# Extended Euclidean Algorithm

Extended Euclidean Algorithm is an extension of https://forthright48.com/2015/07/euclidean-algorithm.html which finds two things for integer $a$ and $b$:

- It finds the value of $GCD(a,b)$.
- It finds two integers $x$ and $y$ such that, $ax + by = gcd(a,b)$.

The expression $ax + by = gcd(a,b)$ is known as Bezout’s identity and the pair $(x,y)$ that satisfies the identity is called Bezout coefficients. We will look into Bezout’s identity at the end of this post. For now, just know the name.

# How It Works

In Euclidean Algorithm we worked with remainders only. In Extended Euclidean Algorithm (ext_gcd) we will use the quotient and few other extra variables. Suppose we are finding $GCD(240,46)$. Using Euclidean Algorithm the steps for finding $GCD$ would be:

$GCD(240,46) = GCD(46, 240 \ \% \ 46) = GCD(46,10)$

$GCD(46,10) = GCD(10, 46 \ \% \ 10) = GCD(10,6)$

$GCD(10,6) = GCD(6, 10 \ \% \ 6) = GCD(6,4)$

$GCD(6,4) = GCD(4, 6 \ \% \ 4) = GCD(4,2)$

$GCD(4,2) = GCD(2, 4 \ \% \ 2) = GCD(2,0)$

$GCD(2,0) = 2$

## Introducing $r$ and $q$

We will slowly move towards ext_gcd algorithm. Let us add two new variables. $r$ represents remainder and $q$ means quotient.

Let $r_0$ be $a$ and $r_1$ be $b$. At each step, we will calculate $r_i$. Let $q_i = \lfloor \frac{r_{i-2}}{ r_{i-1}} \rfloor$. Therefore, $r_i = r_{i-2} – q_i \times r_{i-1}$.

Then the above steps will look like the following:

Index i | Quotient $q_i$ | Remainder $r_i$ |
---|---|---|

$0$ | $240$ | |

$1$ | $46$ | |

$2$ | $240 / 46 = 5$ | $240 – 5 \times 46 = 10$ |

$3$ | $46 / 10 = 4$ | $46 – 4 \times 10 = 6$ |

$4$ | $10 / 6 = 1$ | $10 – 1 \times 6 = 4$ |

$5$ | $6 / 4 = 1$ | $6 – 1 \times 4 = 2$ |

$6$ | $4 / 2 = 2$ | $4 – 2 \times 2 = 0$ |

This table is same as the calculation for Euclidean algorithm except for few extra details. Note that the line before last ( index $5$ ) contains the $GCD(a,b)$.

## Introducing $x$ and $y$

We are trying to express $GCD(a,b) = ax + by$. So the variable $x$ and $y$ will hold the coefficients. To be exact we will write each row as terms of $a$ and $b$, i.e, $r_i = ax_i + by_i$.

Initially, $(x_0, y_0) = (1,0)$ and $(x_1, y_1) = (0,1)$. But how do we calculate $(x_i,y_i)$?

We know that $r_i = r_{i-2} – q_i \times r_{i-1}$. We also claimed that $r_i = ax_i + by_i$. By combining this two we get

$r_i = ( ax_{i-2} + by_{i-2} ) – q_i \times ( ax_{i-1} + by_{i-1} )$

$r_i = ax_{i-2} – q_i \times ax_{i-1} + by_{i-2} – q_i \times by_{i-1}$

$r_i = a ( x_{i-2} – q_i \times x_{i-1} ) + b ( y_{i-2} – q_i \times y_{i-1})$

$r_i = a x_i + b y_i$

$\therefore x_i = x_{i-2} – q_i \times x_{i-1} \ \text{and} \ y_i = y_{i-2} – q_i \times y_{i-1}$.

Our new table enhanced with $x$ and $y$ will now look like:

Index | Quotient $q_i$ | Remainder $r_i$ | $x_i$ | $y_i$ |
---|---|---|---|---|

$0$ | $240$ | $1$ | $0$ | |

$1$ | $46$ | $0$ | $1$ | |

$2$ | $240 / 46 = 5$ | $240 – 5 \times 46 = 10$ | $1 – 5 \times 0 = 1$ | $0 – 5 \times 1 = -5$ |

$3$ | $46 / 10 = 4$ | $46 – 4 \times 10 = 6$ | $0 – 4 \times 1 = -4$ | $1 – 4 \times -5 = 21$ |

$4$ | $10 / 6 = 1$ | $10 – 1 \times 6 = 4$ | $1 − 1 × −4 = 5$ | $−5 − 1 × 21 = −26$ |

$5$ | $6 / 4 = 1$ | $6 – 1 \times 4 = 2$ | $−4 − 1 × 5 = −9$ | $21 − 1 × −26 = 47$ |

$6$ | $4 / 2 = 2$ | $4 – 2 \times 2 = 0$ | $5 − 2 × -9 = 23$ | $−26 − 2 × 47 = −120$ |

Our answer lies on the line before last. $240 \times -9 + 46 \times 47 = 2$.

So all we need to do now is implement these steps in code.

# Code

Even though we will be calculating many rows in ext_gcd algorithm, in order to calculate any row we just need information from previous two rows. So we can save memory by simply storing $r_{i-2}, r_{i-1}, x_{i-2}, x_{i-1}, y_{i-2}, y_{i-1}$.

In our code, $x2$ is $x_{i-2}$, $x1$ is $x_{i-1}$ and $x$ is $x_i$. Same style is followed for other variable.

int ext_gcd ( int A, int B, int *X, int *Y ){ int x2, y2, x1, y1, x, y, r2, r1, q, r; x2 = 1; y2 = 0; x1 = 0; y1 = 1; for (r2 = A, r1 = B; r1 != 0; r2 = r1, r1 = r, x2 = x1, y2 = y1, x1 = x, y1 = y ) { q = r2 / r1; r = r2 % r1; x = x2 - (q * x1); y = y2 - (q * y1); } *X = x2; *Y = y2; return r2; }

In line $1$ we define the function. The function `ext_gcd()`

takes 4 parameters. The first two parameter $A$ and $B$ represents the two integers whose gcd we wish to find. The next two parameters `*X`

and `*Y`

are two integer pointers. Our function returns us $GCD(A,B)$ but we also want to find the coefficients $x,y$ such that $ax + by = GCD(A,B)$. So we extract those values using pointers.

In line $2$, we declare all necessary variables. We initiate some variables in line $3$ and $4$. Then a for loop starts in line $5$. We initiate few more variables in first section, $r2$ and $r1$. The loop runs till $r1$ becomes 0. In the last section of for loop, we make transitions of variables, such as $r2$ becomes $r1$ and $r1$ gets the new value $r$.

Inside the for loop we calculate values needed for the current row, $q, r, x, y$.

When the loop finishes, $r1$ contains $0$ and $r2$ is the row before it containing $GCD(A,B)$. So we send $x2,y2$ as coefficients.

# Complexity

Same as Euclidean Algorithm.

# Uniqueness of Solution

Using ext_gcd we found $(x,y)$ pair which satisfies $ax + by = gcd(a,b)$. But is it unique?

No. Once we find a pair $(x,y)$ using ext_gcd, we can generate infinite pairs of Bezout coefficients using the formula:

$$( x + k \frac{ b } { \text{gcd}(a,b)}, y – k \frac{ a } { \text{gcd}(a,b)} )$$

Using any integer value for $k$ we can generate a different pair of Bezout coefficients $(x,y)$ which will satisfy the Bezout’s identity. Here is why it works:

$a ( x + \frac{ kb } { \text{gcd}(a,b)} ) + b ( y – \frac{ ka } { \text{gcd}(a,b)} )$

$ax + \frac{ kab } { \text{gcd}(a,b)} + by – \frac{ kab } { \text{gcd}(a,b)}$

$ax + by$

As you can see above, the terms with $k$ in them cancel each other out. They don’t change the expression $ax + by$ in any way, therefore, there are infinite pairs of Bezout coefficients.

# Smallest Positive Integer of form $ax + by$

We showed that it is possible to write $ax + by = gcd(a,b)$, now we need to find a positive number smaller than $gcd(a,b)$ of form $ax + by$. Is that even possible?

No. $gcd(a,b)$ divides both $a$ and $b$. Therefore, if a number is of form $ax + by$ it will be divisible by $gcd(a,b)$ since $ax$ and $by$ are both divisible by $gcd(a,b)$. And the smallest positive number which is divisible by $gcd(a,b)$ is $gcd(a,b)$ itself.

So $gcd(a,b)$ is the smallest positive number of the form $ax + by$.

# Bezout’s Identity

This was mentioned at the beginning of the post. Almost everything related to Bezout’s Identity has been explained. I will still mention them once more for the sake of completeness.

Bézout’s identity (also called Bézout’s lemma) is a theorem in the elementary theory of numbers: let a and b be nonzero integers and let d be their greatest common divisor. Then there exist integers x and y such that $ax + by = d$

In addition,

- the greatest common divisor d is the smallest positive integer that can be written as $ax + by$
- every integer of the form ax + by is a multiple of the greatest common divisor d.
–Wiki

Bezout’s Lemma simply states that $ax + by = gcd(a,b)$ exists. We need to use the Extended Euclidean Algorithm to find Bezout’s Coefficients.

# Coding Pitfalls

Careful about the equation $ax + by = \text{gcd} (a,b)$. Here $\text{gcd}()$ means the result from Euclidean Algorithm, not what we mean mathematically. For example, when $a = 4$ and $b = -2$, Extended Euclidean Algorithm finds solution for $4x – 2y = -2$. According to Euclidean algorithm $gcd(4,-2)$ returns $-2$, though in common sense it should be $2$.

# Reference

- forthright48 – Euclidean Algorithm – https://forthright48.com/2015/07/euclidean-algorithm.html
- Wiki – Extended Euclidean algorithm – https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
- Wiki – Bezout’s identity – https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity