# Extended Euclidean Algorithm (The Pulveriser) - Explanation and Implementation

## A little Background

*The Extended Euclidean Algorithm* is, as the name suggests, an extension to the Euclidean algorithm and in addition to ** gcd,** it also computes the coefficients of Bézout's identity. The algorithm has roots dating back to ancient India where it was called

**Kuṭṭaka**, meaning

**The Pulveriser**, considered as the precursor to what is today known as the

*Extended Euclidean Algorithm*.

## Applications

One of the most common uses of the *Extended Euclidean Algorithm* is to solve linear *Diophantine* equations.

It is also used extensively in the field of cryptography, *Extended Euclidean Algorithm* forms the basis of computing *Modular Multiplicative Inverse,* which is a key step required in the famous **RSA** public key cryptographic algorithm to derive the key pairs.

## Bézout's Identity

Bézout's Identity is the following theorem in Number Theory(from Wikipedia).

Let

aandbbe integers with greatest common divisord. Then there exist integersxandysuch thatax+by=d. More generally, the integers of the formax+byare exactly the multiples ofd.

or informally

of two integersgcdandacan be expressed as an integer linear combination ofbanda.b

## The Algorithm

### Euclid's Algorithm

According to Euclid's algorithm

gcd(a,b) = gcd(b, a%b)

this essentially means that **gcd** of **a** and **b** is the same as **gcd** of **b** and the remainder of **a/b**, if we compute this recursively until **b** becomes **0**(this also happens to be the base case) then we are left with **a**, and this new value of **a** is the **gcd** of the two original numbers **a** and **b**.

Here are the high-level steps of Euclid's Algorithm

Step 1(base case): if

`b`

is 0, then`a`

is thegcdStep 2: else set

`a = previous b`

and set`b = a mod b`

.Step 3: Repeat Step 2 until we arrive at the base case.

Here's a Java implementation of Euclid's Algorithm.

```
public int gcd(int a, int b){
if(b==0)
return a;
return gcd(b,a%b);
}
```

### Extended Algorithm

One of the key concepts that we'll be using is the basic *Division Algorithm* i.e

For any given two positive integers

andx, there exist two integersyandqwhich satisfy the equationr

`x = qy + r where 0 <= r <= y`

`x -> Dividend, y -> Divisor`

`q -> Quotient, r -> remainder`

If you think about the previously shown Euclid's Algorithm again, you'll realize that **gcd** is actually the last **non-zero** remainder from all of the performed steps.

And, from the *Division Algorithm* we just saw, we have a way to represent the remainder in terms of a linear combination of the **quotient, dividend, and divisor**, all we need to do is rearrange the equation the way we want it.

`x = qy + r`

can also be written as`r = x - qy`

As part of the Extended Euclidean Algorithm, we just need to maintain some extra information at every step of the basic Euclid's Algorithm.

So, we now have the following important observations/tools that will help us compute Bézout's coefficients.

1.gcd(a,b) = gcd(b, a mod b)

2. gcd(a,b) is the last non-zero remainder.

3. r = x - qy

Now, all we need to do is represent the remainder **r** as an integer linear combination of **a** and **b** at every step of Euclid's algorithm and keep track of the coefficients and keep on updating them, once we are at the final step of the algorithm, we will have the last non-zero remainder expressed as the linear combination, which is also the **gcd** and that is exactly what we need.

It will be easier to understand with an example.

### Example

Let's take two integers **a=785646, b=252**

Step 1)

```
785646 = (3117 * 252) + 162
// can also be written as
162 = 785646 - (3117 * 252)
//replace values with a and b
162 = 1*a - 3117*b
```

Step 2) We now have **a=252, b=162**

```
252 = (1 * 162) + 90
// or
90 = 252 - (1 * 162)
// from previous step we know that
b=252
162 = 1*a -3117*b
//after substituting these values we get
90 = b - (a - 3117b)
90 = -a + 3118b
```

Step 3) **a=162, b=90**

```
162 = 90 + 72
// or
72 = 162 - 90
// from previous steps
162 = 1*a -3117*b
90 = -a + 3118b
// substitue values
72 = (a - 3117b) - (-a + 3118b)
72 = 2a - 6235b
```

After repeating the steps few more times you'll finally end up with the following result.

Step | a | b | r | r as a linear combination |
---|---|---|---|---|

1. | 785646 | 252 | 162 | a - 3117b |

2. | 252 | 162 | 90 | -a + 3118b |

3. | 162 | 90 | 72 | 2a - 6235b |

4. | 90 | 72 | 18 | -3a + 9353b |

5. | 72 | 18 | 0 |

At **Step 5** we get the remainder as 0, this means the **gcd** of **a** and **b** is **18** and from **Step 4** we have **18** expressed as the linear combination of **a** and **b.**

So here we have it, gcd expressed as the linear combination `-3a + 9353b`

Bézout's coefficients:

s= -3

t = 9353

gcd(785646,252) = (-3 * 785646) + (9353 * 252) = 18

## Implementation in Java

```
public static void extendedEuclidean(long a, long b){
long oldR = a, r=b;
long oldS = 1, s = 0;
long oldT = 0, t = 1;
while(r != 0){
var quotient = oldR / r;
var rTemp = r;
r = oldR - quotient * r; oldR = rTemp;
var sTemp = s;
s = oldS - quotient * s; oldS = sTemp;
var tTemp = t;
t = oldT - quotient * t; oldT = tTemp;
}
System.out.printf("Coefficients: s:%s, t:%s%n", oldS,oldT);
System.out.printf("GCD:%s%n", oldR);
}
```