A E F 10 11 12 13 14 15arrow_forward
- security pr As we mentioned in class, a universal hash function is a function UH(K, M) that takes a key K, a message M and produces a fixed length digest. The universal hash is defined to be "secure" if for any two messages M₁ and M2, if K is selected uniformly at random, then the probability that UH(K, M₁) UH(K, M₂) is approximately zero. == Suppose that H is a secure hash function. Is UH(K, M) = H(K||M) a secure universal hash function? Either prove the answer is "yes" using the security properties of H (which can be assumed), or show how the security of UH could be violated. VOarrow_forwardGiven input {71, 22, 21, 99, 53, 69, 39} and a hash function h(x) = x mod 10, show the resulting: Hash table with double hash function (Use: h(x)=5-(x mod 5) ) and give the time complexity of search and insert operation with big-oh notation.arrow_forwardThis is question which I want you to solve but solve with this step and draw Q/Given input (4371, 1323, 6173, 4199, 4344, 9679, 1989) and hash function h(x):=xmod 10 show the resulting a. Separate chaining hash table.. B.Hash table using linear probing. C. Hash table using quadratic probing This is examples pls follow step same step when you solve questions up EX: given input (89, 18, 49, 58, 69) and a hash function h(x) = x mod 10, show the resulting: 1. Hash table using linear probing. 2. Hash table using quadratic probing. Sol: 1- Linear probing 9 89 0 49 في حالة وجود تصادم نتحرك بمقدار واحد لتفادي التصادم H(x)=x mod 10 1 2 3 4 58 89 2- quadratic probing. H(x)=x mod 10 0 1 2 49 58 3 69 5 6 H(x)=x + imod 10 4 5 7 6 نبدي من 1 وفي حاله حصل تصادم نزيد واحد 8 7 18 8 18 9 89arrow_forward
- Consider the following sequence of keys to be inserted in turn in a hash table of size 7 that uses quadratic probing to resolve collisions (only use +i2 in probing, and do not use -i2 in probing) : 19, 26, 3, 20, 2, 10 Select the choice that shows what the hash table looks like after all keys are inserted (X indicates an empty slot). Question 29 options: [20, 3, X, 26, x, 19, 2] [x, 2, 3, 20, X, 19, 26] [20, X, 2, 3, x, 19, 26] [x, 3, 2, 26, X, 19, 20] [20, X, 2, 3, 19, x, 26]arrow_forwardConsider a hash table with `n` slots and a hash function that distributes keys uniformly across the slots. Suppose `m` keys are inserted into the hash table, where `m` is much larger than `n`. What is the expected number of collisions? Assume that each key is equally likely to be hashed to any of the `n` slots, and that each key is independent of all other keys. Express your answer in terms of `n` and `m`.arrow_forwardFor the operation below, provide the worst case running time in terms of n (Big O notation). Briefly justify your answer. >>> Find the maximum value in hash table where linear probing (+1 probe) is used to resolve collisions. Assume: table size = n2 and there are currently n items in the hash tablearrow_forward
- Consider a hash table of size M using separate chaining with ordered lists and the hash function hash(k) = k mod M. a) Assume N items have already been inserted in the table. What is, in average, the cost (number of operations) needed for searching an item in the table? Justify your answer. b) Assume N items have already been inserted in the table. What is, in average, the cost (number of operations) needed for inserting the next (N+1)th item? Justify your answer.arrow_forwardConsider to use RSA with a known key IK to construct a cryptographic hash function H as follow: Encrypt the first block, XOR the result with the second block and encrypt again, etc. Then, the last ciphertext block is the hash value. For example, H(M,M2) = Enc(IK, Enc(IK, M,) O M2) = h. Show that this H does not satisfy the property of second image resistance. That is, we can find N1 and N2 such that H(N,N2)=h.arrow_forwarda) Consider inserting the keys 10, 22, 31, 4, 15, 28, 17, 88, 59 into a hash table of length m = 11 using open addressing with linear probing, we use the hash function h(k, i) = (k+ i) mod m.arrow_forward
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