by Anuj Pahade

How to implement the Chinese Remainder Theorem in Java

“black bunting flags” by Adi Goldstein

This post assumes that you know what Chinese Remainder Theorem (CRT) is and focuses on its implementation in Java. If you don’t, I’d recommend you read about it here.

You can find the link to the complete code at the end of this post. So let’s get started.

What do we need to find?

We need to find X. ?

The statement goes as follows:

There exists a minimum positive number X such that:

X % number[0]    =  remainder[0], X % number[1]    =  remainder[1], ...............X % number[k-1]  =  remainder[k-1]

So, here we have two arrays.

  1. The array of numbers: All the numbers in this array are pairwise relatively prime. Which means, pick any two numbers from the array, you’ll find that their greatest common divisor is 1.
  2. The array of remainders: As you can see in the expressions above, when X is divided by a number n from the number array it leaves a respective remainder from the remainder array.

Steps to implement the CRT

These are the steps, or as we engineers say, the ‘algorithm’, to implement CRT.

Step 1: Find the product of all the numbers in the first array.

for(int i=0; i<number.length; i++ ){   product *= number[i];}

Step 2: Find the partial product of each number.

Partial product of n= product/n

for(int i=0; i<num.length; i++){   partialProduct[i] = product/number[i];}

3. Find the modular multiplicative inverse of number[i] modulo partialProduct[i].

Here we find the inverse using the extended Euclidean algorithm. So, we call computeInverse(partialProduct[i],num[i])

public static int computeInverse(int a, int b){         int m = b, t, q;         int x = 0, y = 1;               if (b == 1)             return 0;               // Apply extended Euclid Algorithm         while (a > 1)         {             // q is quotient             q = a / b;             t = b;                   // now proceed same as Euclid's algorithm             b = a % b;a = t;             t = x;             x = y - q * x;             y = t;         }               // Make x1 positive         if (y < 0)          y += m;               return y;     }

Step 4: Final Sum

sum += partialProduct[i] * inverse[i] * rem[i];

Step 5: Return the smallest X

In order to find the smallest of all solutions, we divide the sum from step 4 by the product from step 2.

return sum % product;

Thus, we have found our X. I’d recommend you to try to implement the code on your own before looking at the code in the link below.

Thanks for reading the post. I hope it helped you. Leave suggestions in the comments below or reach out to me with a better version of this code or queries on anujp5678[at]gmail[dot]com or connect with me on LinkedIn.

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Have fun and happy coding! :)