merge: Implement Shor's factorization algorithm (#1070)

* Updated Documentation in README.md

* merge: Fix GetEuclidGCD (#1068) (#1069)

* Fix GetEuclidGCD

Implement the actual Euclidean Algorithm

* Replace == with ===

* Lua > JS

* Standard sucks

* Oops

* Update GetEuclidGCD.js

* Updated Documentation in README.md

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

Co-authored-by: Lars Müller <34514239+appgurueu@users.noreply.github.com>
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

* feat: implement Shor's Algorithm

* chore: add tests

* Updated Documentation in README.md

* chore: fix spelling

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
Co-authored-by: Lars Müller <34514239+appgurueu@users.noreply.github.com>

DIRECTORY.md

@@ -195,6 +195,7 @@
   * [RadianToDegree](Maths/RadianToDegree.js)
   * [ReverseNumber](Maths/ReverseNumber.js)
   * [ReversePolishNotation](Maths/ReversePolishNotation.js)
+  * [ShorsAlgorithm](Maths/ShorsAlgorithm.js)
   * [SieveOfEratosthenes](Maths/SieveOfEratosthenes.js)
   * [SimpsonIntegration](Maths/SimpsonIntegration.js)
   * [Softmax](Maths/Softmax.js)

Maths/ShorsAlgorithm.js

@@ -0,0 +1,98 @@
+/**
+ * @function ShorsAlgorithm
+ * @description Classical implementation of Shor's Algorithm.
+ * @param {Integer} num - Find a non-trivial factor of this number.
+ * @returns {Integer} - A non-trivial factor of num.
+ * @see https://en.wikipedia.org/wiki/Shor%27s_algorithm
+ * @see https://www.youtube.com/watch?v=lvTqbM5Dq4Q
+ *
+ * Shor's algorithm is a quantum algorithm for integer factorization. This
+ * function implements a version of the algorithm which is computable using
+ * a classical computer, but is not as efficient as the quantum algorithm.
+ *
+ * The algorithm basically consists of guessing a number g which may share
+ * factors with our target number N, and then use Euclid's GCD algorithm to
+ * find the common factor.
+ *
+ * The algorithm starts with a random guess for g, and then improves the
+ * guess by using the fact that for two coprimes A and B, A^p = mB + 1.
+ * For our purposes, this means that g^p = mN + 1. This mathematical
+ * identity can be rearranged into (g^(p/2) + 1)(g^(p/2) - 1) = mN.
+ * Provided that p/2 is an integer, and neither g^(p/2) + 1 nor g^(p/2) - 1
+ * are a multiple of N, either g^(p/2) + 1 or g^(p/2) - 1 must share a
+ * factor with N, which can then be found using Euclid's GCD algorithm.
+ */
+function ShorsAlgorithm (num) {
+  const N = BigInt(num)
+
+  while (true) {
+    // generate random g such that 1 < g < N
+    const g = BigInt(Math.floor(Math.random() * (num - 1)) + 2)
+
+    // check if g shares a factor with N
+    // if it does, find and return the factor
+    let K = gcd(g, N)
+    if (K !== 1) return K
+
+    // find p such that g^p = mN + 1
+    const p = findP(g, N)
+
+    // p needs to be even for it's half to be an integer
+    if (p % 2n === 1n) continue
+
+    const base = g ** (p / 2n) // g^(p/2)
+    const upper = base + 1n // g^(p/2) + 1
+    const lower = base - 1n // g^(p/2) - 1
+
+    // upper and lower can't be a multiple of N
+    if (upper % N === 0n || lower % N === 0n) continue
+
+    // either upper or lower must share a factor with N
+    K = gcd(upper, N)
+    if (K !== 1) return K // upper shares a factor
+    return gcd(lower, N) // otherwise lower shares a factor
+  }
+}
+
+/**
+ * @function findP
+ * @description Finds a value p such that A^p = mB + 1.
+ * @param {BigInt} A
+ * @param {BigInt} B
+ * @returns The value p.
+ */
+function findP (A, B) {
+  let p = 1n
+  while (!isValidP(A, B, p)) p++
+  return p
+}
+
+/**
+ * @function isValidP
+ * @description Checks if A, B, and p fulfill A^p = mB + 1.
+ * @param {BigInt} A
+ * @param {BigInt} B
+ * @param {BigInt} p
+ * @returns Whether A, B, and p fulfill A^p = mB + 1.
+ */
+function isValidP (A, B, p) {
+  // A^p = mB + 1 => A^p - 1 = 0 (mod B)
+  return (A ** p - 1n) % B === 0n
+}
+
+/**
+ * @function gcd
+ * @description Euclid's GCD algorithm.
+ * @param {BigInt} A
+ * @param {BigInt} B
+ * @returns Greatest Common Divisor between A and B.
+ */
+function gcd (A, B) {
+  while (B !== 0n) {
+    [A, B] = [B, A % B]
+  }
+
+  return Number(A)
+}
+
+export { ShorsAlgorithm }

Maths/test/ShorsAlgorithm.test.js

@@ -0,0 +1,29 @@
+import { ShorsAlgorithm } from '../ShorsAlgorithm'
+import { fermatPrimeCheck } from '../FermatPrimalityTest'
+
+describe("Shor's Algorithm", () => {
+  const N = 10 // number of tests
+  const max = 35000 // max value to factorize
+  const min = 1000 // min value to factorize
+
+  for (let i = 0; i < N; i++) {
+    while (true) {
+      const num = Math.floor(Math.random() * max) + min
+      // num must be composite, don't care for false negatives
+      if (fermatPrimeCheck(num, 1)) continue
+
+      it('should find a non-trivial factor of ' + num, () => {
+        const f = ShorsAlgorithm(num)
+
+        // should not be trivial
+        expect(f).not.toEqual(1)
+        expect(f).not.toEqual(num)
+
+        // should be a factor
+        expect(num % f).toEqual(0)
+      })
+
+      break
+    }
+  }
+})