Sunday, July 26, 2015

Extended Euclidean Algorithm

Extended Euclidean Algorithm is an extension of Euclidean Algorithm which finds two things for integer $a$ and $b$:
  1. It finds the value of $GCD(a,b)$
  2. 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 i 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 parameter $*X$ 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 $r$.

In side the for loop we calculate values needed for 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 $ax + by$ is 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 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


1 comment:

  1. Good explanatory resource for extended euclid is hard to come by. Thanks :)

    ReplyDelete

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