1. Euclidean Algorithm for GCD
CSCI 284/162
Spring 2007
GWU

2. Euclidean Algorithm considered first non-trivial algorithm
Algorithm_gcd(m, a) /* m > a */
(X, Y) := (m, a) /* Initialize */
while (Y0) (X, Y) := (Y, X rem Y)
return(X)
Works because:
gcd (X, Y) = gcd(Y, X rem Y)
gcd(X, Y) = Y if Y|X
Stops because:
(X, Y) always decreasing and non-negative
4/16/2019
CS /Spring07/GWU/Vora/ Euuclidean Algorithm for GCD

3. CS284-162/Spring07/GWU/Vora/ Euuclidean Algorithm for GCD
Try
gcd(17, 101)
gcd(57, 93)
4/16/2019
CS /Spring07/GWU/Vora/ Euuclidean Algorithm for GCD

4. Theorem: gcd(X, Y) = gcd(Y, X rem Y); X, Y0
Proof:
Let X rem Y = r  r = X – qY for integer q
Consider a factor g of X and Y
g|X and g|Y
X = q1g, Y = q2g, for some integers q1 and q2
r = g(q1 – qq2)
g|Y and g|r
4/16/2019
CS /Spring07/GWU/Vora/ Euuclidean Algorithm for GCD

5. Theorem: gcd(X, Y) = gcd(Y, X rem Y); X, Y0
Proof contd:
Consider a factor h of Y and r
h|Y and h|r
Y = q3h, r = q4h for some integers q3 and q4
X = h(q4 + qq3)
h|Y and h|X
Note: all integers are factors of 0, see, for example, example 2.80 in Chapter 2, Handbook of Applied Cryptography,
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CS /Spring07/GWU/Vora/ Euuclidean Algorithm for GCD

6. Theorem: gcd(X, Y) = gcd(Y, X rem Y) Proof contd.
Hence all common factors of X and Y are common factors of Y and r and vice versa.
Hence, in particular, the pair (X, Y) and the pair (Y, r) have the same gcd.
4/16/2019
CS /Spring07/GWU/Vora/ Euuclidean Algorithm for GCD

7. Euclidean Algorithm: Correctness Proof
Theorem: Algorithm_gcd (m, a) returns gcd(m, a)
Proof: Let the ith update of (X, Y) be denoted (Xi Yi)
Then
(m, a) = (X0 Y0)
and the algorithm returns XN if the algorithm performs N (a finite number) updates.
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CS /Spring07/GWU/Vora/ Euuclidean Algorithm for GCD

8. Euclidean Algorithm: Correctness Proof contd.
From previous theorem, gcd (X, Y) = gcd(Y, X rem Y)
 gcd(m, a) = gcd(X0 Y0) = gcd(X1 Y1) = gcd(X2 Y2) = …. gcd(XN-1 YN-1)
YN-1|XN-1
gcd(XN-1 YN-1) = YN-1 = XN = Algorithm_gcd (m, a)
Hence gcd(m, a) = Algorithm_gcd (m, a)
If N finite
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CS /Spring07/GWU/Vora/ Euuclidean Algorithm for GCD

9. Euclidean Algorithm: Correctness Proof contd.
a=Y0 > Y1 > Y2 > …. > YN-1 > YN = 0
As Y decreases by at least 1 each iteration
N  a
N is finite.
4/16/2019
CS /Spring07/GWU/Vora/ Euuclidean Algorithm for GCD