Show that this algorithm determines the number of 1 bits in the bit string S:
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a) Show that this
bits in the bit string S:
procedure bit count(S: bit string)
count := 0
while S ≠ 0
count := count + 1
S := S ∧ (S − 1)
return count {count is the number of 1s in S}
Here S − 1 is the bit string obtained by changing the
rightmost 1 bit of S to a 0 and all the 0 bits to the
right of this to 1s. [Recall that S ∧ (S − 1) is the bitwise AND of S and S − 1.]
b) How many bitwise AND operations are needed to find
the number of 1 bits in a string S using the algorithm
in part (a)?
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- CRYPTOGRAPHY.In this problem we will work through a round of DES. For notational simplicity, assume it is the first round. Please enter all answers as strings of 0's and 1's. The input is the 64 bit block 00000000000000100000000000000010000001000000010001000000000000000000000000000010000000000000001000000100000001000100000000000000 Suppose that the subkey for the current round is this 48 bit number: 000000000000000001000000000000000000000000000000000000000000000001000000000000000000000000000000 What does the 64 bit state look like after the IP transformation is applied to the input? Now find L0L0 and R0R0, the left and right halves of the state.L0=L0= R0=R0= What is the result of applying the expansion box to R0R0? E(R0)=E(R0)= What is the result of XORing the subkey with E(R0)E(R0)? k1⊕E(R0)=k1⊕E(R0)= We now apply the S-box transformation. S(k1⊕E(R0))=S(k1⊕E(R0))= Finally we apply the permutation box to complete the function ff. f(R0)=P(S(k1⊕E(R0)))=f(R0)=P(S(k1⊕E(R0)))= We…How do I write a C language program that takes in a positive integer as input, and outputs a string of 1's and O's representing the integer in reverse binary. For an integer x, the algorithm is As long as x is greater than O Output x $ 2 (remainder is either 0 or 1) x = x / 2 Note: The above algorithm outputs the 0's and 1's in reverse order. Ex: If the input is the output is: 011 6 in binary is 110; the algorithm outputs the bits in reverseAnswer questions (a) and (b) below: (a) How many times exactly is the code block below executed? For (i = 1, n) { For (j = 1, i) { For (k = 1, j) { code block } } } Hint: You have to start with n=1, then make assumption what you make expect for any given n = N, and check if the formula you found works for n =N+1. This is what we call prove by induction. (b) What is the theta value of this code segment?
- procedure Horner(c, a_(0), a_(1), a_(2),… , a_(n): real numbers)y := a_(n)for i := 1 to ny := y ∗ c + a_(n−i)return y {y = a_(n)c^(n) + a_(n−1)c^(n−1) + ⋯ + a_(1)c + a_(0)}a) Evaluate 3x^(2) + x + 1 at x = 2 by working througheach step of the algorithm showing the values assigned at each assignment step.b) Exactly how many multiplications and additions areused by this algorithm to evaluate a polynomial ofdegree n at x = c? (Do not count additions used toincrement the loop variable.)We assume that the standard input contains a sequence of non-zero integers between - 121 and 121, which ends with 0. This sequence will be given by the user. 1. Write an algorithm, called Decomposition_Powers_Three, which produces the decomposition of each integer using powers of 3, namely 1, 3, 9, 27, and 81, and the + and – operators. Each power of 3 should appear at most once in the decomposition. Examples: 1 = 1 2 = 3 – 1 3 = 3 4 = 3 + 1 7 = 9 – 3 + 1 14 = 27 – 9 – 3 – 1 43 = 81 – 27 – 9 – 3 + 1 121 = 81 + 27 + 9 + 3 + 1 2. Show that the algorithm Decomposition_Powers_Three is correct using an informal proof (i.e., discussion). 3. Give a program corresponding to Decomposition_Powers_Three, using any of your favorite programming languages. Observation: The intervals [-121,-41], [-40,-14], [-13,-5], [-4,-2], [-1,-1], [1,1], [2,4], [5,13], [14,40], and [41,121] play a particular role. explain the algorithm and output as well. expected output for the program is 1 = 1 2 = 3 – 1 3 = 3…The following questions are independent. 1. A palindrome is a string that reads the same forward and backward. Using our algorithmic language, propose an algorithm that determines whether a string of n characters is a palindrome. 2. Let x be a real number, and n be an integer. Devise an algorithm that computes xn. [Hint: First, give a procedure for computing xn when n is nonnegative by successive multiplication by x, starting with 1 until we reach n. Then, extend this procedure and use the fact that x–n = 1/xn to compute xn when n is negative.] 3. The median of a set of integers is the middle element in the list when these integers are listed in increasing order. The mean of a set of integers is the sum of integers divided by the number of integers in the set. Write an algorithm that produces the maximum, median, mean, and minimum of a set of three integers. Answer only 2 question
- : The following questions are independent. 1. A palindrome is a string that reads the same forward and backward. Using our algorithmic language, propose an algorithm that determines whether a string of n characters is a palindrome. 2. Let x be a real number, and n be an integer. Devise an algorithm that computes xn. [Hint: First, give a procedure for computing xn when n is nonnegative by successive multiplication by x, starting with 1 until we reach n. Then, extend this procedure and use the fact that x–n = 1/xn to compute xn when n is negative.] 3. The median of a set of integers is the middle element in the list when these integers are listed in increasing order. The mean of a set of integers is the sum of integers divided by the number of integers in the set. Write an algorithm that produces the maximum, median, mean, and minimum of a set of three integers.The following questions are independent. 1. A palindrome is a string that reads the same forward and backward. Using our algorithmic language, propose an algorithm that determines whether a string of n characters is a palindrome. 2. Let x be a real number, and n be an integer. Devise an algorithm that computes xn. [Hint: First, give a procedure for computing xn when n is nonnegative by successive multiplication by x, starting with 1 until we reach n. Then, extend this procedure and use the fact that x–n = 1/xn to compute xn when n is negative.] 3. The median of a set of integers is the middle element in the list when these integers are listed in increasing order. The mean of a set of integers is the sum of integers divided by the number of integers in the set. Write an algorithm that produces the maximum, median, mean, and minimum of a set of three integers. Answer 2 and 3 questionsThe following questions are independent. 1. A palindrome is a string that reads the same forward and backward. Using our algorithmic language, propose an algorithm that determines whether a string of n characters is a palindrome. 2. Let x be a real number, and n be an integer. Devise an algorithm that computes xn. [Hint: First, give a procedure for computing xn when n is nonnegative by successive multiplication by x, starting with 1 until we reach n. Then, extend this procedure and use the fact that x–n = 1/xn to compute xn when n is negative.] 3. The median of a set of integers is the middle element in the list when these integers are listed in increasing order. The mean of a set of integers is the sum of integers divided by the number of integers in the set. Write an algorithm that produces the maximum, median, mean, and minimum of a set of three integers. Answer question 3 only with details
- Consider the following algorithm and answer the questions. ALGORITHM X(A[0..n − 1])// Input: A contains n real numbers for i ← 0 to n − 2 do for j ← i + 1 to n − 1 do if A[j] > A[i] swap A[i] and A[j] 1. What does this algorithm compute? 2. What is the input size? 3. What is the basic operation? 4. How many times is the basic operation executed? (Set up a sum, and simplify the sum, to find the function expressing the number of repetitions of the basic operation.) 5. What is the efficiency class of this algorithm?(a) Suppose we agree to represent integers using 8-bit signed integers (also known as the two’scomplement method).(i) Use set-roster notation and decimal notation to describe the set of integers that may be representedusing 8-bit signed integers.Write a program that takes in a positive integer as input, and outputs a string of 1's and 0's representing the integer in reverse binary. For an integer x, the algorithm is: As long as x is greater than 0 Output x % 2 (remainder is either 0 or 1) x = x / 2 Note: The above algorithm outputs the 0's and 1's in reverse order. Ex: If the input is: 6 the output is: 011 6 in binary is 110; the algorithm outputs the bits in reverse.