Add Fermat Primality Test

Author: arthurvergacas

Maths/FermatPrimalityTest.test.js

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+import { modularExponentiation, fermatPrimeCheck } 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, 50)).toBe(true)
+ expect(fermatPrimeCheck(10, 50)).toBe(false)
+ expect(fermatPrimeCheck(94286167, 50)).toBe(true)
+ expect(fermatPrimeCheck(83165867, 50)).toBe(true)
+ expect(fermatPrimeCheck(13268774, 50)).toBe(false)
+ expect(fermatPrimeCheck(13233852, 50)).toBe(false)
+ })
+})

Maths/test/FermatPrimalityTest.js

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+/**
+ * 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 Liar (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!
+ *
+ * You can find more about the Fermat primality test and its flaws here:
+ * https://en.wikipedia.org/wiki/Fermat_primality_test
+ */
+
+/**
+ * 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) => {
+ // first check for corner cases
+<<<<<<< HEAD
+ if (n <= 1 || n === 4) return false
+=======
+ if (n <= 1 || n == 4) return false
+>>>>>>> 951c7258323a057041c0d128880982ddab303ee5
+ if (n <= 3) return true // 2 and 3 are included here
+
+ for (let i = 0; i < numberOfIterations; i++) {
+ // pick a random number between 2 and n - 2
+<<<<<<< HEAD
+ const randomNumber = Math.floor(Math.random() * (n - 1 - 2) + 2)
+=======
+ let randomNumber = Math.floor(Math.random() * (n - 1 - 2) + 2)
+>>>>>>> 951c7258323a057041c0d128880982ddab303ee5
+
+ // 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 }