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sqrtm1.cc
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sqrtm1.cc
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// Precompute "sqrtm1 = sqrt(-1) (mod p)" for 10000-9383761 digit primes:
//
// g++ sqrtm1.cc -lgmp -lgmpxx -O3 -Wall -pedantic -Wextra -o sqrtm1
// (cpplinted and cppchecked)
//
#include <time.h>
#include <math.h>
#include <gmpxx.h>
#include <assert.h>
#include <iostream>
// 0: 10000-digit https://t5k.org/curios/page.php?number_id=9680
// 1: 36401-digit https://t5k.org/curios/page.php?number_id=3658
// 2: 100355-digit https://t5k.org/primes/page.php?id=89650
// 3: 200700-digit https://t5k.org/primes/page.php?id=103792
// 4: 272770-digit https://t5k.org/primes/lists/all.txt
// 5: 330855-digit https://t5k.org/primes/lists/all.txt
// 6: 388342-digit https://t5k.org/primes/page.php?id=122213
// 7: 2165 10^999999+308267*10^292000+1 1000000 CH10 2021
// https://t5k.org/primes/lists/all.txt
// 8: 10 10223*2^31172165+1 9383761 SB12 2016
// 9: 4794 456551^98304-456551^49152+1 556351 L4142 2017
// >9: 2^n+9 isprime https://oeis.org/A057196
struct row { std::string f; unsigned b, e, a; } r[] = {
{ "65516468355", 2, 333333, 1 }, // 100355-digit
{ "3756801695685", 2, 666669, 1 }, // 200700-digit
{ "1705", 2, 906110, 1 }, // 272770-digit
{ "2145", 2, 1099064, 1 }, // 330855-digit
{ "2996863034895", 2, 1290000, 1 } // 388342-digit
};
float roundf_(float f, int p) {
float q = powf(10., p);
return roundf(q * f) / q;
}
int main(int argc, char *argv[]) {
const int sday = 24*60*60;
mpz_class a, b, c, p;
char *buf;
if (argc < 2) {
std::cerr << "Format: ./sqrtm1 i [div] with 0<=i<=8 or i>8, div>=1\n";
exit(1);
}
int d = 1, bits, u = atoi(argv[1]);
float dt;
assert(u >= 0);
buf = new char[32000000];
assert(buf);
switch (u) {
case 0: {
mpz_ui_pow_ui(a.get_mpz_t(), 10, 10000);
a -= 1;
a /= 3;
mpz_ui_pow_ui(b.get_mpz_t(), 10, 6333);
a -= b;
break;
}
case 1: {
mpz_ui_pow_ui(a.get_mpz_t(), 10, 36400);
a -= 1;
a /= 9;
b = a;
a *= 34;
mpz_ui_pow_ui(c.get_mpz_t(), 10, 2264);
c *= b;
c *= mpz_class("42000040044444004000024");
mpz_ui_pow_ui(b.get_mpz_t(), 10, 4550);
b -= 1;
b /= 9;
c /= b;
a -= c;
a -= 1;
break;
}
case 2:
case 3:
case 4:
case 5:
case 6: {
u -= 2;
mpz_ui_pow_ui(a.get_mpz_t(), r[u].b, r[u].e);
a *= mpz_class(r[u].f);
a += r[u].a;
break;
}
case 7: {
mpz_ui_pow_ui(a.get_mpz_t(), 10, 999999);
mpz_ui_pow_ui(b.get_mpz_t(), 10, 292000);
b *= mpz_class("308267");
a += b;
a += 1;
break;
}
case 8: {
mpz_ui_pow_ui(a.get_mpz_t(), 2, 31172165);
a *= mpz_class("10223");
a += 1;
break;
}
case 9: {
mpz_ui_pow_ui(b.get_mpz_t(), 456551, 49152);
a = b*b - b + 1;
break;
}
default: {
assert(u > 9 || !"wrong selection (0-9, >9)");
mpz_ui_pow_ui(a.get_mpz_t(), 2, u);
a += 9;
break;
}
}
p = a;
c = a / 4;
std::cerr
<< strlen(mpz_get_str(buf, 10, p.get_mpz_t())) << "-digit prime p ("
<< (bits = strlen(mpz_get_str(buf, 2, p.get_mpz_t()))) << " bits)\n";
if (argc > 2) {
b = 2;
c = d = 1 + (strlen(buf) - 1) / atoi(argv[2]);
mpz_powm(c.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t(), p.get_mpz_t());
}
// deterministic fast search for smallest quadratic non-residue
b = 2;
while (mpz_kronecker(b.get_mpz_t(), p.get_mpz_t()) != -1) {
mpz_nextprime(b.get_mpz_t(), b.get_mpz_t());
}
std::cerr << "smallest quadratic non-residue prime: " << b << "\n";
clock_t start = clock();
mpz_powm(a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t(), p.get_mpz_t());
dt = static_cast<float>(clock() - start) / CLOCKS_PER_SEC;
if (argc > 2) {
std::cerr << dt << "s (" << roundf_(dt/d*bits/sday, 2) << " days)\n";
} else {
std::cerr << dt << "s\n";
std::cout << a << "\n";
assert(a * a % p == p - 1);
}
std::cerr << "done\n";
delete buf;
return 0;
}