米勒-拉宾素数检验对于大数失败
本文关键字：于大数 失败 检验 -拉 米勒 | 更新日期: 2023-09-27 18:36:49
在研究了与米勒-拉宾素数测试相关的其他 SO 答案后，我用 C# 实现了一个版本，但它偶尔会在 30 亿左右失败，当它达到 40 亿时，它停止识别任何素数。我怀疑我正在遭受溢出，但不知道在哪里。我的目标是让它适用于 0 <= n <= 2^63 - 1 范围内的任何值。
我创建了一个小提琴：https://dotnetfiddle.net/3F7P97
我尝试过的想法包括：
-
使用预先计算的基数 2、325、9375、28178、450775、9780504，1795265022本网站宣传为适用于小于 2^64 的数字：http://miller-rabin.appspot.com/这是由这个问题的回答者推荐的：米勒·拉宾素数检验精度
-
编写一个抗溢出的power-mod函数来计算a^b mod n。
-
编写一个防溢出乘法函数来计算 a*b mod n（使用俄罗斯农民算法）。
这是我创建这个问题时小提琴的代码：
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;
}
}
}
终于找到了问题所在：俄罗斯农民。我没有测试每个边缘情况。我的溢出限制应该是 2^31，而不是 2^32，以解释符号位。这是更正的方法：
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;
}