Explain step by step and provide all correct answers (1 point) The Chinese Remainder Theorem is often used as a way to speed up modular exponentiation. Inthis problem we go through the procedure of using CRT.Suppose we want to compute xd mod N, where x = 2956, d=3323, and N = 5191.To use the CRT technique we must know the factorization of N. In the case of this problem, Np =179 and q = 29.=pq whereStep 1)We first compute xp= x mod Ρ and = xᏆ qmod q.xpx q=Step 2)We compute the exponents dp = d mod p 1 and da:=d mod q-1. Notice that this step usesFermat's Little Theorem.dpda==Step 3)In this stage we do the exponentiation in the smaller groups.=Ур xpdp mod p ==Уд xqda mod q=29Step 4)We now return to the big group using the formula y=qCpYpPcqyq mod N.In this formula, Cp =q-1 mod Ρ1and =СаΡmod q.Ср=Са=And finally,y ==

The Chinese Remainder Theorem (CRT) is a powerful tool in number theory, often used to simplify…

Answered: (1 point) The Chinese Remainder Theorem…

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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USING PYTHON A tridiagonal matrix is one where the only nonzero elements are the ones on the main diagonal (i.e., ai,j where j = i) and the ones immediately above and belowit(i.e.,ai,j wherej=i+1orj=i−1). Write a function that solves a linear system whose coefficient matrix is tridiag- onal. In this case, Gauss elimination can be made much more efficient because most elements are already zero and don’t need to be modified or added. Please show steps and explain.
Determine φ (m), for m=12,15, 26, according to the definition: Check for each positive integer n smaller m whether gcd(n,m) = 1. (You do not have to apply Euclid’s algorithm.)
Write pseudocode for an algorithm for finding the real roots of equation ax2 + bx + c for arbitrary real coefficients a, b, and c. (You may assume the availability of the square root function sqrt(x).)
Wilson's Theorem states that for any natural number n >1, n is prime if and only if (n – 1)! = -1 (mod n) Write a function wilson (n) that accepts a natural number n, and returns the remainder of (n – 1)! +1 after division by n. Note: you cannot use numpy here.
Consider nonnegative integer solutions of the equation x1+x2+x3+x4+x5+x6=30. How many different solutions are there? How many solutions also satisfy: for every i∈{1,2,3,4,5,6}, xi is positive and even?
Program the Gaussian elimination method with no partial pivoting for solving a linear system of the form Ax=b, where b is a single column vector. Your function should take in a coefficient matrix A, and a single input vector b. Your function should return the solution vector x. Your code should also return the appropriate error message. The first line of your function should look like: function x = gaussElimination (A,b)
Step 3) In this stage we do the exponentiation in the smaller groups. Xp Ур x q = Cp Cq da = Yq dp Step 4) We now return to the big group using the formula x = qcp xp + pcqq mod N. qx In this formula, Cp Cq = p¯¹ = = mod p = And finally, X = mod q = = mod q. q-¹ mod p and
suppose a computer solves a 100x100 matrix using Gauss elimination with partial pivoting in 1 second, how long will it take to solve a 300x300 matrix using Gauss elimination with partial pivoting on the same computer? and if you have a limit of 100 seconds to solve a matrix of size (N x N) using Gauss elimination with partial pivoting, what is the largest N can you do? show all the steps of the solution
The stock span problem is a financial problem where we have a series of n daily price quotes for a stock and we need to calculate the span of stocks price for all n days. The span S; of the stocks price on a given day i is defined as the maximum number of consecutive days just before the given day, for which the price of the stock on the current day is less than or equal to its price on the given day. For example, if an array of 7 days prices is given as {100, 80, 60, 70, 60, 75, 85}, then the span values for corresponding 7 days are {1, 1, 1, 2, 1, 4, 6}. Example 1: Input: N = 7, price[] Output: 1 1 1 2 1 4 6 = [100 80 60 70 60 75 85]
You suppose If you pass n=2 to your function your 2nd term of Fibonacci series is 1 and (n-1)th= 1" term of Fibonacci series is 0 and (n+1)th= 3rd term of Fibonacci series is 1..
Let x and y be integers such that x = 3 (mod 10) and y = 5 (mod 10). Find the integer z such that 97x + 3y³ z (mod 10) and 0 ≤ z ≤9.
a) Arrange the following functions into increasing order; that is, by asymptotic growth rate. 2181gn + (√n)¹gn 8lgn 0⁰, (logn)!, 21gn, 32n log(n!), lg(2¹8lgn), lg(n+10)", (1+2+3+.+ n) b) Your lecturer is a funny guy; he wants you to find out from a sorted list that contains distinct integers, whether there is an index i for which the value of the element i is also i. For example, given a sorted list of distinct integers A[1], A[2], A[3],..., A[n], find whether there is an index i for which A[i] = i. Your lecturer wants you to device an algorithm, that performs the described function, that runs in complexity time 0(log n). An example, A = [-1, 0, 1, 3, 7, 8, 9, 10] does have such an index, that is, A[3] = 3.

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