Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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For positive integer n and given integer a that satisfies gcd(a, n) = 1,
the order of a modulo n is the smallest positive integer k that satisfies
pow (a, k) % n = 1. In other words, (a^k) = 1 (mod n).
Order of certain number may or may not be exist. If so, return -1.
def find_order(a, n):
||||||
Find order for positive integer n and given integer a that satisfies gcd(a, n) = 1.
Time complexity O(nlog(n))
if (a
1) & (n
1):
# Exception Handeling: 1 is the order of of 1
return 1
==
if math.gcd(a, n) != 1:
print ("a and n should be relative prime!")
return -1
for i in range(1, n):
if pow(a, i) % n == 1:
return i
return -1
==
Euler's totient function, also known as phi-function (n),
counts the number of integers between 1 and n inclusive,
which are coprime to n.
(Two numbers are coprime if their greatest common divisor (GCD) equals 1).
Code from /algorithms/maths/euler_totient.py, written by 'goswami-rahul'
def euler_totient(n):
'Euler's totient function or Phi function.
||||||
Time Complexity: O(sqrt(n))."
result = n
for i in range(2, int(n** 0.5) + 1):
if n % i == 0:
while n % i == 0:
n//=i
result = result // i
if n > 1:
result result // n
return result
For positive integer n and given integer a that satisfies gcd(a, n) = 1,
a is the primitive root of n, if a's order k for n satisfies k = o(n).
Primitive roots of certain number may or may not exist.
If so, return empty list.
def find_primitive_root(n):
if n == 1:
# Exception Handeling : 0 is the only primitive root of
return [0]
phi = euler_totient(n)
p_root_list =
||||||
It will return every primitive roots of n.
for i in range (1, n):
#To have order, a and n must be relative prime with each other.
if math.gcd(i, n) == 1:
order = find_order(i, n)
if order == phi:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fff8bc427-4dd5-458f-a0de-0d1350ec468a%2F6e1d4520-7ab0-49da-b5d8-ce396c5de6ee%2Fn2dwuft_processed.jpeg&w=3840&q=75)
Transcribed Image Text:||||||
For positive integer n and given integer a that satisfies gcd(a, n) = 1,
the order of a modulo n is the smallest positive integer k that satisfies
pow (a, k) % n = 1. In other words, (a^k) = 1 (mod n).
Order of certain number may or may not be exist. If so, return -1.
def find_order(a, n):
||||||
Find order for positive integer n and given integer a that satisfies gcd(a, n) = 1.
Time complexity O(nlog(n))
if (a
1) & (n
1):
# Exception Handeling: 1 is the order of of 1
return 1
==
if math.gcd(a, n) != 1:
print ("a and n should be relative prime!")
return -1
for i in range(1, n):
if pow(a, i) % n == 1:
return i
return -1
==
Euler's totient function, also known as phi-function (n),
counts the number of integers between 1 and n inclusive,
which are coprime to n.
(Two numbers are coprime if their greatest common divisor (GCD) equals 1).
Code from /algorithms/maths/euler_totient.py, written by 'goswami-rahul'
def euler_totient(n):
'Euler's totient function or Phi function.
||||||
Time Complexity: O(sqrt(n))."
result = n
for i in range(2, int(n** 0.5) + 1):
if n % i == 0:
while n % i == 0:
n//=i
result = result // i
if n > 1:
result result // n
return result
For positive integer n and given integer a that satisfies gcd(a, n) = 1,
a is the primitive root of n, if a's order k for n satisfies k = o(n).
Primitive roots of certain number may or may not exist.
If so, return empty list.
def find_primitive_root(n):
if n == 1:
# Exception Handeling : 0 is the only primitive root of
return [0]
phi = euler_totient(n)
p_root_list =
||||||
It will return every primitive roots of n.
for i in range (1, n):
#To have order, a and n must be relative prime with each other.
if math.gcd(i, n) == 1:
order = find_order(i, n)
if order == phi:
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