Question

why do I get this error message, when I’m using these parameters(pow(1,3,3)) ?:

sage: pow(1,3,3)                                    
3
3/2
3/2
---------------------------------------------------------------------------
ZeroDivisionError                         Traceback (most recent call last)

/home/kai/<ipython console> in <module>()

/home/kai/<ipython console> in pow(a, e, n)

/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/rings/rational.so in sage.rings.rational.Rational.__mod__ (sage/rings/rational.c:19891)()

/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/rings/integer.so in sage.rings.integer.Integer.inverse_mod (sage/rings/integer.c:32726)()

ZeroDivisionError: Inverse does not exist.

pow():

def pow(a,e,n):
....:     num = 1
....:     while e >= 1:
....:         print(e)
....:         if (e%2) == 1:
....:             num = (num*a) % n  
....:         e = e/2
....:         print(e)
....:         a = (a*a) % n
....:     return num
....:

Answer 1

The modular exponentiation by squaring algorithm you implemented should use integers.

e = e/2 returns a Rational 3/2.

You should convert this to an integer e = (e/2).floor().