Question

You may import the following library functions in your module:

from fractions import gcd

from math import floor

from random import randint

You may also use:

• the built-in pow() function to compute modular exponents efficiently (i.e., ak mod n can be written in Python as pow(a,k,n)),

• the sum() function returns the sum of a list of integers (sum(1,2,3,4) returns 10).

problem 1

a. Implement a function invPrime(a, p) that takes two integers a and p > 1 where p is prime. The function should return the multiplicative inverse of a ∈ ℤ/pℤ (if a ≡ 0, it should return None). Your solution must use Fermat's little theorem.

>>> [invPrime(i, 7) for i in range(0,7)] [None, 1, 4, 5, 2, 3, 6]

>>> [invPrime(i, 13) for i in range(1,13)] [1, 7, 9, 10, 8, 11, 2, 5, 3, 4, 6, 12]

b. Include the following definition in your code. This non-recursive implementation of the extended Euclidean algorithm avoids a stack overflow error on large inputs.

def egcd(a, b):

(x, s, y, t) = (0, 1, 1, 0)

while b != 0:

k = a // b

(a, b) = (b, a % b)

(x, s, y, t) = (s - k*x, x, t - k*y, y)

return (s, t)

Given two inputs a and b, egcd(a, b) returns a solution (s, t) to the following instance of Bézout's identity: a ⋅ s + b ⋅ t = gcd(a,b)

Using egcd(), implement a function inv(a, m) that takes two integers a and m > 1. If a and m are coprime, it should return the multiplicative inverse of a ∈ ℤ/mℤ. If a and m are not coprime, it should return None.

>>> [inv(i, 13) for i in range(1,13)] [1, 7, 9, 10, 8, 11, 2, 5, 3, 4, 6, 12]

>>> [inv(i, 8) for i in range(1,8)] [1, None, 3, None, 5, None, 7]

Answer 1

Code :-

1. #initialiation as mentioned in the question
   from fractions import gcd
   from math import floor
   from random import randint

   #extended euclid without recursion
   def egcd (x1, x2) :
   res = []
   x, y, e1, e2 = 0, 1, 1, 0
   while x2 != 0 :
   const = x1//x2
   x1, x2 = x2, x1%x2
   x, y, e1, e2 = e1-const*x, e2-const*y, x, y
   res = [e1, e2]
   return res

   #inverse modular using egcd
   def inv (a, p) :
   if (gcd(a, p) != 1) :
   return None
   else :
   x, y = egcd(a, p)
   return (x%p+p)%p

   #inverse modular using fermat
   def invPrime(a, p) :
   if (gcd(a, p) != 1) :
   return None
   else :
   return pow(a, p-2, p)
     
   if __name__ == '__main__' :
   n = randint(5, 13)
   print ('fermat', n, [invPrime(i, n) for i in range(1, n)])
   print ('egcd', n, [inv(i, n) for i in range(1, n)])

Images :-

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Note :-

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