Extended Miller-Rabin Primality test [code]

* Adds a more readable documentation and clear naming for variables to the Miller-Rabin Primality Test example
   * Miller-Rabin can fail under certain situations. These have to throw an error.
* Updates the relevant playground files
* Add name to documentation

Author: Simon C. Krüger

Miller-Rabin Primality Test/MRPrimality.playground/Contents.swift

@@ -5,25 +5,27 @@
 print("Hello, Swift 4!")
 #endif
 
-// Real primes
-mrPrimalityTest(5)
-mrPrimalityTest(439)
-mrPrimalityTest(1201)
-mrPrimalityTest(143477)
-mrPrimalityTest(1299869)
-mrPrimalityTest(15487361)
-mrPrimalityTest(179426363)
-mrPrimalityTest(32416187747)
-
-// Fake primes
-mrPrimalityTest(15)
-mrPrimalityTest(435)
-mrPrimalityTest(1207)
-mrPrimalityTest(143473)
-mrPrimalityTest(1291869)
-mrPrimalityTest(15487161)
-mrPrimalityTest(178426363)
-mrPrimalityTest(32415187747)
-
-// With iteration
-mrPrimalityTest(32416190071, iteration: 10)
+do {
+    // Real primes
+    try checkWithMillerRabin(5)
+    try checkWithMillerRabin(439)
+    try checkWithMillerRabin(1201)
+    try checkWithMillerRabin(143477)
+    try checkWithMillerRabin(1299869)
+    try checkWithMillerRabin(15487361)
+    try checkWithMillerRabin(179426363)
+    
+    // Fake primes
+    try checkWithMillerRabin(15)
+    try checkWithMillerRabin(435)
+    try checkWithMillerRabin(1207)
+    try checkWithMillerRabin(143473)
+    try checkWithMillerRabin(1291869)
+    try checkWithMillerRabin(15487161)
+    try checkWithMillerRabin(178426363)
+    
+    // Specifying accuracy
+    try checkWithMillerRabin(179426363, accuracy: 10)
+} catch {
+    dump(error)
+}

Miller-Rabin Primality Test/MRPrimality.playground/Sources/MRPrimality.swift

@@ -0,0 +1,100 @@
+//
+//  MRPrimality.swift
+//
+//
+//  Created by Sahn Cha on 2016. 10. 18..
+//
+//
+
+import Foundation
+
+enum MillerRabinError: Error {
+    case primeLowAccuracy
+    case primeLowerBorder
+    case uIntOverflow
+}
+
+/*
+ The Miller–Rabin test relies on an equality or set of equalities that
+ hold true for prime values, then checks whether or not they hold for
+ a number that we want to test for primality.
+ 
+ - Parameter n: an odd integer to be tested for primality;
+ - Parameter k: a parameter that determines the accuracy of the test
+ - throws: Can throw an error of type `MillerRabinError`.
+ - Returns: composite if n is composite, otherwise probably prime
+ */
+public func checkWithMillerRabin(_ n: UInt, accuracy k: UInt = 1) throws -> Bool {
+    guard k > 0 else { throw MillerRabinError.primeLowAccuracy }
+    guard n > 0 else { throw MillerRabinError.primeLowerBorder }
+    guard n > 3 else { return true }
+    
+    // return false for all even numbers bigger than 2
+    if n % 2 == 0 {
+        return false
+    }
+    
+    let s: UInt = UInt((n - 1).trailingZeroBitCount)
+    let d: UInt = (n - 1) >> s
+    
+    guard UInt(pow(2.0, Double(s))) * d == n - 1 else { throw MillerRabinError.primeLowerBorder }
+    
+    /// Inspect whether a given witness will reveal the true identity of n.
+    func tryComposite(_ a: UInt, d: UInt, n: UInt) throws -> Bool? {
+        var x = try calculateModularExponentiation(base: a, exponent: d, modulus: n)
+        if x == 1 || x == (n - 1) {
+            return nil
+        }
+        for _ in 1..<s {
+            x = try calculateModularExponentiation(base: x, exponent: 2, modulus: n)
+            if x == 1 {
+                return false
+            } else if x == (n - 1) {
+                return nil
+            }
+        }
+        return false
+    }
+    
+    for _ in 0..<k {
+        let a = UInt.random(in: 2..<n-2)
+        if let composite = try tryComposite(a, d: d, n: n) {
+            return composite
+        }
+    }
+    
+    return true
+}
+
+/*
+ Calculates the modular exponentiation based on `Applied Cryptography by Bruce Schneier.`
+ in `Schneier, Bruce (1996). Applied Cryptography: Protocols, Algorithms,
+ and Source Code in C, Second Edition (2nd ed.). Wiley. ISBN 978-0-471-11709-4.`
+ 
+ - Parameter base: The natural base b.
+ - Parameter base: The natural exponent e.
+ - Parameter base: The natural modulus m.
+ - Throws: Can throw a `uIntOverflow` if the modulus' square exceeds the memory
+ limitations of UInt on the current system.
+ - Returns: The modular exponentiation c.
+ */
+private func calculateModularExponentiation(base: UInt, exponent: UInt, modulus: UInt) throws -> UInt {
+    guard modulus > 1 else { return 0 }
+    guard !(modulus-1).multipliedReportingOverflow(by: (modulus-1)).overflow else {
+        throw MillerRabinError.uIntOverflow
+    }
+    
+    var result: UInt = 1
+    var exponentCopy = exponent
+    var baseCopy = base % modulus
+    
+    while exponentCopy > 0 {
+        if exponentCopy % 2 == 1 {
+            result = (result * baseCopy) % modulus
+        }
+        exponentCopy = exponentCopy >> 1
+        baseCopy = (baseCopy * baseCopy) % modulus
+    }
+    
+    return result
+}

Miller-Rabin Primality Test/MRPrimality.playground/Sources/MillerRabin.swift

@@ -0,0 +1,8 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<!DOCTYPE plist PUBLIC "-//Apple//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd">
+<plist version="1.0">
+<dict>
+	<key>IDEDidComputeMac32BitWarning</key>
+	<true/>
+</dict>
+</plist>