大数的素数因子分解

Question

我想找到小于10 ^ 12的大数的素数因子分解.我得到了这段代码(在java中):

public static List<Long> primeFactors(long numbers) {
        long n = numbers;
        List<Long> factors = new ArrayList<Long>();
        for (long i = 2; i <= n / i; i++) {
            while (n % i == 0) {
                factors.add(i);
                n /= i;
            }
        }
        if (n > 1) {
            factors.add(n);
        }
        return factors;
    }

首先,上述算法的复杂性是什么？我很难找到它？

对于素数较大的数字来说,它也会太慢.

是否有更好的算法,或者如何优化这个算法？

Answer 1

如果你想要分解许多大数,那么你可能最好先找到素数sqrt(n)(例如使用Eratosthenes的Sieve).然后你必须只检查那些素数是否是因子而不是全部测试i <= sqrt(n).

Answer 2

复杂性是O(sqrt(n)).检查后的数字是没有意义的sqrt(n).

这意味着,对于10^12,它将在大多数1 000 000迭代中进行,这并不慢.

Answer 3

对大数进行因子分解是一个难题,这也是加密算法使用大素数因子使加密难以破解的部分原因.

public static void main(String... args)  {
    int nums = 100;
    for (int i = 0; i < nums; i++) {
        long start = System.nanoTime();
        primeFactors(Long.MAX_VALUE - i);
        long time = System.nanoTime() - start;
        if (time > 100e6)
            System.out.println((Long.MAX_VALUE-i) + " took "+time/1000000+" ms.");
    }
}

public static List<Long> primeFactors(long n) {
    List<Long> factors = new ArrayList<Long>();
    while (n % 2 == 0 && n > 0) {
        factors.add(2L);
        n /= 2;
    }

    for (long i = 3; i * i <= n; i+=2) {
        while (n % i == 0) {
            factors.add(i);
            n /= i;
        }
    }
    if (n > 1)
        factors.add(n);

    return factors;
}

版画

9223372036854775806 took 3902 ms.
9223372036854775805 took 287 ms.
9223372036854775804 took 8356 ms.
9223372036854775797 took 9519 ms.
9223372036854775796 took 1507 ms.
9223372036854775794 took 111 ms.
9223372036854775788 took 184 ms.

如果将Long.MAX_VALUE替换为1000000000000L,则它们都会在20 ms内进行分解.