# Revision history [back]

### Extended Euclidean Algorithm for Univariate Polynomials with Coefficients in a Finite Field

Consider the following snippet, intended to compute the extended euclidean algorithm for polynomials in $F_q[x]$.

def extended_euclides(a,b,quo=lambda a,b:a//b):
r0 = a; r1 = b
s0 = 1; s1 = 0
t0 = 0; t1 = 1

while r1 != 0:
q = quo(r0, r1)
r0, r1 = r1, r0 - q * r1
s0, s1 = s1, s0 - q * s1
t0, t1 = t1, t0 - q * t1

return r0, s0, t0

GFX.<x> = GF(5)[x]
g = x^3 + 2*x^2 - x - 2
h = x - 2
r,s,t = extended_euclides(f,g)
print("The GCD of {} and {} is {}".format(g,h,r))
print("Its Bézout coefficients are {} and {}".format(s,t))
assert r == gcd(g,h), "The gcd should be {}!".format(gcd(g,h))
assert r == s*g + t*h, "The cofactors are wrong!"


Executing this snippet produces the following output:

The GCD of x^3 + 2*x^2 + 4*x + 3 and x + 3 is 3*x + 1
Its Bézout coefficients are 2*x + 4 and 3*x^2 + 4

Error in lines 45-45
Traceback (most recent call last):
File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute
flags=compile_flags) in namespace, locals
File "", line 1, in <module>
AssertionError: The gcd should be 1!


That is, my code disagrees with sagemath's gcd function, and I think sagemath's result is correct.

Why is mine wrong?

### Extended Euclidean Algorithm for Univariate Polynomials with Coefficients in a Finite Field

Consider the following snippet, intended to compute the extended euclidean algorithm for polynomials in $F_q[x]$.

def extended_euclides(a,b,quo=lambda a,b:a//b):
r0 = a; r1 = b
s0 = 1; s1 = 0
t0 = 0; t1 = 1

while r1 != 0:
q = quo(r0, r1)
r0, r1 = r1, r0 - q * r1
s0, s1 = s1, s0 - q * s1
t0, t1 = t1, t0 - q * t1

return r0, s0, t0

GFX.<x> = GF(5)[x]
g = x^3 + 2*x^2 - x - 2
h = x - 2
r,s,t = extended_euclides(f,g)
print("The GCD of {} and {} is {}".format(g,h,r))
print("Its Bézout coefficients are {} and {}".format(s,t))
assert r == gcd(g,h), "The gcd should be {}!".format(gcd(g,h))
assert r == s*g + t*h, "The cofactors are wrong!"


Executing this snippet produces the following output:

The GCD of x^3 + 2*x^2 + 4*x + 3 and x + 3 is 3*x + 1
Its Bézout coefficients are 2*x + 4 and 3*x^2 + 4

Error in lines 45-45
Traceback (most recent call last):
File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute
flags=compile_flags) in namespace, locals
File "", line 1, in <module>
AssertionError: The gcd should be 1!


That is, my code disagrees with sagemath's gcd function, and I think sagemath's result is correct.correct; g(2)!=0, so g and h should not have common factors.

Why is mine wrong?