Removed redundant greatest_common_divisor code

blockchain/diophantine_equation.py

@@ -1,11 +1,13 @@
 from __future__ import annotations
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
  """
  Diophantine Equation : Given integers a,b,c ( at least one of a and b != 0), the
  diophantine equation a*x + b*y = c has a solution (where x and y are integers)
- iff gcd(a,b) divides c.
+ iff greatest_common_divisor(a,b) divides c.
 
  GCD ( Greatest Common Divisor ) or HCF ( Highest Common Factor )
 
@@ -22,7 +24,7 @@ def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
 
  assert (
  c % greatest_common_divisor(a, b) == 0
- ) # greatest_common_divisor(a,b) function implemented below
+ ) # greatest_common_divisor(a,b) is in maths directory
  (d, x, y) = extended_gcd(a, b) # extended_gcd(a,b) function implemented below
  r = c / d
  return (r * x, r * y)
@@ -69,32 +71,6 @@ def diophantine_all_soln(a: int, b: int, c: int, n: int = 2) -> None:
  print(x, y)
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Euclid's Lemma : d divides a and b, if and only if d divides a-b and b
-
- Euclid's Algorithm
-
- >>> greatest_common_divisor(7,5)
- 1
-
- Note : In number theory, two integers a and b are said to be relatively prime,
- mutually prime, or co-prime if the only positive integer (factor) that
- divides both of them is 1 i.e., gcd(a,b) = 1.
-
- >>> greatest_common_divisor(121, 11)
- 11
-
- """
- if a < b:
- a, b = b, a
-
- while a % b != 0:
- a, b = b, a % b
-
- return b
-
-
 def extended_gcd(a: int, b: int) -> tuple[int, int, int]:
  """
  Extended Euclid's Algorithm : If d divides a and b and d = a*x + b*y for integers

ciphers/affine_cipher.py

@@ -1,6 +1,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module as cryptomath
 
 SYMBOLS = (
@@ -26,7 +28,7 @@ def check_keys(key_a: int, key_b: int, mode: str) -> None:
  "Key A must be greater than 0 and key B must "
  f"be between 0 and {len(SYMBOLS) - 1}."
  )
- if cryptomath.gcd(key_a, len(SYMBOLS)) != 1:
+ if gcd_by_iterative(key_a, len(SYMBOLS)) != 1:
  sys.exit(
  f"Key A {key_a} and the symbol set size {len(SYMBOLS)} "
  "are not relatively prime. Choose a different key."
@@ -76,7 +78,7 @@ def get_random_key() -> int:
  while True:
  key_b = random.randint(2, len(SYMBOLS))
  key_b = random.randint(2, len(SYMBOLS))
- if cryptomath.gcd(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
+ if gcd_by_iterative(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
  return key_b * len(SYMBOLS) + key_b
 
 

ciphers/cryptomath_module.py

@@ -1,11 +1,8 @@
-def gcd(a: int, b: int) -> int:
- while a != 0:
- a, b = b % a, a
- return b
+from maths.greatest_common_divisor import gcd_by_iterative
 
 
 def find_mod_inverse(a: int, m: int) -> int:
- if gcd(a, m) != 1:
+ if gcd_by_iterative(a, m) != 1:
  msg = f"mod inverse of {a!r} and {m!r} does not exist"
  raise ValueError(msg)
  u1, u2, u3 = 1, 0, a

ciphers/hill_cipher.py

@@ -39,19 +39,7 @@
 
 import numpy
 
-
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- >>> greatest_common_divisor(4, 8)
- 4
- >>> greatest_common_divisor(8, 4)
- 4
- >>> greatest_common_divisor(4, 7)
- 1
- >>> greatest_common_divisor(0, 10)
- 10
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 class HillCipher:

ciphers/rsa_key_generator.py

@@ -2,6 +2,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module, rabin_miller
 
 
@@ -27,7 +29,7 @@ def generate_key(key_size: int) -> tuple[tuple[int, int], tuple[int, int]]:
  # Generate e that is relatively prime to (p - 1) * (q - 1)
  while True:
  e = random.randrange(2 ** (key_size - 1), 2 ** (key_size))
- if cryptomath_module.gcd(e, (p - 1) * (q - 1)) == 1:
+ if gcd_by_iterative(e, (p - 1) * (q - 1)) == 1:
  break
 
  # Calculate d that is mod inverse of e

maths/carmichael_number.py

@@ -10,14 +10,7 @@
 Examples of Carmichael Numbers: 561, 1105, ...
 https://en.wikipedia.org/wiki/Carmichael_number
 """
-
-
-def gcd(a: int, b: int) -> int:
- if a < b:
- return gcd(b, a)
- if a % b == 0:
- return b
- return gcd(b, a % b)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 def power(x: int, y: int, mod: int) -> int:
@@ -33,7 +26,7 @@ def power(x: int, y: int, mod: int) -> int:
 def is_carmichael_number(n: int) -> bool:
  b = 2
  while b < n:
- if gcd(b, n) == 1 and power(b, n - 1, n) != 1:
+ if greatest_common_divisor(b, n) == 1 and power(b, n - 1, n) != 1:
  return False
  b += 1
  return True

maths/least_common_multiple.py

@@ -1,6 +1,8 @@
 import unittest
 from timeit import timeit
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  """
@@ -20,26 +22,6 @@ def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  return common_mult
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Calculate Greatest Common Divisor (GCD).
- see greatest_common_divisor.py
- >>> greatest_common_divisor(24, 40)
- 8
- >>> greatest_common_divisor(1, 1)
- 1
- >>> greatest_common_divisor(1, 800)
- 1
- >>> greatest_common_divisor(11, 37)
- 1
- >>> greatest_common_divisor(3, 5)
- 1
- >>> greatest_common_divisor(16, 4)
- 4
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
-
-
 def least_common_multiple_fast(first_num: int, second_num: int) -> int:
  """
  Find the least common multiple of two numbers.

blockchain/diophantine_equation.py

@@ -1,11 +1,13 @@
 from __future__ import annotations
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
  """
  Diophantine Equation : Given integers a,b,c ( at least one of a and b != 0), the
  diophantine equation a*x + b*y = c has a solution (where x and y are integers)
- iff gcd(a,b) divides c.
+ iff greatest_common_divisor(a,b) divides c.
 
  GCD ( Greatest Common Divisor ) or HCF ( Highest Common Factor )
 
@@ -22,7 +24,7 @@ def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
 
  assert (
  c % greatest_common_divisor(a, b) == 0
- ) # greatest_common_divisor(a,b) function implemented below
+ ) # greatest_common_divisor(a,b) is in maths directory
  (d, x, y) = extended_gcd(a, b) # extended_gcd(a,b) function implemented below
  r = c / d
  return (r * x, r * y)
@@ -69,32 +71,6 @@ def diophantine_all_soln(a: int, b: int, c: int, n: int = 2) -> None:
  print(x, y)
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Euclid's Lemma : d divides a and b, if and only if d divides a-b and b
-
- Euclid's Algorithm
-
- >>> greatest_common_divisor(7,5)
- 1
-
- Note : In number theory, two integers a and b are said to be relatively prime,
- mutually prime, or co-prime if the only positive integer (factor) that
- divides both of them is 1 i.e., gcd(a,b) = 1.
-
- >>> greatest_common_divisor(121, 11)
- 11
-
- """
- if a < b:
- a, b = b, a
-
- while a % b != 0:
- a, b = b, a % b
-
- return b
-
-
 def extended_gcd(a: int, b: int) -> tuple[int, int, int]:
  """
  Extended Euclid's Algorithm : If d divides a and b and d = a*x + b*y for integers

ciphers/affine_cipher.py

@@ -1,6 +1,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module as cryptomath
 
 SYMBOLS = (
@@ -26,7 +28,7 @@ def check_keys(key_a: int, key_b: int, mode: str) -> None:
  "Key A must be greater than 0 and key B must "
  f"be between 0 and {len(SYMBOLS) - 1}."
  )
- if cryptomath.gcd(key_a, len(SYMBOLS)) != 1:
+ if gcd_by_iterative(key_a, len(SYMBOLS)) != 1:
  sys.exit(
  f"Key A {key_a} and the symbol set size {len(SYMBOLS)} "
  "are not relatively prime. Choose a different key."
@@ -76,7 +78,7 @@ def get_random_key() -> int:
  while True:
  key_b = random.randint(2, len(SYMBOLS))
  key_b = random.randint(2, len(SYMBOLS))
- if cryptomath.gcd(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
+ if gcd_by_iterative(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
  return key_b * len(SYMBOLS) + key_b
 
 

ciphers/cryptomath_module.py

@@ -1,11 +1,8 @@
-def gcd(a: int, b: int) -> int:
- while a != 0:
- a, b = b % a, a
- return b
+from maths.greatest_common_divisor import gcd_by_iterative
 
 
 def find_mod_inverse(a: int, m: int) -> int:
- if gcd(a, m) != 1:
+ if gcd_by_iterative(a, m) != 1:
  msg = f"mod inverse of {a!r} and {m!r} does not exist"
  raise ValueError(msg)
  u1, u2, u3 = 1, 0, a

ciphers/hill_cipher.py

@@ -39,19 +39,7 @@
 
 import numpy
 
-
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- >>> greatest_common_divisor(4, 8)
- 4
- >>> greatest_common_divisor(8, 4)
- 4
- >>> greatest_common_divisor(4, 7)
- 1
- >>> greatest_common_divisor(0, 10)
- 10
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 class HillCipher:

ciphers/rsa_key_generator.py

@@ -2,6 +2,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module, rabin_miller
 
 
@@ -27,7 +29,7 @@ def generate_key(key_size: int) -> tuple[tuple[int, int], tuple[int, int]]:
  # Generate e that is relatively prime to (p - 1) * (q - 1)
  while True:
  e = random.randrange(2 ** (key_size - 1), 2 ** (key_size))
- if cryptomath_module.gcd(e, (p - 1) * (q - 1)) == 1:
+ if gcd_by_iterative(e, (p - 1) * (q - 1)) == 1:
  break
 
  # Calculate d that is mod inverse of e

maths/carmichael_number.py

@@ -10,14 +10,7 @@
 Examples of Carmichael Numbers: 561, 1105, ...
 https://en.wikipedia.org/wiki/Carmichael_number
 """
-
-
-def gcd(a: int, b: int) -> int:
- if a < b:
- return gcd(b, a)
- if a % b == 0:
- return b
- return gcd(b, a % b)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 def power(x: int, y: int, mod: int) -> int:
@@ -33,7 +26,7 @@ def power(x: int, y: int, mod: int) -> int:
 def is_carmichael_number(n: int) -> bool:
  b = 2
  while b < n:
- if gcd(b, n) == 1 and power(b, n - 1, n) != 1:
+ if greatest_common_divisor(b, n) == 1 and power(b, n - 1, n) != 1:
  return False
  b += 1
  return True

maths/least_common_multiple.py

@@ -1,6 +1,8 @@
 import unittest
 from timeit import timeit
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  """
@@ -20,26 +22,6 @@ def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  return common_mult
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Calculate Greatest Common Divisor (GCD).
- see greatest_common_divisor.py
- >>> greatest_common_divisor(24, 40)
- 8
- >>> greatest_common_divisor(1, 1)
- 1
- >>> greatest_common_divisor(1, 800)
- 1
- >>> greatest_common_divisor(11, 37)
- 1
- >>> greatest_common_divisor(3, 5)
- 1
- >>> greatest_common_divisor(16, 4)
- 4
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
-
-
 def least_common_multiple_fast(first_num: int, second_num: int) -> int:
  """
  Find the least common multiple of two numbers.

blockchain/diophantine_equation.py

@@ -1,11 +1,13 @@
 from __future__ import annotations
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
  """
  Diophantine Equation : Given integers a,b,c ( at least one of a and b != 0), the
  diophantine equation a*x + b*y = c has a solution (where x and y are integers)
- iff gcd(a,b) divides c.
+ iff greatest_common_divisor(a,b) divides c.
 
  GCD ( Greatest Common Divisor ) or HCF ( Highest Common Factor )
 
@@ -22,7 +24,7 @@ def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
 
  assert (
  c % greatest_common_divisor(a, b) == 0
- ) # greatest_common_divisor(a,b) function implemented below
+ ) # greatest_common_divisor(a,b) is in maths directory
  (d, x, y) = extended_gcd(a, b) # extended_gcd(a,b) function implemented below
  r = c / d
  return (r * x, r * y)
@@ -69,32 +71,6 @@ def diophantine_all_soln(a: int, b: int, c: int, n: int = 2) -> None:
  print(x, y)
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Euclid's Lemma : d divides a and b, if and only if d divides a-b and b
-
- Euclid's Algorithm
-
- >>> greatest_common_divisor(7,5)
- 1
-
- Note : In number theory, two integers a and b are said to be relatively prime,
- mutually prime, or co-prime if the only positive integer (factor) that
- divides both of them is 1 i.e., gcd(a,b) = 1.
-
- >>> greatest_common_divisor(121, 11)
- 11
-
- """
- if a < b:
- a, b = b, a
-
- while a % b != 0:
- a, b = b, a % b
-
- return b
-
-
 def extended_gcd(a: int, b: int) -> tuple[int, int, int]:
  """
  Extended Euclid's Algorithm : If d divides a and b and d = a*x + b*y for integers

ciphers/affine_cipher.py

@@ -1,6 +1,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module as cryptomath
 
 SYMBOLS = (
@@ -26,7 +28,7 @@ def check_keys(key_a: int, key_b: int, mode: str) -> None:
  "Key A must be greater than 0 and key B must "
  f"be between 0 and {len(SYMBOLS) - 1}."
  )
- if cryptomath.gcd(key_a, len(SYMBOLS)) != 1:
+ if gcd_by_iterative(key_a, len(SYMBOLS)) != 1:
  sys.exit(
  f"Key A {key_a} and the symbol set size {len(SYMBOLS)} "
  "are not relatively prime. Choose a different key."
@@ -76,7 +78,7 @@ def get_random_key() -> int:
  while True:
  key_b = random.randint(2, len(SYMBOLS))
  key_b = random.randint(2, len(SYMBOLS))
- if cryptomath.gcd(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
+ if gcd_by_iterative(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
  return key_b * len(SYMBOLS) + key_b
 
 

ciphers/cryptomath_module.py

@@ -1,11 +1,8 @@
-def gcd(a: int, b: int) -> int:
- while a != 0:
- a, b = b % a, a
- return b
+from maths.greatest_common_divisor import gcd_by_iterative
 
 
 def find_mod_inverse(a: int, m: int) -> int:
- if gcd(a, m) != 1:
+ if gcd_by_iterative(a, m) != 1:
  msg = f"mod inverse of {a!r} and {m!r} does not exist"
  raise ValueError(msg)
  u1, u2, u3 = 1, 0, a

ciphers/hill_cipher.py

@@ -39,19 +39,7 @@
 
 import numpy
 
-
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- >>> greatest_common_divisor(4, 8)
- 4
- >>> greatest_common_divisor(8, 4)
- 4
- >>> greatest_common_divisor(4, 7)
- 1
- >>> greatest_common_divisor(0, 10)
- 10
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 class HillCipher:

ciphers/rsa_key_generator.py

@@ -2,6 +2,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module, rabin_miller
 
 
@@ -27,7 +29,7 @@ def generate_key(key_size: int) -> tuple[tuple[int, int], tuple[int, int]]:
  # Generate e that is relatively prime to (p - 1) * (q - 1)
  while True:
  e = random.randrange(2 ** (key_size - 1), 2 ** (key_size))
- if cryptomath_module.gcd(e, (p - 1) * (q - 1)) == 1:
+ if gcd_by_iterative(e, (p - 1) * (q - 1)) == 1:
  break
 
  # Calculate d that is mod inverse of e

maths/carmichael_number.py

@@ -10,14 +10,7 @@
 Examples of Carmichael Numbers: 561, 1105, ...
 https://en.wikipedia.org/wiki/Carmichael_number
 """
-
-
-def gcd(a: int, b: int) -> int:
- if a < b:
- return gcd(b, a)
- if a % b == 0:
- return b
- return gcd(b, a % b)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 def power(x: int, y: int, mod: int) -> int:
@@ -33,7 +26,7 @@ def power(x: int, y: int, mod: int) -> int:
 def is_carmichael_number(n: int) -> bool:
  b = 2
  while b < n:
- if gcd(b, n) == 1 and power(b, n - 1, n) != 1:
+ if greatest_common_divisor(b, n) == 1 and power(b, n - 1, n) != 1:
  return False
  b += 1
  return True

maths/least_common_multiple.py

@@ -1,6 +1,8 @@
 import unittest
 from timeit import timeit
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  """
@@ -20,26 +22,6 @@ def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  return common_mult
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Calculate Greatest Common Divisor (GCD).
- see greatest_common_divisor.py
- >>> greatest_common_divisor(24, 40)
- 8
- >>> greatest_common_divisor(1, 1)
- 1
- >>> greatest_common_divisor(1, 800)
- 1
- >>> greatest_common_divisor(11, 37)
- 1
- >>> greatest_common_divisor(3, 5)
- 1
- >>> greatest_common_divisor(16, 4)
- 4
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
-
-
 def least_common_multiple_fast(first_num: int, second_num: int) -> int:
  """
  Find the least common multiple of two numbers.

blockchain/diophantine_equation.py

@@ -1,11 +1,13 @@
 from __future__ import annotations
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
  """
  Diophantine Equation : Given integers a,b,c ( at least one of a and b != 0), the
  diophantine equation a*x + b*y = c has a solution (where x and y are integers)
- iff gcd(a,b) divides c.
+ iff greatest_common_divisor(a,b) divides c.
 
  GCD ( Greatest Common Divisor ) or HCF ( Highest Common Factor )
 
@@ -22,7 +24,7 @@ def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
 
  assert (
  c % greatest_common_divisor(a, b) == 0
- ) # greatest_common_divisor(a,b) function implemented below
+ ) # greatest_common_divisor(a,b) is in maths directory
  (d, x, y) = extended_gcd(a, b) # extended_gcd(a,b) function implemented below
  r = c / d
  return (r * x, r * y)
@@ -69,32 +71,6 @@ def diophantine_all_soln(a: int, b: int, c: int, n: int = 2) -> None:
  print(x, y)
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Euclid's Lemma : d divides a and b, if and only if d divides a-b and b
-
- Euclid's Algorithm
-
- >>> greatest_common_divisor(7,5)
- 1
-
- Note : In number theory, two integers a and b are said to be relatively prime,
- mutually prime, or co-prime if the only positive integer (factor) that
- divides both of them is 1 i.e., gcd(a,b) = 1.
-
- >>> greatest_common_divisor(121, 11)
- 11
-
- """
- if a < b:
- a, b = b, a
-
- while a % b != 0:
- a, b = b, a % b
-
- return b
-
-
 def extended_gcd(a: int, b: int) -> tuple[int, int, int]:
  """
  Extended Euclid's Algorithm : If d divides a and b and d = a*x + b*y for integers

ciphers/affine_cipher.py

@@ -1,6 +1,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module as cryptomath
 
 SYMBOLS = (
@@ -26,7 +28,7 @@ def check_keys(key_a: int, key_b: int, mode: str) -> None:
  "Key A must be greater than 0 and key B must "
  f"be between 0 and {len(SYMBOLS) - 1}."
  )
- if cryptomath.gcd(key_a, len(SYMBOLS)) != 1:
+ if gcd_by_iterative(key_a, len(SYMBOLS)) != 1:
  sys.exit(
  f"Key A {key_a} and the symbol set size {len(SYMBOLS)} "
  "are not relatively prime. Choose a different key."
@@ -76,7 +78,7 @@ def get_random_key() -> int:
  while True:
  key_b = random.randint(2, len(SYMBOLS))
  key_b = random.randint(2, len(SYMBOLS))
- if cryptomath.gcd(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
+ if gcd_by_iterative(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
  return key_b * len(SYMBOLS) + key_b
 
 

ciphers/cryptomath_module.py

@@ -1,11 +1,8 @@
-def gcd(a: int, b: int) -> int:
- while a != 0:
- a, b = b % a, a
- return b
+from maths.greatest_common_divisor import gcd_by_iterative
 
 
 def find_mod_inverse(a: int, m: int) -> int:
- if gcd(a, m) != 1:
+ if gcd_by_iterative(a, m) != 1:
  msg = f"mod inverse of {a!r} and {m!r} does not exist"
  raise ValueError(msg)
  u1, u2, u3 = 1, 0, a

ciphers/hill_cipher.py

@@ -39,19 +39,7 @@
 
 import numpy
 
-
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- >>> greatest_common_divisor(4, 8)
- 4
- >>> greatest_common_divisor(8, 4)
- 4
- >>> greatest_common_divisor(4, 7)
- 1
- >>> greatest_common_divisor(0, 10)
- 10
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 class HillCipher:

ciphers/rsa_key_generator.py

@@ -2,6 +2,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module, rabin_miller
 
 
@@ -27,7 +29,7 @@ def generate_key(key_size: int) -> tuple[tuple[int, int], tuple[int, int]]:
  # Generate e that is relatively prime to (p - 1) * (q - 1)
  while True:
  e = random.randrange(2 ** (key_size - 1), 2 ** (key_size))
- if cryptomath_module.gcd(e, (p - 1) * (q - 1)) == 1:
+ if gcd_by_iterative(e, (p - 1) * (q - 1)) == 1:
  break
 
  # Calculate d that is mod inverse of e

maths/carmichael_number.py

@@ -10,14 +10,7 @@
 Examples of Carmichael Numbers: 561, 1105, ...
 https://en.wikipedia.org/wiki/Carmichael_number
 """
-
-
-def gcd(a: int, b: int) -> int:
- if a < b:
- return gcd(b, a)
- if a % b == 0:
- return b
- return gcd(b, a % b)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 def power(x: int, y: int, mod: int) -> int:
@@ -33,7 +26,7 @@ def power(x: int, y: int, mod: int) -> int:
 def is_carmichael_number(n: int) -> bool:
  b = 2
  while b < n:
- if gcd(b, n) == 1 and power(b, n - 1, n) != 1:
+ if greatest_common_divisor(b, n) == 1 and power(b, n - 1, n) != 1:
  return False
  b += 1
  return True

maths/least_common_multiple.py

@@ -1,6 +1,8 @@
 import unittest
 from timeit import timeit
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  """
@@ -20,26 +22,6 @@ def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  return common_mult
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Calculate Greatest Common Divisor (GCD).
- see greatest_common_divisor.py
- >>> greatest_common_divisor(24, 40)
- 8
- >>> greatest_common_divisor(1, 1)
- 1
- >>> greatest_common_divisor(1, 800)
- 1
- >>> greatest_common_divisor(11, 37)
- 1
- >>> greatest_common_divisor(3, 5)
- 1
- >>> greatest_common_divisor(16, 4)
- 4
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
-
-
 def least_common_multiple_fast(first_num: int, second_num: int) -> int:
  """
  Find the least common multiple of two numbers.

blockchain/diophantine_equation.py

@@ -1,11 +1,13 @@
 from __future__ import annotations
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
  """
  Diophantine Equation : Given integers a,b,c ( at least one of a and b != 0), the
  diophantine equation a*x + b*y = c has a solution (where x and y are integers)
- iff gcd(a,b) divides c.
+ iff greatest_common_divisor(a,b) divides c.
 
  GCD ( Greatest Common Divisor ) or HCF ( Highest Common Factor )
 
@@ -22,7 +24,7 @@ def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
 
  assert (
  c % greatest_common_divisor(a, b) == 0
- ) # greatest_common_divisor(a,b) function implemented below
+ ) # greatest_common_divisor(a,b) is in maths directory
  (d, x, y) = extended_gcd(a, b) # extended_gcd(a,b) function implemented below
  r = c / d
  return (r * x, r * y)
@@ -69,32 +71,6 @@ def diophantine_all_soln(a: int, b: int, c: int, n: int = 2) -> None:
  print(x, y)
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Euclid's Lemma : d divides a and b, if and only if d divides a-b and b
-
- Euclid's Algorithm
-
- >>> greatest_common_divisor(7,5)
- 1
-
- Note : In number theory, two integers a and b are said to be relatively prime,
- mutually prime, or co-prime if the only positive integer (factor) that
- divides both of them is 1 i.e., gcd(a,b) = 1.
-
- >>> greatest_common_divisor(121, 11)
- 11
-
- """
- if a < b:
- a, b = b, a
-
- while a % b != 0:
- a, b = b, a % b
-
- return b
-
-
 def extended_gcd(a: int, b: int) -> tuple[int, int, int]:
  """
  Extended Euclid's Algorithm : If d divides a and b and d = a*x + b*y for integers

ciphers/affine_cipher.py

@@ -1,6 +1,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module as cryptomath
 
 SYMBOLS = (
@@ -26,7 +28,7 @@ def check_keys(key_a: int, key_b: int, mode: str) -> None:
  "Key A must be greater than 0 and key B must "
  f"be between 0 and {len(SYMBOLS) - 1}."
  )
- if cryptomath.gcd(key_a, len(SYMBOLS)) != 1:
+ if gcd_by_iterative(key_a, len(SYMBOLS)) != 1:
  sys.exit(
  f"Key A {key_a} and the symbol set size {len(SYMBOLS)} "
  "are not relatively prime. Choose a different key."
@@ -76,7 +78,7 @@ def get_random_key() -> int:
  while True:
  key_b = random.randint(2, len(SYMBOLS))
  key_b = random.randint(2, len(SYMBOLS))
- if cryptomath.gcd(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
+ if gcd_by_iterative(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
  return key_b * len(SYMBOLS) + key_b
 
 

ciphers/cryptomath_module.py

@@ -1,11 +1,8 @@
-def gcd(a: int, b: int) -> int:
- while a != 0:
- a, b = b % a, a
- return b
+from maths.greatest_common_divisor import gcd_by_iterative
 
 
 def find_mod_inverse(a: int, m: int) -> int:
- if gcd(a, m) != 1:
+ if gcd_by_iterative(a, m) != 1:
  msg = f"mod inverse of {a!r} and {m!r} does not exist"
  raise ValueError(msg)
  u1, u2, u3 = 1, 0, a

ciphers/hill_cipher.py

@@ -39,19 +39,7 @@
 
 import numpy
 
-
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- >>> greatest_common_divisor(4, 8)
- 4
- >>> greatest_common_divisor(8, 4)
- 4
- >>> greatest_common_divisor(4, 7)
- 1
- >>> greatest_common_divisor(0, 10)
- 10
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 class HillCipher:

ciphers/rsa_key_generator.py

@@ -2,6 +2,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module, rabin_miller
 
 
@@ -27,7 +29,7 @@ def generate_key(key_size: int) -> tuple[tuple[int, int], tuple[int, int]]:
  # Generate e that is relatively prime to (p - 1) * (q - 1)
  while True:
  e = random.randrange(2 ** (key_size - 1), 2 ** (key_size))
- if cryptomath_module.gcd(e, (p - 1) * (q - 1)) == 1:
+ if gcd_by_iterative(e, (p - 1) * (q - 1)) == 1:
  break
 
  # Calculate d that is mod inverse of e

maths/carmichael_number.py

@@ -10,14 +10,7 @@
 Examples of Carmichael Numbers: 561, 1105, ...
 https://en.wikipedia.org/wiki/Carmichael_number
 """
-
-
-def gcd(a: int, b: int) -> int:
- if a < b:
- return gcd(b, a)
- if a % b == 0:
- return b
- return gcd(b, a % b)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 def power(x: int, y: int, mod: int) -> int:
@@ -33,7 +26,7 @@ def power(x: int, y: int, mod: int) -> int:
 def is_carmichael_number(n: int) -> bool:
  b = 2
  while b < n:
- if gcd(b, n) == 1 and power(b, n - 1, n) != 1:
+ if greatest_common_divisor(b, n) == 1 and power(b, n - 1, n) != 1:
  return False
  b += 1
  return True

maths/least_common_multiple.py

@@ -1,6 +1,8 @@
 import unittest
 from timeit import timeit
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  """
@@ -20,26 +22,6 @@ def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  return common_mult
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Calculate Greatest Common Divisor (GCD).
- see greatest_common_divisor.py
- >>> greatest_common_divisor(24, 40)
- 8
- >>> greatest_common_divisor(1, 1)
- 1
- >>> greatest_common_divisor(1, 800)
- 1
- >>> greatest_common_divisor(11, 37)
- 1
- >>> greatest_common_divisor(3, 5)
- 1
- >>> greatest_common_divisor(16, 4)
- 4
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
-
-
 def least_common_multiple_fast(first_num: int, second_num: int) -> int:
  """
  Find the least common multiple of two numbers.

blockchain/diophantine_equation.py

@@ -1,11 +1,13 @@
 from __future__ import annotations
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
  """
  Diophantine Equation : Given integers a,b,c ( at least one of a and b != 0), the
  diophantine equation a*x + b*y = c has a solution (where x and y are integers)
- iff gcd(a,b) divides c.
+ iff greatest_common_divisor(a,b) divides c.
 
  GCD ( Greatest Common Divisor ) or HCF ( Highest Common Factor )
 
@@ -22,7 +24,7 @@ def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
 
  assert (
  c % greatest_common_divisor(a, b) == 0
- ) # greatest_common_divisor(a,b) function implemented below
+ ) # greatest_common_divisor(a,b) is in maths directory
  (d, x, y) = extended_gcd(a, b) # extended_gcd(a,b) function implemented below
  r = c / d
  return (r * x, r * y)
@@ -69,32 +71,6 @@ def diophantine_all_soln(a: int, b: int, c: int, n: int = 2) -> None:
  print(x, y)
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Euclid's Lemma : d divides a and b, if and only if d divides a-b and b
-
- Euclid's Algorithm
-
- >>> greatest_common_divisor(7,5)
- 1
-
- Note : In number theory, two integers a and b are said to be relatively prime,
- mutually prime, or co-prime if the only positive integer (factor) that
- divides both of them is 1 i.e., gcd(a,b) = 1.
-
- >>> greatest_common_divisor(121, 11)
- 11
-
- """
- if a < b:
- a, b = b, a
-
- while a % b != 0:
- a, b = b, a % b
-
- return b
-
-
 def extended_gcd(a: int, b: int) -> tuple[int, int, int]:
  """
  Extended Euclid's Algorithm : If d divides a and b and d = a*x + b*y for integers

ciphers/affine_cipher.py

@@ -1,6 +1,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module as cryptomath
 
 SYMBOLS = (
@@ -26,7 +28,7 @@ def check_keys(key_a: int, key_b: int, mode: str) -> None:
  "Key A must be greater than 0 and key B must "
  f"be between 0 and {len(SYMBOLS) - 1}."
  )
- if cryptomath.gcd(key_a, len(SYMBOLS)) != 1:
+ if gcd_by_iterative(key_a, len(SYMBOLS)) != 1:
  sys.exit(
  f"Key A {key_a} and the symbol set size {len(SYMBOLS)} "
  "are not relatively prime. Choose a different key."
@@ -76,7 +78,7 @@ def get_random_key() -> int:
  while True:
  key_b = random.randint(2, len(SYMBOLS))
  key_b = random.randint(2, len(SYMBOLS))
- if cryptomath.gcd(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
+ if gcd_by_iterative(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
  return key_b * len(SYMBOLS) + key_b
 
 

ciphers/cryptomath_module.py

@@ -1,11 +1,8 @@
-def gcd(a: int, b: int) -> int:
- while a != 0:
- a, b = b % a, a
- return b
+from maths.greatest_common_divisor import gcd_by_iterative
 
 
 def find_mod_inverse(a: int, m: int) -> int:
- if gcd(a, m) != 1:
+ if gcd_by_iterative(a, m) != 1:
  msg = f"mod inverse of {a!r} and {m!r} does not exist"
  raise ValueError(msg)
  u1, u2, u3 = 1, 0, a

ciphers/hill_cipher.py

@@ -39,19 +39,7 @@
 
 import numpy
 
-
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- >>> greatest_common_divisor(4, 8)
- 4
- >>> greatest_common_divisor(8, 4)
- 4
- >>> greatest_common_divisor(4, 7)
- 1
- >>> greatest_common_divisor(0, 10)
- 10
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 class HillCipher:

ciphers/rsa_key_generator.py

@@ -2,6 +2,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module, rabin_miller
 
 
@@ -27,7 +29,7 @@ def generate_key(key_size: int) -> tuple[tuple[int, int], tuple[int, int]]:
  # Generate e that is relatively prime to (p - 1) * (q - 1)
  while True:
  e = random.randrange(2 ** (key_size - 1), 2 ** (key_size))
- if cryptomath_module.gcd(e, (p - 1) * (q - 1)) == 1:
+ if gcd_by_iterative(e, (p - 1) * (q - 1)) == 1:
  break
 
  # Calculate d that is mod inverse of e

maths/carmichael_number.py

@@ -10,14 +10,7 @@
 Examples of Carmichael Numbers: 561, 1105, ...
 https://en.wikipedia.org/wiki/Carmichael_number
 """
-
-
-def gcd(a: int, b: int) -> int:
- if a < b:
- return gcd(b, a)
- if a % b == 0:
- return b
- return gcd(b, a % b)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 def power(x: int, y: int, mod: int) -> int:
@@ -33,7 +26,7 @@ def power(x: int, y: int, mod: int) -> int:
 def is_carmichael_number(n: int) -> bool:
  b = 2
  while b < n:
- if gcd(b, n) == 1 and power(b, n - 1, n) != 1:
+ if greatest_common_divisor(b, n) == 1 and power(b, n - 1, n) != 1:
  return False
  b += 1
  return True

maths/least_common_multiple.py

@@ -1,6 +1,8 @@
 import unittest
 from timeit import timeit
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  """
@@ -20,26 +22,6 @@ def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  return common_mult
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Calculate Greatest Common Divisor (GCD).
- see greatest_common_divisor.py
- >>> greatest_common_divisor(24, 40)
- 8
- >>> greatest_common_divisor(1, 1)
- 1
- >>> greatest_common_divisor(1, 800)
- 1
- >>> greatest_common_divisor(11, 37)
- 1
- >>> greatest_common_divisor(3, 5)
- 1
- >>> greatest_common_divisor(16, 4)
- 4
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
-
-
 def least_common_multiple_fast(first_num: int, second_num: int) -> int:
  """
  Find the least common multiple of two numbers.

blockchain/diophantine_equation.py

@@ -1,11 +1,13 @@
 from __future__ import annotations
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
  """
  Diophantine Equation : Given integers a,b,c ( at least one of a and b != 0), the
  diophantine equation a*x + b*y = c has a solution (where x and y are integers)
- iff gcd(a,b) divides c.
+ iff greatest_common_divisor(a,b) divides c.
 
  GCD ( Greatest Common Divisor ) or HCF ( Highest Common Factor )
 
@@ -22,7 +24,7 @@ def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
 
  assert (
  c % greatest_common_divisor(a, b) == 0
- ) # greatest_common_divisor(a,b) function implemented below
+ ) # greatest_common_divisor(a,b) is in maths directory
  (d, x, y) = extended_gcd(a, b) # extended_gcd(a,b) function implemented below
  r = c / d
  return (r * x, r * y)
@@ -69,32 +71,6 @@ def diophantine_all_soln(a: int, b: int, c: int, n: int = 2) -> None:
  print(x, y)
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Euclid's Lemma : d divides a and b, if and only if d divides a-b and b
-
- Euclid's Algorithm
-
- >>> greatest_common_divisor(7,5)
- 1
-
- Note : In number theory, two integers a and b are said to be relatively prime,
- mutually prime, or co-prime if the only positive integer (factor) that
- divides both of them is 1 i.e., gcd(a,b) = 1.
-
- >>> greatest_common_divisor(121, 11)
- 11
-
- """
- if a < b:
- a, b = b, a
-
- while a % b != 0:
- a, b = b, a % b
-
- return b
-
-
 def extended_gcd(a: int, b: int) -> tuple[int, int, int]:
  """
  Extended Euclid's Algorithm : If d divides a and b and d = a*x + b*y for integers

ciphers/affine_cipher.py

@@ -1,6 +1,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module as cryptomath
 
 SYMBOLS = (
@@ -26,7 +28,7 @@ def check_keys(key_a: int, key_b: int, mode: str) -> None:
  "Key A must be greater than 0 and key B must "
  f"be between 0 and {len(SYMBOLS) - 1}."
  )
- if cryptomath.gcd(key_a, len(SYMBOLS)) != 1:
+ if gcd_by_iterative(key_a, len(SYMBOLS)) != 1:
  sys.exit(
  f"Key A {key_a} and the symbol set size {len(SYMBOLS)} "
  "are not relatively prime. Choose a different key."
@@ -76,7 +78,7 @@ def get_random_key() -> int:
  while True:
  key_b = random.randint(2, len(SYMBOLS))
  key_b = random.randint(2, len(SYMBOLS))
- if cryptomath.gcd(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
+ if gcd_by_iterative(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
  return key_b * len(SYMBOLS) + key_b
 
 

ciphers/cryptomath_module.py

@@ -1,11 +1,8 @@
-def gcd(a: int, b: int) -> int:
- while a != 0:
- a, b = b % a, a
- return b
+from maths.greatest_common_divisor import gcd_by_iterative
 
 
 def find_mod_inverse(a: int, m: int) -> int:
- if gcd(a, m) != 1:
+ if gcd_by_iterative(a, m) != 1:
  msg = f"mod inverse of {a!r} and {m!r} does not exist"
  raise ValueError(msg)
  u1, u2, u3 = 1, 0, a

ciphers/hill_cipher.py

@@ -39,19 +39,7 @@
 
 import numpy
 
-
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- >>> greatest_common_divisor(4, 8)
- 4
- >>> greatest_common_divisor(8, 4)
- 4
- >>> greatest_common_divisor(4, 7)
- 1
- >>> greatest_common_divisor(0, 10)
- 10
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 class HillCipher:

ciphers/rsa_key_generator.py

@@ -2,6 +2,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module, rabin_miller
 
 
@@ -27,7 +29,7 @@ def generate_key(key_size: int) -> tuple[tuple[int, int], tuple[int, int]]:
  # Generate e that is relatively prime to (p - 1) * (q - 1)
  while True:
  e = random.randrange(2 ** (key_size - 1), 2 ** (key_size))
- if cryptomath_module.gcd(e, (p - 1) * (q - 1)) == 1:
+ if gcd_by_iterative(e, (p - 1) * (q - 1)) == 1:
  break
 
  # Calculate d that is mod inverse of e

maths/carmichael_number.py

@@ -10,14 +10,7 @@
 Examples of Carmichael Numbers: 561, 1105, ...
 https://en.wikipedia.org/wiki/Carmichael_number
 """
-
-
-def gcd(a: int, b: int) -> int:
- if a < b:
- return gcd(b, a)
- if a % b == 0:
- return b
- return gcd(b, a % b)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 def power(x: int, y: int, mod: int) -> int:
@@ -33,7 +26,7 @@ def power(x: int, y: int, mod: int) -> int:
 def is_carmichael_number(n: int) -> bool:
  b = 2
  while b < n:
- if gcd(b, n) == 1 and power(b, n - 1, n) != 1:
+ if greatest_common_divisor(b, n) == 1 and power(b, n - 1, n) != 1:
  return False
  b += 1
  return True

maths/least_common_multiple.py

@@ -1,6 +1,8 @@
 import unittest
 from timeit import timeit
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  """
@@ -20,26 +22,6 @@ def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  return common_mult
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Calculate Greatest Common Divisor (GCD).
- see greatest_common_divisor.py
- >>> greatest_common_divisor(24, 40)
- 8
- >>> greatest_common_divisor(1, 1)
- 1
- >>> greatest_common_divisor(1, 800)
- 1
- >>> greatest_common_divisor(11, 37)
- 1
- >>> greatest_common_divisor(3, 5)
- 1
- >>> greatest_common_divisor(16, 4)
- 4
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
-
-
 def least_common_multiple_fast(first_num: int, second_num: int) -> int:
  """
  Find the least common multiple of two numbers.

blockchain/diophantine_equation.py

@@ -1,11 +1,13 @@
 from __future__ import annotations
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
  """
  Diophantine Equation : Given integers a,b,c ( at least one of a and b != 0), the
  diophantine equation a*x + b*y = c has a solution (where x and y are integers)
- iff gcd(a,b) divides c.
+ iff greatest_common_divisor(a,b) divides c.
 
  GCD ( Greatest Common Divisor ) or HCF ( Highest Common Factor )
 
@@ -22,7 +24,7 @@ def diophantine(a: int, b: int, c: int) -> tuple[float, float]:
 
  assert (
  c % greatest_common_divisor(a, b) == 0
- ) # greatest_common_divisor(a,b) function implemented below
+ ) # greatest_common_divisor(a,b) is in maths directory
  (d, x, y) = extended_gcd(a, b) # extended_gcd(a,b) function implemented below
  r = c / d
  return (r * x, r * y)
@@ -69,32 +71,6 @@ def diophantine_all_soln(a: int, b: int, c: int, n: int = 2) -> None:
  print(x, y)
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Euclid's Lemma : d divides a and b, if and only if d divides a-b and b
-
- Euclid's Algorithm
-
- >>> greatest_common_divisor(7,5)
- 1
-
- Note : In number theory, two integers a and b are said to be relatively prime,
- mutually prime, or co-prime if the only positive integer (factor) that
- divides both of them is 1 i.e., gcd(a,b) = 1.
-
- >>> greatest_common_divisor(121, 11)
- 11
-
- """
- if a < b:
- a, b = b, a
-
- while a % b != 0:
- a, b = b, a % b
-
- return b
-
-
 def extended_gcd(a: int, b: int) -> tuple[int, int, int]:
  """
  Extended Euclid's Algorithm : If d divides a and b and d = a*x + b*y for integers

ciphers/affine_cipher.py

@@ -1,6 +1,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module as cryptomath
 
 SYMBOLS = (
@@ -26,7 +28,7 @@ def check_keys(key_a: int, key_b: int, mode: str) -> None:
  "Key A must be greater than 0 and key B must "
  f"be between 0 and {len(SYMBOLS) - 1}."
  )
- if cryptomath.gcd(key_a, len(SYMBOLS)) != 1:
+ if gcd_by_iterative(key_a, len(SYMBOLS)) != 1:
  sys.exit(
  f"Key A {key_a} and the symbol set size {len(SYMBOLS)} "
  "are not relatively prime. Choose a different key."
@@ -76,7 +78,7 @@ def get_random_key() -> int:
  while True:
  key_b = random.randint(2, len(SYMBOLS))
  key_b = random.randint(2, len(SYMBOLS))
- if cryptomath.gcd(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
+ if gcd_by_iterative(key_b, len(SYMBOLS)) == 1 and key_b % len(SYMBOLS) != 0:
  return key_b * len(SYMBOLS) + key_b
 
 

ciphers/cryptomath_module.py

@@ -1,11 +1,8 @@
-def gcd(a: int, b: int) -> int:
- while a != 0:
- a, b = b % a, a
- return b
+from maths.greatest_common_divisor import gcd_by_iterative
 
 
 def find_mod_inverse(a: int, m: int) -> int:
- if gcd(a, m) != 1:
+ if gcd_by_iterative(a, m) != 1:
  msg = f"mod inverse of {a!r} and {m!r} does not exist"
  raise ValueError(msg)
  u1, u2, u3 = 1, 0, a

ciphers/hill_cipher.py

@@ -39,19 +39,7 @@
 
 import numpy
 
-
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- >>> greatest_common_divisor(4, 8)
- 4
- >>> greatest_common_divisor(8, 4)
- 4
- >>> greatest_common_divisor(4, 7)
- 1
- >>> greatest_common_divisor(0, 10)
- 10
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 class HillCipher:

ciphers/rsa_key_generator.py

@@ -2,6 +2,8 @@
 import random
 import sys
 
+from maths.greatest_common_divisor import gcd_by_iterative
+
 from . import cryptomath_module, rabin_miller
 
 
@@ -27,7 +29,7 @@ def generate_key(key_size: int) -> tuple[tuple[int, int], tuple[int, int]]:
  # Generate e that is relatively prime to (p - 1) * (q - 1)
  while True:
  e = random.randrange(2 ** (key_size - 1), 2 ** (key_size))
- if cryptomath_module.gcd(e, (p - 1) * (q - 1)) == 1:
+ if gcd_by_iterative(e, (p - 1) * (q - 1)) == 1:
  break
 
  # Calculate d that is mod inverse of e

maths/carmichael_number.py

@@ -10,14 +10,7 @@
 Examples of Carmichael Numbers: 561, 1105, ...
 https://en.wikipedia.org/wiki/Carmichael_number
 """
-
-
-def gcd(a: int, b: int) -> int:
- if a < b:
- return gcd(b, a)
- if a % b == 0:
- return b
- return gcd(b, a % b)
+from maths.greatest_common_divisor import greatest_common_divisor
 
 
 def power(x: int, y: int, mod: int) -> int:
@@ -33,7 +26,7 @@ def power(x: int, y: int, mod: int) -> int:
 def is_carmichael_number(n: int) -> bool:
  b = 2
  while b < n:
- if gcd(b, n) == 1 and power(b, n - 1, n) != 1:
+ if greatest_common_divisor(b, n) == 1 and power(b, n - 1, n) != 1:
  return False
  b += 1
  return True

maths/least_common_multiple.py

@@ -1,6 +1,8 @@
 import unittest
 from timeit import timeit
 
+from maths.greatest_common_divisor import greatest_common_divisor
+
 
 def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  """
@@ -20,26 +22,6 @@ def least_common_multiple_slow(first_num: int, second_num: int) -> int:
  return common_mult
 
 
-def greatest_common_divisor(a: int, b: int) -> int:
- """
- Calculate Greatest Common Divisor (GCD).
- see greatest_common_divisor.py
- >>> greatest_common_divisor(24, 40)
- 8
- >>> greatest_common_divisor(1, 1)
- 1
- >>> greatest_common_divisor(1, 800)
- 1
- >>> greatest_common_divisor(11, 37)
- 1
- >>> greatest_common_divisor(3, 5)
- 1
- >>> greatest_common_divisor(16, 4)
- 4
- """
- return b if a == 0 else greatest_common_divisor(b % a, a)
-
-
 def least_common_multiple_fast(first_num: int, second_num: int) -> int:
  """
  Find the least common multiple of two numbers.