Тест на первичность Миллера-Рабина терпит неудачу для больших чисел

Question

После изучения других ответов SO, связанных с критерием Миллера-Рабина, для ясности, я реализовал версию на C#, но она начинает иногда терпеть неудачу где-то в районе трех миллиардов, а время, которое доходит до четырех миллиардов, перестает признавать любые простые числа. Я подозреваю, что я страдаю от переполнения, но не могу понять, где. Моя цель состоит в том, чтобы заставить это работать для любого значения в диапазоне от 0 < = п < = 2^63 - 1.

Я создал скрипку: https://dotnetfiddle.net/3F7P97

Среди идей, которые я пробовал были:

1. Использование Precalculated основы 2, 325, 9375, 28178, 450775, 9780504, 1795265022, рекламируемые, как хорошо работает для чисел меньше 2^64 с этого сайта: http://miller-rabin.appspot.com/ Это был рекомендован с помощью отвечающего этого вопроса: Miller Rabin Primality test accuracy

2. Написание перегрузочной функции power-mod для вычисления a^b mod n.

3. Запись функции умножения при переполнении для вычисления a * b mod n (с использованием русского крестьянского алгоритма).

Вот код из скрипки, как в то время я создал этот вопрос:

using System; 
using System.Collections.Generic; 
using System.Linq; 

// AUTHOR: Paul A. Chernoch 
// 
// Purpose: Use Rabin-Miller algorithm to test if numbers are prime. 
// Problem: Somewhere between 2 billion and 4,194,304,903 it stops working and always says the number is not prime. 
// Hypothesis: The code should work for all 64-bit values, but suspiciously breaks near the maximum value for a signed 32-bit integer. 
public class Program 
{ 
    public static void Main() 
    { 
     // These cases always succeed. 
     for (long n = 0; n < 20; n++) 
     { 
      TestRabinMiller(n); 
     } 

     TestRabinMiller(2000000011L); 
     TestRabinMiller(2147483647L); // 2^31 - 1 is prime. 
     TestRabinMiller(2147483659L); // 2^31 + 11 is prime. 

     // These cases fail! I think it has to do with overflow on a multiplication or something. 

     TestRabinMiller(3042000007L); // Sometimes succeeds, sometimes fails 
     TestRabinMiller(3043000003L); // Sometimes succeeds, sometimes fails 
     TestRabinMiller(3045000031L); // Sometimes succeeds, sometimes fails 
     TestRabinMiller(4000000007L); // Always fails 
     TestRabinMiller(4194304903L); // Always fails 
     TestRabinMiller(4294967291L); // Always fails 
     TestRabinMiller(4294967311L); // Always fails 
    } 

    public static void TestRabinMiller(long n) 
    { 
     var factors = BuggyCode.RabinMiller.Factor(n); 
     var expectedIsPrime = factors.Count() == 1 && n >= 2; 
     var expectedWords = expectedIsPrime ? "IS A PRIME. " : "IS NOT PRIME."; 
     var actualIsPrime = BuggyCode.RabinMiller.IsPrime(n,20); 
     var actualWords = actualIsPrime ? "IS A PRIME. " : "IS NOT PRIME."; 
     var results = actualIsPrime == expectedIsPrime ? "SUCCEEDED." : "FAILED. "; 
     Console.WriteLine(String.Format("Test of RabinMiller {0} It says that {1} {2} In reality, the number {1} {3}", results, n, actualWords, expectedWords)); 
    } 
} 

namespace BuggyCode 
{ 

    /// <summary> 
    /// Test if a number is prime using the Rabin-Miller primality test. 
    /// </summary> 
    public class RabinMiller 
    { 
     private static HashSet<long> KnownPrimes = new HashSet<long>() 
     { 
      2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 
      31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 
      73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 
      127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 
      179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 
      233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 
      283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 
      353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 
      419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 
      467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 
      547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 
      607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 
      661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 
      739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 
      811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 
      877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 
      947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 
      1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 
      1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 
      1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 
      1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 
      1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 
      1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 
      1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 
      1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 
      1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 
      1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 
      1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 
      1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 
      1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 
      1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 
      2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 
      2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 
      2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 
      2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 
      2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 
      2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 
      2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 
      2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 
      2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 
      2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 
      2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 
      2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 
      3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 
      3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 
      3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 
      3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 
      3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 
      3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 
      3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 
      3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 
      3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 
      3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 
      3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 
      3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 
      4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 
      4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 
      4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 
      4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 
      4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 
      4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 
      4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 
      4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 
      4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 
      4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 
      4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 
      4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 
      5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 
      5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 
      5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 
      5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 
      5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 
      5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 
      5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 
      5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 
      5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 
      5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 
      5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 
      5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 
      6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 
      6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 
      6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 
      6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 
      6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 
      6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 
      6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 
      6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 
      6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 
      6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 
      6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 
      7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 
      7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 
      7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 
      7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 
      7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 
      7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 
      7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 
      7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 
      7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 
      7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919 
     }; 

     private static long MaxKnownPrime { get; set; } 

     static RabinMiller() 
     { 
      MaxKnownPrime = KnownPrimes.Max(); 
     } 

     /// <summary> 
     /// For the deterministic Rabin-Miller test, these are the best bases for numbers below 2^64. 
     /// 
     /// See http://miller-rabin.appspot.com/ 
     /// </summary> 
     private static long[] BestRabinMillerBases = new long[] { 2, 325, 9375, 28178, 450775, 9780504, 1795265022 }; 

     /// <summary> 
     /// The smallest prime factor for small numbers. 
     /// </summary> 
     private static long[] FactorsForSmallNumbers = new long[] { 0, 1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2 }; 


     /// <summary> 
     /// Rabin-Miller primality test. 
     /// 
     /// The error rate of false results is (1/4)^k. 
     /// </summary> 
     /// <param name="n">Number to test for primality.</param> 
     /// <param name="k">Number of different bases to test. 
     /// The higher the number, the more accurate the test and the longer the running time.</param> 
     /// <returns><c>true</c> if n is prime; otherwise, <c>false</c>. 
     /// Note: Zero and one are not considered prime. 
     /// </returns> 
     public static bool IsPrime(long n, int k) 
     { 
      if(n < 2) 
      { 
       return false; // Zero and one are not prime. 
      } 

      // Speedup for low values that also improves accuracy. 
      if (n <= MaxKnownPrime) 
       return KnownPrimes.Contains (n); 

      foreach(var knownPrime in KnownPrimes) 
      { 
       if (n % knownPrime == 0) return false; 
      } 

      var s = n - 1L; 
      while((s & 1L) == 0L) 
      { 
       s >>= 1; 
      } 
      Random r = new Random(); 
      for (int i = 0; i < k; i++) 
      { 
       long a; 
       if (i < BestRabinMillerBases.Length) 
        a = BestRabinMillerBases [i]; 
       else // Random choice of base. 
        a = (long)(r.NextDouble() * (n - 1L)) + 1L; 
       var temp = s; 
       var mod = ModuloPower(a, temp, n); 
       while(temp != n - 1L && mod != 1L && mod != n - 1L) 
       { 
        mod = RussianPeasant(mod, mod, n); 
        temp = temp << 1; 
       } 
       if(mod != n - 1L && (temp & 1L) == 0L) 
       { 
        return false; 
       } 
      } 
      return true; 
     } 

     public static bool IsPrime(long n) 
     { 
      var k = 1; 
      var temp = n; 
      while (temp > 0L) 
      { 
       temp /= 10L; 
       k++; 
      } 
      k = Math.Max (5, k); 
      return IsPrime (n, k); 
     } 

     /// <summary> 
     /// Return a^b mod n but guard against overflow. 
     /// 
     /// Use repeated squarings to reduce the number of operations. 
     /// Special case: Assume 0^0 = 1 to be consistenct with Math.Pow. 
     /// 
     /// See https://helloacm.com/compute-powermod-abn/ 
     /// </summary> 
     /// <param name="a">Base to be exponentiated.</param> 
     /// <param name="b">The exponent.</param> 
     /// <param name="n">Modulus.</param> 
     /// <returns>a^b mod n.</returns> 
     public static long ModuloPower(long a, long b, long n) 
     { 
      // return (a^b)%n -> Simple calculation that would often overflow 
      // Example: For a^19, there are five squarings, two multipications and seven modulos, instead of 18 multiplications and eighteen modulos 
      //  a^19 -> (a^2)^9 * a -> (((a^2)^2)^4 * (a^2)) * a -> ((((a^2)^2)^2)^2 * (a^2)) * a 
      if (b == 0L) return 1L; 
      if (a == 0L) return 0L; 
      if (b == 1L) return a % n; 
      var r = ModuloPower (a, b >> 1, n); 
      r = RussianPeasant(r, r, n); 
      if ((b & 1L) == 1L) 
       r = RussianPeasant(r, a, n); 
      return r; 
     } 

     /// <summary> 
     /// Russian peasant multiplication of a*b mod c, which avoids overflow. 
     /// </summary> 
     /// <param name="a">First multiplicand.</param> 
     /// <param name="b">Second multiplicand.</param> 
     /// <param name="c">Modulus.</param> 
     /// <returns>a * b mod c</returns> 
     public static long RussianPeasant(long a, long b, long c) 
     { 
      const long _2_32 = 1L << 32; 
      a = Math.Abs (a); 
      b = Math.Abs (b); 
      if (a < _2_32 && b < _2_32) 
       return (a * b % c); // No possibility of overflow. 
      if (c < _2_32) 
       return (a % c) * (b % c) % c; 
      long ret = 0; 
      while(b != 0) { 
       if((b&1L) != 0L) { 
        ret += a; 
        ret %= c; 
       } 
       a *= 2; 
       a %= c; 
       b /= 2; 
      } 
      return ret; 
     } 



     /// <summary> 
     /// Slow, exhaustive but simple method of finding prime factors, useful for testing against the more complex methods. 
     /// 
     /// Its only speedup is a table of known primes. 
     /// </summary> 
     /// <param name="n">The number to be factored.</param> 
     /// <returns>Prime factors of n, sorted frmo low to high.</returns> 
     public static List<long> Factor(long n) 
     { 
      var factors = new List<long>(); 
      var lowFactor = 2; 
      var factorFound = true; 
      while (factorFound) 
      { 
       if (n <= MaxKnownPrime && KnownPrimes.Contains (n)) 
        break; 

       factorFound = false; 
       var maxFactor = (long) Math.Sqrt (n); 
       for (var fac = lowFactor; fac <= maxFactor; fac++) 
       { 
        if (n % fac == 0) 
        { 
         factors.Add (fac); 
         n /= fac; 
         lowFactor = fac; 
         factorFound = true; 
         break; 
        } 
       } 
      } 
      factors.Add (n); 
      return factors; 
     } 
    } 

} 

Answer 1

Наконец нашел проблему: RussianPeasant. Я не тестировал каждый край. Мой лимит переполнения должен составлять 2^31, а не 2^32, для учета знакового бита. Вот скорректированный метод:

public static long RussianPeasant(long a, long b, long c) 
    { 
     const long overflow_limit = 1L << 31; 
     a = Math.Abs (a); 
     b = Math.Abs (b); 
     if (a < overflow_limit && b < overflow_limit) 
      return (a * b % c); // No possibility of overflow. 
     if (c < overflow_limit) 
      return (a % c) * (b % c) % c; 
     long ret = 0; 
     while(b != 0) { 
      if((b&1L) != 0L) { 
       ret += a; 
       ret %= c; 
      } 
      a *= 2; 
      a %= c; 
      b /= 2; 
     } 
     return ret; 
    }