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2026-01-11 09:19:22 +01:00
commit 50f7b1159e
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+#!/usr/bin/env python3
+
+from Cryptotools.Numbers.primeNumber import gcd, isPrimeNumber
+from Cryptotools.lcm import lcm
+from Cryptotools.Numbers.coprime import phi
+from Cryptotools.Utils.utils import gcd
+from sympy import factorint
+from functools import reduce
+
+
+def carmichael_lambda(n: int) -> int:
+    """
+        This function is a carmichael lambda for identifying the smallest integer m
+        and saisfy this condition: a ** m congrue 1 modulo n
+        This function is relatively closed with the euler's totient (phi of n)
+
+        Args:
+            n (Integer): found the smallest integer of n
+
+        Returns:
+            REturn the smallest integer of n        
+    """
+
+    # First, factorizing n number
+    # factorint return a list of prime factors of n
+    # Base on that theorem:
+    #    https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic
+    factorsN = factorint(n)
+
+    """
+        The result of the factorint, we have the following:
+        n = p ** e1, p ** e2, ..., p ** ek
+    """
+
+    factors = list()
+
+    for p, e in factorsN.items():
+        # Base on the theorem of arithmetic, p is always prime
+        if isPrimeNumber(p):
+            if p == 2 and e >= 3:
+                factors.append(phi(pow(p, e)) // 2)
+            elif p >= 2 and e > 0:
+                factors.append(phi(pow(p, e)))
+            elif p > 2:
+                factors.append(phi(pow(p, e)))
+
+    # Now, we can compute lcm 
+    result = reduce(lcm, factors)
+
+    return result
+
+def carmichael_lambda2(n):
+    """
+        Deprecated function
+    """
+    # Get all coprimes with n
+    coprimes = list()
+    for a in range(1, n):
+        if gcd(a, n) == 1:
+            coprimes.append(a)
+
+            # Need to find the smallest value
+            """
+                Need to find the smallest value m. Must to satisfy this condition:
+                a ** m congrus 1 mod n
+            """
+            for m in range(1, n):
+                print(f"{m} {a ** m % n}")
+
+    print(coprimes)
+
+def carmi_numbers(n):
+    """
+        This function get all GCD of the number if these number are prime or not
+        
+        Args:
+            n (Integer): Iterate to n elements
+
+        Returns:
+            Return a list of prime numbers
+    """
+    # https://kconrad.math.uconn.edu/blurbs/ugradnumthy/carmichaelkorselt.pdf
+    primes = list()
+    for i in range(2, n - 1):
+        if gcd(n, i) == i:
+            if isPrimeNumber(i):
+                primes.append(i)
+
+    return primes
+
+def is_carmichael(n, primes):
+    """
+        This function check if the n number is a carmichael number. The arguments primes at least 3 prime numbers
+        As it's said in the Carmichael theorem, a carmichael number has three factors and they are all primes: https://en.wikipedia.org/wiki/Carmichael_number
+        With the Korselt's criterion, to check if the number is a carmichael number p - 1 / n - 1.
+
+        Args:
+            n (Integer): the carmichael number
+            primes (list): list of prime numbers
+
+        Returns:
+            Boolean of the carmichael's number result
+    """
+    r = True
+    n = n - 1
+
+    if len(primes) < 3:
+        return False
+
+    # Korselt's criterion
+    for p in primes:
+        p = p - 1
+        if n % p != 0:
+            r = False
+            break
+    return r
+
+def is_carmi_number(n) -> bool:
+    """
+        This function check if the number n is a carmichael number (pseudoprime)
+
+        Args:
+            n (Integer): Iterate to n elements
+
+        Returns:
+            Return a Boolean if n is a carmichael number or not. True if yes
+    """
+    j = 0
+    for i in range(n):
+        if i ** n % n == i:
+            j += 1
+    if j == n:
+        return True
+    return False
+
+def generate_carmi_numbers(nbr) -> list:
+    """
+        This function generate carmichael numbers, they are called pseudoprimes
+        For instance: a = 545, n = 561
+        gcd(a, n) = 1
+        pow(a, n - 1) % n = 1
+
+        Args:
+            nbr (Integer): Find all pseudoprimes until nbr is achieves
+
+        Returns:
+            Return the list of pseudoprimes
+    """
+    carmi = list()
+
+    count = 0
+    n = 2
+    while count < nbr:
+        # First, we need to find a, coprime with n
+        for a in range(1, n):
+            # We check if n is coprime with a
+            # All integers must be coprime with n, a belong to n
+            if gcd(a, n) == 1:
+                # Check if is a carmichael number
+                # Fermat's little theorem, a ** (n - 1) % n, must be congruent to 1
+                if pow(a, (n - 1)) % n == 1:
+                #if (a ** (n - 1)) % n == 1:
+                    #print(f"{gcd(a, n)} {a} {n}")
+                    if n not in carmi:
+                        count += 1
+                        carmi.append(n)
+            else:
+                print(f"{gcd(a, n)} {a} {n}")
+
+                break
+        n += 1
+    print(carmi)
+    #for cop in range(2, n):
+    #    if gcd(cop, n) == 1:
+    #        coprimes.append(cop)
+
+    return carmi