Anul 3 Semestrul 1

Anul 3/Semestrul 1/Public Key Cryptography/Crypto2.py

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+#7. Algorithm for determining all Carmichael numbers less than a given bound.
+
+import math
+
+# Helper function to find all prime factors of a number
+def prime_factors(n):
+    factors = set()
+    # Check for divisibility by 2
+    while n % 2 == 0:
+        factors.add(2)
+        n //= 2
+    # Check for odd factors from 3 upwards
+    for i in range(3, int(math.sqrt(n)) + 1, 2):
+        while n % i == 0:
+            factors.add(i)
+            n //= i
+    # If n is prime and greater than 2 then add it to the set
+    if n > 2:
+        factors.add(n)
+    return factors
+
+# Function to check if a number is square-free (no repeated prime factors)
+def is_square_free(n, prime_factors):
+    for p in prime_factors:
+        if n % (p * p) == 0:
+            return False
+    return True
+
+# Function to check for Carmichael numbers
+def is_carmichael(n):
+    prime_factors_of_n = prime_factors(n)
+    if(len(prime_factors_of_n)<2):
+        return False
+    if n < 3 or not is_square_free(n, prime_factors_of_n):
+        return False
+    for p in prime_factors_of_n:
+        if (n - 1) % (p - 1) != 0:
+            return False
+    return True
+
+# Function to find all Carmichael numbers below a given bound
+def carmichael_numbers_below(bound):
+    carmichaels = []
+    for n in range(3, bound):
+        if is_carmichael(n):
+            carmichaels.append(n)
+    return carmichaels
+
+bound = 552722
+carmichaels = carmichael_numbers_below(bound)
+print(f"Carmichael numbers below {bound}:")
+print(carmichaels)
+
+#https://oeis.org/A002997