Logical Implementation of Shor Algorithm (#12317)

* Added Shor Algorithm for RSA n factorization

* updating DIRECTORY.md

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* Added referred website and handle type hints

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* Developed the Logical implementation of Shor Algo #12318

* Apply suggestion from @cclauss

---------

Co-authored-by: joelkurien <joelkurien@users.noreply.github.com>
Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>

Author: 3 people

DIRECTORY.md

@@ -1408,6 +1408,7 @@
 
 ## [Quantum](quantum)
   * [Q Fourier Transform](quantum/q_fourier_transform.py)
+  * [Shor Algorithm](quantum/shor_algorithm.py)
 
 ## [Scheduling](scheduling)
   * [Cpuschedulingalgorithms](scheduling/cpuschedulingalgorithms.py)

quantum/shor_algorithm.py

@@ -0,0 +1,67 @@
+import math
+import random
+
+"""
+Shor Algorithm is one of the basic quantum computing algorithm
+that is used in breaking the RSA cryptography protocol, by finding the
+prime numbers that are used to create the public key value, n
+
+In this implementation, I have used a very simple construct without
+the use of qiskit or cirq to help understand how Shor algorithm's
+idea actually works.
+
+Website referred for shor algorithm:
+https://www.geeksforgeeks.org/shors-factorization-algorithm/
+
+"""
+
+
+class Shor:
+    def period_find(self, num: int, number: int) -> int:
+        """
+        Find the period of a^x mod N.
+
+        >>> shor = Shor()
+        >>> shor.period_find(2, 15)
+        4
+        >>> shor.period_find(3, 7)
+        6
+        """
+        start: int = 1
+        while pow(num, start, number) != 1:
+            start += 1
+        return start
+
+    def shor_algorithm(self, number: int) -> tuple[int, int]:
+        """
+        Run Shor's algorithm to factor a number.
+        >>> shor = Shor()
+        >>> random.seed(0)
+        >>> factors = shor.shor_algorithm(15)
+        >>> isinstance(factors, tuple) and len(factors) == 2
+        True
+        >>> factors
+        (3, 5)
+        """
+        if number % 2 == 0:
+            return 2, number // 2
+        while True:
+            random.seed(0)
+            num: int = random.randint(2, number - 1)
+            gcd_number_num: int = math.gcd(number, num)
+            if gcd_number_num > 1:
+                return gcd_number_num, number // gcd_number_num
+
+            result: int = self.period_find(num, number)
+            if not result % 2:
+                start: int = pow(num, result // 2, number)
+                if start != number - 1:
+                    p_value: int = math.gcd(start - 1, number)
+                    q_value: int = math.gcd(start + 1, number)
+                    if p_value > 1 and q_value > 1:
+                        return p_value, q_value
+
+
+if __name__ == "__main__":
+    shor = Shor()
+    print(shor.shor_algorithm(15))