/* prime.c -- programming with prime numbers
 * compile as: gcc -O3 prime.c -lgmp -o prime
 * run as: ./prime */

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <string.h>
#include <time.h>
#include <gmp.h>

/* bit arrays */

#define ISBITSET(x, i) (( x[i>>3] & (1<<(i&7)) ) != 0)
#define SETBIT(x, i)   x[i>>3] |= (1<<(i&7));
#define CLEARBIT(x, i) x[i>>3] &= (1<<(i&7)) ^ 0xFF;

/* basic list library */

typedef struct list {
    void *data;
    struct list *next;
} List;

List *insert(void *data, List *next)
{
    List *new;

    new = malloc(sizeof(List));
    new->data = data;
    new->next = next;
    return new;
}

List *reverse(List *list) {
    List *new = NULL;
    List *next;

    while (list != NULL)
    {
        next = list->next;
        list->next = new;
        new = list;
        list = next;
    }

    return new;
}

int length(List *xs)
{
    int len = 0;
    while (xs != NULL)
    {
        len += 1;
        xs = xs->next;
    }
    return len;
}

/* sieve of eratosthenes */

List *primes(long n)
{
    int m = (n-1) / 2;
    unsigned char b[(m+7)/8];
    int i = 0;
    int p = 3;
    List *ps = NULL;
    int j;

    ps = insert((void *) 2, ps);

    memset(b, 0xFF, sizeof(b));

    while (p*p <= n)
    {
        if (ISBITSET(b,i))
        {
            ps = insert((void *) p, ps);
            j = (p*p - 3) / 2;
            while (j < m)
            {
                CLEARBIT(b, j);
                j += p;
            }
        }
        i += 1; p += 2;
    }

    while (i < m)
    {
        if (ISBITSET(b,i))
        {
            ps = insert((void *) p, ps);
        }
        i += 1; p += 2;
    }

    return reverse(ps);
}

/* miller-rabin primality checker */

void power_mod(mpz_t r, mpz_t base, mpz_t exp, mpz_t m)
{
    mpz_t b, e;
    mpz_init_set(b, base);
    mpz_init_set(e, exp);
    mpz_set_ui(r, 1);
    while (mpz_cmp_ui(e, 0) > 0)
    {
        if (mpz_odd_p(e))
        {
            mpz_mul(r, r, b);
            mpz_mod(r, r, m);
        }
        mpz_fdiv_q_2exp(e, e, 1);
        mpz_mul(b, b, b);
        mpz_mod(b, b, m);
    }
    mpz_clears(b, e, NULL);
}

int is_spsp(mpz_t n, mpz_t a)
{
    mpz_t d, n1, t;
    mpz_inits(d, n1, t, NULL);
    mpz_sub_ui(n1, n, 1);
    mpz_set(d, n1);
    int s = 0;

    while (mpz_even_p(d))
    {
        mpz_divexact_ui(d, d, 2);
        s += 1;
    }

    mpz_powm(t, a, d, n);
    if (mpz_cmp_ui(t, 1) == 0 || mpz_cmp(t, n1) == 0)
    {
        mpz_clears(d, n1, t, NULL);
        return 1;
    }

    while (--s > 0)
    {
        mpz_mul(t, t, t);
        mpz_mod(t, t, n);
        if (mpz_cmp(t, n1) == 0)
        {
            mpz_clears(d, n1, t, NULL);
            return 1;
        }
    }

    mpz_clears(d, n1, t, NULL);
    return 0;
}

int is_prime(mpz_t n)
{
    static gmp_randstate_t gmpRandState;
    static int is_seeded = 0;

    if (! is_seeded)
    {
        gmp_randinit_default(gmpRandState);
        gmp_randseed_ui(gmpRandState, time(NULL));
        is_seeded = 1;
    }

    mpz_t a, n3, t;
    mpz_inits(a, n3, t, NULL);
    mpz_sub_ui(n3, n, 3);
    int i;
    int k = 25;
    long unsigned ps[] = { 2, 3, 5, 7 };

    if (mpz_cmp_ui(n, 2) < 0)
    {
        mpz_clears(a, n3, t, NULL);
        return 0;
    }

    for (i = 0; i < sizeof(ps) / sizeof(long unsigned); i++)
    {
        mpz_mod_ui(t, n, ps[i]);
        if (mpz_cmp_ui(t, 0) == 0)
        {
            mpz_clears(a, n3, t, NULL);
            return mpz_cmp_ui(n, ps[i]) == 0;
        }
    }

    while (k > 0)
    {
        mpz_urandomm(a, gmpRandState, n3);
        mpz_add_ui(a, a, 2);
        if (! is_spsp(n, a))
        {
            mpz_clears(a, n3, t, NULL);
            return 0;
        }
        k -= 1;
    }

    mpz_clears(a, n3, t, NULL);
    return 1;
}

/* integer factorization by trial division */

void td_factors(List **fs, mpz_t n)
{
    mpz_t d, d2, q, r;
    mpz_inits(d, d2, q, r, NULL);
    mpz_set_ui(d, 2);

    while (mpz_even_p(n))
    {
        mpz_t *f = malloc(sizeof(*f));
        mpz_init_set(*f, d);
        *fs = insert(*f, *fs);
        mpz_divexact(n, n, d);
    }

    mpz_set_ui(d, 3);
    mpz_set_ui(d2, 9);

    while (mpz_cmp(n, d2) >= 0)
    {
        mpz_fdiv_qr(q, r, n, d);
        if (mpz_cmp_ui(r, 0) > 0)
        {
            mpz_add_ui(d, d, 2);
            mpz_mul(d2, d, d);
        }
        else
        {
            mpz_t *f = malloc(sizeof(*f));
            mpz_init_set(*f, d);
            *fs = insert(*f, *fs);
            mpz_divexact(n, n, d);
        } 
    }

    if (mpz_cmp_ui(n, 1) != 0)
    {
        *fs = reverse(insert(n, *fs));
    }
    mpz_clears(d, d2, q, r, NULL);
}

/* integer factorization by pollard rho */

List *insert_in_order(void *x, List *xs)
{
    if (xs == NULL || mpz_cmp(x, xs->data) < 0)
    {
        return insert(x, xs);
    }
    else
    {
        List *head = xs;
        while (xs->next != NULL && mpz_cmp(x, xs->next->data) > 0)
        {
            xs = xs->next;
        }
        xs->next = insert(x, xs->next);
        return head;
    }
}

void rho_factor(mpz_t f, mpz_t n, long unsigned c)
{
    mpz_t t, h, d, r;

    mpz_init_set_ui(t, 2);
    mpz_init_set_ui(h, 2);
    mpz_init_set_ui(d, 1);
    mpz_init_set_ui(r, 0);

    while (mpz_cmp_si(d, 1) == 0)
    {
        mpz_mul(t, t, t);
        mpz_add_ui(t, t, c);
        mpz_mod(t, t, n);

        mpz_mul(h, h, h);
        mpz_add_ui(h, h, c);
        mpz_mod(h, h, n);

        mpz_mul(h, h, h);
        mpz_add_ui(h, h, c);
        mpz_mod(h, h, n);

        mpz_sub(r, t, h);
        mpz_gcd(d, r, n);
    }

    if (mpz_cmp(d, n) == 0) /* cycle */
    {
        rho_factor(f, n, c+1);
    }
    else if (mpz_probab_prime_p(d, 25)) /* success */
    {
        mpz_set(f, d);
    }
    else /* found composite factor */
    {
        rho_factor(f, d, c+1);
    }

    mpz_clears(t, h, d, r, NULL);
}

void rho_factors(List **fs, mpz_t n)
{

    while (mpz_even_p(n))
    {
        mpz_t *f = malloc(sizeof(*f));
        mpz_init_set_ui(*f, 2);
        *fs = insert(*f, *fs);
        mpz_divexact_ui(n, n, 2);
    }

    if (mpz_cmp_ui(n, 1) == 0) return;

    while (! (mpz_probab_prime_p(n, 25)))
    {
        mpz_t *f = malloc(sizeof(*f));
        mpz_init_set_ui(*f, 0);

        rho_factor(*f, n, 1);
        *fs = insert_in_order(*f, *fs);
        mpz_divexact(n, n, *f);
    }

    *fs = insert_in_order(n, *fs);
}

/* demonstration */

int main(int argc, char *argv[])
{
    mpz_t n;
    mpz_init(n);
    List *ps = NULL;
    List *fs = NULL;

    printf("Primes less than a hundred:");
    ps = primes(100);
    while (ps != NULL)
    {
        printf(" %ld", (long) ps->data);
        ps = ps->next;
    }
    printf("\n");

    printf("Count of primes less than a million: %d\n",
            length(primes(1000000)));

    mpz_t b, e, m, t;
    mpz_init_set_ui(b, 437);
    mpz_init_set_ui(e, 13);
    mpz_init_set_ui(m, 1741);
    mpz_init(t);
    power_mod(t, b, e, m);
    printf("Power_mod of 437^13 (mod 1741) is %s\n",
            mpz_get_str(NULL, 10, t));

    mpz_t a;
    mpz_init(a);
    mpz_set_str(n, "2047", 10);
    mpz_set_str(a, "2", 10);
    printf("Is_spsp(%s,%s) is %d\n", mpz_get_str(NULL, 10, n),
            mpz_get_str(NULL, 10, a), is_spsp(n, a));

    mpz_set_str(n, "600851475143", 10); /* composite */
    printf("By 25 iterations of Miller-Rabin, 600851475143 is %s\n",
            is_prime(n) ? "prime" : "composite");

    mpz_set_str(n, "2305843009213693951", 10); /* prime */
    printf("By 25 iterations of Miller-Rabin, ");
    printf("2^61-1 = 2305843009213693951 is %s\n",
            is_prime(n) ? "prime" : "composite");

    printf("Factors of 600851475143 by trial division:");
    mpz_set_str(n, "600851475143", 10);
    td_factors(&fs, n);
    while (fs != NULL) {
        printf(" %s", mpz_get_str(NULL, 10, fs->data));
        fs = fs->next;
    }
    printf("\n");

    printf("Factors of 600851475143 by Pollard rho:");
    mpz_set_str(n, "600851475143", 10);
    rho_factors(&fs, n);
    while (fs != NULL) {
        printf(" %s", mpz_get_str(NULL, 10, fs->data));
        fs = fs->next;
    }
    printf("\n");
}