greatest-common-divisor

I looking for the information about fast GCD computation algorithms. Especially, I would like to take a look at the realizations of that. The most interesting for me: - Lehmer GCD algorithm, - Accelerated GCD algorithm, - k-ary algorithm, - Knuth-Schonhage with FFT. I have completely NO information about the accelerated GCD algorithm, I just have seen a few articles where it was mentioned as the most effective and fast gcd computation method on the medium inputs (~1000 bits) They looks much difficult for me to understand from the theory view. Could anybody please share the code (preferable on

I can't figure out what Knuth meant in his instructions for an exercise 8 from Chapter 1.1. The task is to make an efficient gcd algorithm of two positive integers m and n using his notation theta[j] , phi[j] , b[j] and a[j] where theta and phi are strings and a and b - positive integers which represent computational steps in this case. Let an input be the string of the form a^mb^n . An excellent explanation of Knuth's algorithm is given by schnaader here . My question is how this may be connected with the direction given in the exercise to use his Algorithm E given in the book with original r

Given an array of positive numbers. I want to split the array into 2 different subset(s) such that the sum of their gcd (greatest common divisor) is maximum. Example array: {6,7,6,7} . Answer: The two required subsets are: {6,6} and {7,7} ; their respective gcd(s) are 6 and 7, their sum = 6+7=13 ; which is the maximum gcd summation possible. Gcd: Gcd of {8,12} is {4} as 4 is the greatest number which divides 8 as well as 12. Note: gcd(X)=X in case the subset contains only one element. My approach: By brute-forcing, I find all possible sub-sequences of the array,then, I find the maximum sum,

I'm a high school student writing a paper on RSA, and I'm doing an example with some very small prime numbers. I understand how the system works, but I can't for the life of me calculate the private key using the extended euclidean algorithm. Here's what I have done so far: I have chosen the prime numbers p=37 and q=89 and calculated N=3293 I have calculated (p-1)(q-1)=3168 I have chosen a number e so that e and 3168 are relatively prime. I'm checking this with the standard euclidean algorithm, and that works very well. My e=25 Now I just have to calculate the private key d, which should

One way is to calculate their gcd and check if it is 1. Is there some faster way? The Euclidean algorithm (computes gcd ) is very fast. When two numbers are drawn uniformly at random from [1, n] , the average number of steps to compute their gcd is O(log n) . The average computation time required for each step is quadratic in the number of digits. There are alternatives that perform somewhat better (i.e., each step is subquadratic in the number of digits), but they are only effective on very large integers. See, for example, On Schönhage's algorithm and subquadratic integer gcd computation .

Given an array A of length n = 10^5. I have to find the sum of GCD of all subarrays of this array efficiently. import math def lgcd(a): g = a[0] for i in range(1,len(a)): g = math.gcd(g,a[i]) return g n = int(input()) A = list(map(int,input().split())) ans = 0 for i in range(n): for j in range(i,n): ans+=lgcd(A[i:j+1]) print(ans) The first thing that should be marked is that GCD(A[i-1 : j]) * d = GCD(A[i : j]) where d is natural number. So for the fixed subarray end, there will be some blocks with equal GCD, and there will be no more than *log n* blocks, since GCD in one block is the divisor