chore: Merge pull request TheAlgorithms#790 from arthurvergacas/Fermat-Primality-Test

Add Fermat Primality Test

Author: raklaptudirm

DIRECTORY.md

@@ -157,6 +157,7 @@
  * [Factorial](https://github.com/TheAlgorithms/Javascript/blob/master/Maths/Factorial.js)
  * [Factors](https://github.com/TheAlgorithms/Javascript/blob/master/Maths/Factors.js)
  * [FareyApproximation](https://github.com/TheAlgorithms/Javascript/blob/master/Maths/FareyApproximation.js)
+ * [FermatPrimalityTest](https://github.com/TheAlgorithms/Javascript/blob/master/Maths/FermatPrimalityTest.js)
  * [Fibonacci](https://github.com/TheAlgorithms/Javascript/blob/master/Maths/Fibonacci.js)
  * [FindHcf](https://github.com/TheAlgorithms/Javascript/blob/master/Maths/FindHcf.js)
  * [FindLcm](https://github.com/TheAlgorithms/Javascript/blob/master/Maths/FindLcm.js)

Maths/FermatPrimalityTest.js

@@ -0,0 +1,97 @@
+/*
+ * The Fermat primality test is a probabilistic test to determine whether a number
+ * is a probable prime.
+ *
+ * It relies on Fermat's Little Theorem, which states that if p is prime and a
+ * is not divisible by p, then
+ *
+ * a^(p - 1) % p = 1
+ *
+ * However, there are certain numbers (so called Fermat Liars) that screw things up;
+ * if a is one of these liars the equation will hold even though p is composite.
+ *
+ * But not everything is lost! It's been proven that at least half of all integers
+ * aren't Fermat Liars (these ones called Fermat Witnesses). Thus, if we keep
+ * testing the primality with random integers, we can achieve higher reliability.
+ *
+ * The interesting about all of this is that since half of all integers are
+ * Fermat Witnesses, the precision gets really high really fast! Suppose that we
+ * make the test 50 times: the chance of getting only Fermat Liars in all runs is
+ *
+ * 1 / 2^50 = 8.8 * 10^-16 (a pretty small number)
+ *
+ * For comparison, the probability of a cosmic ray causing an error to your
+ * infalible program is around 1.4 * 10^-15. An order of magnitude below!
+ *
+ * But because nothing is perfect, there's a major flaw to this algorithm, and
+ * the cause are the so called Carmichael Numbers. These are composite numbers n
+ * that hold the equality from Fermat's Little Theorem for every a < n (excluding
+ * is factors). In other words, if we are trying to determine if a Carmichael Number
+ * is prime or not, the chances of getting a wrong answer are pretty high! Because
+ * of that, the Fermat Primality Test is not used is serious applications. :(
+ *
+ * You can find more about the Fermat primality test and its flaws here:
+ * https://en.wikipedia.org/wiki/Fermat_primality_test
+ *
+ * And about Carmichael Numbers here:
+ * https://primes.utm.edu/glossary/xpage/CarmichaelNumber.html
+ */
+
+/**
+ * Faster exponentiation that capitalize on the fact that we are only interested
+ * in the modulus of the exponentiation.
+ *
+ * Find out more about it here: https://en.wikipedia.org/wiki/Modular_exponentiation
+ *
+ * @param {number} base
+ * @param {number} exponent
+ * @param {number} modulus
+ */
+const modularExponentiation = (base, exponent, modulus) => {
+ if (modulus === 1) return 0 // after all, any x % 1 = 0
+
+ let result = 1
+ base %= modulus // make sure that base < modulus
+
+ while (exponent > 0) {
+ // if exponent is odd, multiply the result by the base
+ if (exponent % 2 === 1) {
+ result = (result * base) % modulus
+ exponent--
+ } else {
+ exponent = exponent / 2 // exponent is even for sure
+ base = (base * base) % modulus
+ }
+ }
+
+ return result
+}
+
+/**
+ * Test if a given number n is prime or not.
+ *
+ * @param {number} n The number to check for primality
+ * @param {number} numberOfIterations The number of times to apply Fermat's Little Theorem
+ * @returns True if prime, false otherwise
+ */
+const fermatPrimeCheck = (n, numberOfIterations = 50) => {
+ // first check for edge cases
+ if (n <= 1 || n === 4) return false
+ if (n <= 3) return true // 2 and 3 are included here
+
+ for (let i = 0; i < numberOfIterations; i++) {
+ // pick a random number a, with 2 <= a < n - 2
+ const randomNumber = Math.floor(Math.random() * (n - 2) + 2)
+
+ // if a^(n - 1) % n is different than 1, n is composite
+ if (modularExponentiation(randomNumber, n - 1, n) !== 1) {
+ return false
+ }
+ }
+
+ // if we arrived here without finding a Fermat Witness, this is almost guaranteed
+ // to be a prime number (or a Carmichael number, if you are unlucky)
+ return true
+}
+
+export { modularExponentiation, fermatPrimeCheck }

Maths/test/FermatPrimalityTest.test.js

@@ -0,0 +1,19 @@
+import { fermatPrimeCheck, modularExponentiation } from '../FermatPrimalityTest'
+
+describe('modularExponentiation', () => {
+ it('should give the correct output for all exponentiations', () => {
+ expect(modularExponentiation(38, 220, 221)).toBe(1)
+ expect(modularExponentiation(24, 220, 221)).toBe(81)
+ })
+})
+
+describe('fermatPrimeCheck', () => {
+ it('should give the correct output for prime and composite numbers', () => {
+ expect(fermatPrimeCheck(2, 35)).toBe(true)
+ expect(fermatPrimeCheck(10, 30)).toBe(false)
+ expect(fermatPrimeCheck(94286167)).toBe(true)
+ expect(fermatPrimeCheck(83165867)).toBe(true)
+ expect(fermatPrimeCheck(13268774)).toBe(false)
+ expect(fermatPrimeCheck(13233852)).toBe(false)
+ })
+})