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/**************************************************************************
* *
* Regina - A Normal Surface Theory Calculator *
* Computational Engine *
* *
* Copyright (c) 1999-2025, Ben Burton *
* For further details contact Ben Burton (bab@debian.org). *
* *
* This program is free software; you can redistribute it and/or *
* modify it under the terms of the GNU General Public License as *
* published by the Free Software Foundation; either version 2 of the *
* License, or (at your option) any later version. *
* *
* As an exception, when this program is distributed through (i) the *
* App Store by Apple Inc.; (ii) the Mac App Store by Apple Inc.; or *
* (iii) Google Play by Google Inc., then that store may impose any *
* digital rights management, device limits and/or redistribution *
* restrictions that are required by its terms of service. *
* *
* This program is distributed in the hope that it will be useful, but *
* WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU *
* General Public License for more details. *
* *
* You should have received a copy of the GNU General Public License *
* along with this program. If not, see <https://www.gnu.org/licenses/>. *
* *
**************************************************************************/
#include <algorithm>
#include <functional>
#include <numeric> // for std::gcd()
#include "maths/numbertheory.h"
namespace regina {
long reducedMod(long k, long modBase) {
if (modBase <= 0)
throw InvalidArgument(
"reducedMod() requires modBase to be strictly positive");
long ans = k % modBase;
if (ans < 0) {
if ((ans + modBase) <= (-ans))
return (ans + modBase);
} else if (-(ans - modBase) < ans)
return (ans - modBase);
return ans;
}
long gcd(long a, long b) {
while (a != b && b != 0) {
long tmp = a;
a = b;
b = tmp % b;
}
return (a >= 0 ? a : -a);
}
namespace {
long gcdWithCoeffsInternal(long a, long b, long& u, long& v) {
// PRE: a and b are non-negative.
// First get the trivial cases out of the way.
if (b == 0 || a == b) {
u = 1; v = 0;
return a;
}
if (a == 0) {
u = 0; v = 1;
return b;
}
u = 1;
v = 0;
long uu = 0;
long vv = 1;
while (b != 0) {
// At each stage:
// a != b and a != 0;
// u*(a_orig) + v*(b_orig) = a;
// uu*(a_orig) + vv*(b_orig) = b;
// u*vv - uu*v = ±1;
// (u,v), (uu,vv), (u,uu), (v,vv) are all coprime pairs with
// opposite signs (we treat 0 as negative for this purpose).
//
// Moreover, if we treat magnitude as distance from 1/2 (so that
// ... > |-1| > |0| == |1| < |2| < ...), then at every stage
// we have |u| ≤ |uu| and |v| ≤ |vv|.
long q = a / b;
// (u,uu) <- (uu, u - q*uu)
long tmp = u;
u = uu;
uu = tmp - (q * uu);
// (v,vv) <- (vv, v - q*vv)
tmp = v;
v = vv;
vv = tmp - (q * vv);
// (a,b) <- (b, a % b)
tmp = a;
a = b;
b = tmp % b;
}
// At this point:
// a = gcd = u*(a_orig) + v*(b_orig);
// (uu, vv) = ±(b_orig, -a_orig)/gcd.
//
// This means we have one of the following two scenarios:
//
// 1: (uu, vv) = (-b_orig, a_orig)/gcd.
// The magnitude result above then gives
// -a_orig/gcd < v ≤ 0 < u ≤ b_orig/gcd + 1,
// and the relation u*a_orig + v*b_orig = gcd then forces
// -a_orig/gcd < v ≤ 0 < u ≤ b_orig/gcd.
//
// 2: (uu, vv) = (b_orig, -a_orig)/gcd.
// The magnitude result above then gives
// -b_orig/gcd < u ≤ 0 < v ≤ a_orig/gcd + 1,
// and the relation u*a_orig + v*b_orig = gcd then forces
// -b_orig/gcd < u ≤ 0 < v ≤ a_orig/gcd.
//
// Our final aim is -a_orig/gcd < v ≤ 0 < u ≤ b_orig/gcd, which
// is easy from here:
if (u <= 0) {
u += uu; // adds (b_orig / gcd)
v += vv; // subtracts (a_orig / gcd)
}
return a;
}
}
std::tuple<long, long, long> gcdWithCoeffs(long a, long b) {
long signA = (a > 0 ? 1 : a == 0 ? 0 : -1);
long signB = (b > 0 ? 1 : b == 0 ? 0 : -1);
std::tuple<long, long, long> ans;
std::get<0>(ans) = gcdWithCoeffsInternal(a >= 0 ? a : -a,
b >= 0 ? b : -b, std::get<1>(ans), std::get<2>(ans));
std::get<1>(ans) *= signA;
std::get<2>(ans) *= signB;
return ans;
}
long lcm(long a, long b) {
if (a == 0 || b == 0)
return 0;
long tmp = std::gcd(a, b);
tmp = (a / tmp) * b;
return (tmp >= 0 ? tmp : -tmp);
}
long modularInverse(long n, long k) {
if (n <= 0)
throw InvalidArgument(
"modularInverse(n, k) requires n to be strictly positive");
if (n == 1)
return 0;
// Compute (d, u, v).
auto ans = gcdWithCoeffs(n, k % n);
// GCD should equal 1, so u*n + k*v = 1.
if (std::get<0>(ans) != 1)
throw InvalidArgument(
"modularInverse(n, k) requires n and k to be coprime");
// Inverse is v; note that -n < (+/-)v <= 0.
// Since n >= 2 now and (n,k) = 1, we know v != 0.
return (std::get<2>(ans) > 0 ? std::get<2>(ans) : std::get<2>(ans) + n);
}
} // namespace regina
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