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359 lines
12 KiB
C++
359 lines
12 KiB
C++
/*
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* Copyright (c) 2020, Ali Mohammad Pur <ali.mpfard@gmail.com>
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* All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions are met:
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*
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* 1. Redistributions of source code must retain the above copyright notice, this
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* list of conditions and the following disclaimer.
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*
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* 2. Redistributions in binary form must reproduce the above copyright notice,
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* this list of conditions and the following disclaimer in the documentation
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* and/or other materials provided with the distribution.
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*
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* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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* AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
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* DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
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* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
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* SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
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* CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
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* OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
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* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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*/
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#include <LibCrypto/NumberTheory/ModularFunctions.h>
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namespace Crypto {
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namespace NumberTheory {
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UnsignedBigInteger ModularInverse(const UnsignedBigInteger& a_, const UnsignedBigInteger& b)
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{
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if (b == 1)
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return { 1 };
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UnsignedBigInteger one { 1 };
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UnsignedBigInteger temp_1;
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UnsignedBigInteger temp_2;
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UnsignedBigInteger temp_3;
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UnsignedBigInteger temp_4;
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UnsignedBigInteger temp_plus;
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UnsignedBigInteger temp_minus;
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UnsignedBigInteger temp_quotient;
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UnsignedBigInteger temp_remainder;
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UnsignedBigInteger d;
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auto a = a_;
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auto u = a;
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if (a.words()[0] % 2 == 0) {
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// u += b
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UnsignedBigInteger::add_without_allocation(u, b, temp_plus);
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u.set_to(temp_plus);
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}
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auto v = b;
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UnsignedBigInteger x { 0 };
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// d = b - 1
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UnsignedBigInteger::subtract_without_allocation(b, one, d);
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while (!(v == 1)) {
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while (v < u) {
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// u -= v
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UnsignedBigInteger::subtract_without_allocation(u, v, temp_minus);
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u.set_to(temp_minus);
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// d += x
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UnsignedBigInteger::add_without_allocation(d, x, temp_plus);
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d.set_to(temp_plus);
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while (u.words()[0] % 2 == 0) {
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if (d.words()[0] % 2 == 1) {
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// d += b
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UnsignedBigInteger::add_without_allocation(d, b, temp_plus);
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d.set_to(temp_plus);
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}
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// u /= 2
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UnsignedBigInteger::divide_u16_without_allocation(u, 2, temp_quotient, temp_remainder);
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u.set_to(temp_quotient);
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// d /= 2
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UnsignedBigInteger::divide_u16_without_allocation(d, 2, temp_quotient, temp_remainder);
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d.set_to(temp_quotient);
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}
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}
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// v -= u
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UnsignedBigInteger::subtract_without_allocation(v, u, temp_minus);
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v.set_to(temp_minus);
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// x += d
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UnsignedBigInteger::add_without_allocation(x, d, temp_plus);
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x.set_to(temp_plus);
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while (v.words()[0] % 2 == 0) {
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if (x.words()[0] % 2 == 1) {
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// x += b
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UnsignedBigInteger::add_without_allocation(x, b, temp_plus);
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x.set_to(temp_plus);
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}
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// v /= 2
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UnsignedBigInteger::divide_u16_without_allocation(v, 2, temp_quotient, temp_remainder);
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v.set_to(temp_quotient);
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// x /= 2
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UnsignedBigInteger::divide_u16_without_allocation(x, 2, temp_quotient, temp_remainder);
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x.set_to(temp_quotient);
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}
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}
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// x % b
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UnsignedBigInteger::divide_without_allocation(x, b, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder);
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return temp_remainder;
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}
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UnsignedBigInteger ModularPower(const UnsignedBigInteger& b, const UnsignedBigInteger& e, const UnsignedBigInteger& m)
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{
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if (m == 1)
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return 0;
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UnsignedBigInteger ep { e };
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UnsignedBigInteger base { b };
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UnsignedBigInteger exp { 1 };
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UnsignedBigInteger temp_1;
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UnsignedBigInteger temp_2;
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UnsignedBigInteger temp_3;
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UnsignedBigInteger temp_4;
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UnsignedBigInteger temp_multiply;
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UnsignedBigInteger temp_quotient;
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UnsignedBigInteger temp_remainder;
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while (!(ep < 1)) {
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if (ep.words()[0] % 2 == 1) {
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// exp = (exp * base) % m;
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UnsignedBigInteger::multiply_without_allocation(exp, base, temp_1, temp_2, temp_3, temp_4, temp_multiply);
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UnsignedBigInteger::divide_without_allocation(temp_multiply, m, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder);
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exp.set_to(temp_remainder);
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}
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// ep = ep / 2;
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UnsignedBigInteger::divide_u16_without_allocation(ep, 2, temp_quotient, temp_remainder);
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ep.set_to(temp_quotient);
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// base = (base * base) % m;
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UnsignedBigInteger::multiply_without_allocation(base, base, temp_1, temp_2, temp_3, temp_4, temp_multiply);
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UnsignedBigInteger::divide_without_allocation(temp_multiply, m, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder);
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base.set_to(temp_remainder);
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}
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return exp;
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}
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static void GCD_without_allocation(
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const UnsignedBigInteger& a,
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const UnsignedBigInteger& b,
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UnsignedBigInteger& temp_a,
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UnsignedBigInteger& temp_b,
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UnsignedBigInteger& temp_1,
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UnsignedBigInteger& temp_2,
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UnsignedBigInteger& temp_3,
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UnsignedBigInteger& temp_4,
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UnsignedBigInteger& temp_quotient,
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UnsignedBigInteger& temp_remainder,
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UnsignedBigInteger& output)
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{
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temp_a.set_to(a);
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temp_b.set_to(b);
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for (;;) {
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if (temp_a == 0) {
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output.set_to(temp_b);
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return;
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}
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// temp_b %= temp_a
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UnsignedBigInteger::divide_without_allocation(temp_b, temp_a, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder);
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temp_b.set_to(temp_remainder);
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if (temp_b == 0) {
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output.set_to(temp_a);
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return;
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}
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// temp_a %= temp_b
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UnsignedBigInteger::divide_without_allocation(temp_a, temp_b, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder);
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temp_a.set_to(temp_remainder);
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}
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}
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UnsignedBigInteger GCD(const UnsignedBigInteger& a, const UnsignedBigInteger& b)
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{
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UnsignedBigInteger temp_a;
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UnsignedBigInteger temp_b;
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UnsignedBigInteger temp_1;
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UnsignedBigInteger temp_2;
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UnsignedBigInteger temp_3;
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UnsignedBigInteger temp_4;
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UnsignedBigInteger temp_quotient;
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UnsignedBigInteger temp_remainder;
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UnsignedBigInteger output;
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GCD_without_allocation(a, b, temp_a, temp_b, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder, output);
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return output;
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}
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UnsignedBigInteger LCM(const UnsignedBigInteger& a, const UnsignedBigInteger& b)
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{
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UnsignedBigInteger temp_a;
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UnsignedBigInteger temp_b;
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UnsignedBigInteger temp_1;
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UnsignedBigInteger temp_2;
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UnsignedBigInteger temp_3;
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UnsignedBigInteger temp_4;
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UnsignedBigInteger temp_quotient;
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UnsignedBigInteger temp_remainder;
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UnsignedBigInteger gcd_output;
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UnsignedBigInteger output { 0 };
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GCD_without_allocation(a, b, temp_a, temp_b, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder, gcd_output);
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if (gcd_output == 0) {
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#ifdef NT_DEBUG
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dbgln("GCD is zero");
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#endif
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return output;
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}
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// output = (a / gcd_output) * b
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UnsignedBigInteger::divide_without_allocation(a, gcd_output, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder);
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UnsignedBigInteger::multiply_without_allocation(temp_quotient, b, temp_1, temp_2, temp_3, temp_4, output);
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#ifdef NT_DEBUG
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dbg() << "quot: " << temp_quotient << " rem: " << temp_remainder << " out: " << output;
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#endif
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return output;
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}
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static bool MR_primality_test(UnsignedBigInteger n, const Vector<UnsignedBigInteger, 256>& tests)
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{
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// Written using Wikipedia:
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// https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Miller%E2%80%93Rabin_test
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ASSERT(!(n < 4));
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auto predecessor = n.minus({ 1 });
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auto d = predecessor;
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size_t r = 0;
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{
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auto div_result = d.divided_by(2);
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while (div_result.remainder == 0) {
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d = div_result.quotient;
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div_result = d.divided_by(2);
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++r;
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}
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}
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if (r == 0) {
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// n - 1 is odd, so n was even. But there is only one even prime:
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return n == 2;
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}
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for (auto a : tests) {
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// Technically: ASSERT(2 <= a && a <= n - 2)
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ASSERT(a < n);
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auto x = ModularPower(a, d, n);
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if (x == 1 || x == predecessor)
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continue;
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bool skip_this_witness = false;
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// r − 1 iterations.
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for (size_t i = 0; i < r - 1; ++i) {
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x = ModularPower(x, 2, n);
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if (x == predecessor) {
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skip_this_witness = true;
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break;
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}
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}
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if (skip_this_witness)
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continue;
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return false; // "composite"
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}
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return true; // "probably prime"
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}
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UnsignedBigInteger random_number(const UnsignedBigInteger& min, const UnsignedBigInteger& max_excluded)
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{
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ASSERT(min < max_excluded);
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auto range = max_excluded.minus(min);
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UnsignedBigInteger base;
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auto size = range.trimmed_length() * sizeof(u32) + 2;
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// "+2" is intentional (see below).
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// Also, if we're about to crash anyway, at least produce a nice error:
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ASSERT(size < 8 * MiB);
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u8 buf[size];
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AK::fill_with_random(buf, size);
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UnsignedBigInteger random { buf, size };
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// At this point, `random` is a large number, in the range [0, 256^size).
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// To get down to the actual range, we could just compute random % range.
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// This introduces "modulo bias". However, since we added 2 to `size`,
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// we know that the generated range is at least 65536 times as large as the
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// required range! This means that the modulo bias is only 0.0015%, if all
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// inputs are chosen adversarially. Let's hope this is good enough.
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auto divmod = random.divided_by(range);
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// The proper way to fix this is to restart if `divmod.quotient` is maximal.
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return divmod.remainder.plus(min);
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}
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bool is_probably_prime(const UnsignedBigInteger& p)
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{
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// Is it a small number?
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if (p < 49) {
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u32 p_value = p.words()[0];
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// Is it a very small prime?
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if (p_value == 2 || p_value == 3 || p_value == 5 || p_value == 7)
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return true;
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// Is it the multiple of a very small prime?
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if (p_value % 2 == 0 || p_value % 3 == 0 || p_value % 5 == 0 || p_value % 7 == 0)
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return false;
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// Then it must be a prime, but not a very small prime, like 37.
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return true;
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}
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Vector<UnsignedBigInteger, 256> tests;
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// Make some good initial guesses that are guaranteed to find all primes < 2^64.
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tests.append(UnsignedBigInteger(2));
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tests.append(UnsignedBigInteger(3));
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tests.append(UnsignedBigInteger(5));
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tests.append(UnsignedBigInteger(7));
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tests.append(UnsignedBigInteger(11));
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tests.append(UnsignedBigInteger(13));
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UnsignedBigInteger seventeen { 17 };
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for (size_t i = tests.size(); i < 256; ++i) {
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tests.append(random_number(seventeen, p.minus(2)));
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}
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// Miller-Rabin's "error" is 8^-k. In adversarial cases, it's 4^-k.
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// With 200 random numbers, this would mean an error of about 2^-400.
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// So we don't need to worry too much about the quality of the random numbers.
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return MR_primality_test(p, tests);
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}
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UnsignedBigInteger random_big_prime(size_t bits)
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{
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ASSERT(bits >= 33);
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UnsignedBigInteger min = UnsignedBigInteger::from_base10("6074001000").shift_left(bits - 33);
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UnsignedBigInteger max = UnsignedBigInteger { 1 }.shift_left(bits).minus(1);
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for (;;) {
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auto p = random_number(min, max);
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if ((p.words()[0] & 1) == 0) {
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// An even number is definitely not a large prime.
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continue;
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}
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if (is_probably_prime(p))
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return p;
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}
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}
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}
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}
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