Table of Contents
What does the Chinese remainder theorem find?
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two …
How do you calculate AB mod n?
If n is a prime number, pow(a,n-1)%n==1 (assuming infinite digit integers). So, basically you need to calculate pow(a,b%(n-1))%n ; According to group theory, you can find e such that every other number is equivalent to a power of e modulo n .
How do you solve a quadratic equation with modular?
This Web application can solve equations of the form a x ² + b x + c ≡ 0 (mod n ) where the integer unknown x is in the range 0 ≤ x < n . In particular, it can find modular square roots by setting a = -1, b = 0, c = number whose root we want to find and n = modulus .
How do you solve modulo equations?
How to calculate the modulo – an example
- Start by choosing the initial number (before performing the modulo operation).
- Choose the divisor.
- Divide one number by the other, rounding down: 250 / 24 = 10 .
- Multiply the divisor by the quotient.
- Subtract this number from your initial number (dividend).
Why is it called Chinese Remainder Theorem?
Chinese remainder theorem, ancient theorem that gives the conditions necessary for multiple equations to have a simultaneous integer solution. The theorem has its origin in the work of the 3rd-century-ad Chinese mathematician Sun Zi, although the complete theorem was first given in 1247 by Qin Jiushao.
How can we calculate AB mod C quickly if B is a power of 2?
How can we calculate A^B mod C quickly for any B?
- Step 1: Divide B into powers of 2 by writing it in binary. Start at the rightmost digit, let k=0 and for each digit:
- Step 2: Calculate mod C of the powers of two ≤ B. 5^1 mod 19 = 5.
- Step 3: Use modular multiplication properties to combine the calculated mod C values.
What is the smallest number that has the Chinese Remainder Theorem?
Input: num [] = {3, 4, 5}, rem [] = {2, 3, 1} Output: 11 Explanation: 11 is the smallest number such that: (1) When we divide it by 3, we get remainder 2. (2) When we divide it by 4, we get remainder 3. (3) When we divide it by 5, we get remainder 1. Chinese Remainder Theorem states that there always exists an x that satisfies given congruences.
When is the Chinese Remainder Theorem used in RSA?
Most implementations of RSA use the Chinese remainder theorem during signing of HTTPS certificates and during decryption.
How do you find the remainder when divided by 60?
And 11 is one solution, other solutions are 71, 131, .. etc. All these solutions produce the same remainder when divided by 60, i.e., they are of form 11 + m*60 where m >= 0. A Naive Approach to find x is to start with 1 and one by one increment it and check if dividing it with given elements in num [] produces corresponding remainders in rem [].
What problem does Gauss illustrate the Chinese Remainder Theorem?
Gauss illustrates the Chinese remainder theorem on a problem involving calendars, namely, “to find the years that have a certain period number with respect to the solar and lunar cycle and the Roman indiction.”