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serenity/Libraries/LibCrypto/NumberTheory/ModularFunctions.h

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/*
* Copyright (c) 2020, Ali Mohammad Pur <ali.mpfard@gmail.com>
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
*
* 1. Redistributions of source code must retain the above copyright notice, this
* list of conditions and the following disclaimer.
*
* 2. Redistributions in binary form must reproduce the above copyright notice,
* this list of conditions and the following disclaimer in the documentation
* and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
* AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
* DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
* SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
* CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
* OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
#pragma once
#include <LibCrypto/BigInt/UnsignedBigInteger.h>
//#define NT_DEBUG
namespace Crypto {
namespace NumberTheory {
static auto ModularInverse(const UnsignedBigInteger& a_, const UnsignedBigInteger& b) -> UnsignedBigInteger
{
if (b == 1)
return { 1 };
UnsignedBigInteger one { 1 };
UnsignedBigInteger temp_1;
UnsignedBigInteger temp_2;
UnsignedBigInteger temp_3;
UnsignedBigInteger temp_4;
UnsignedBigInteger temp_plus;
UnsignedBigInteger temp_minus;
UnsignedBigInteger temp_quotient;
UnsignedBigInteger temp_remainder;
UnsignedBigInteger d;
auto a = a_;
auto u = a;
if (a.words()[0] % 2 == 0) {
// u += b
UnsignedBigInteger::add_without_allocation(u, b, temp_plus);
u.set_to(temp_plus);
}
auto v = b;
UnsignedBigInteger x { 0 };
// d = b - 1
UnsignedBigInteger::subtract_without_allocation(b, one, d);
while (!(v == 1)) {
while (v < u) {
// u -= v
UnsignedBigInteger::subtract_without_allocation(u, v, temp_minus);
u.set_to(temp_minus);
// d += x
UnsignedBigInteger::add_without_allocation(d, x, temp_plus);
d.set_to(temp_plus);
while (u.words()[0] % 2 == 0) {
if (d.words()[0] % 2 == 1) {
// d += b
UnsignedBigInteger::add_without_allocation(d, b, temp_plus);
d.set_to(temp_plus);
}
// u /= 2
UnsignedBigInteger::divide_u16_without_allocation(u, 2, temp_quotient, temp_remainder);
u.set_to(temp_quotient);
// d /= 2
UnsignedBigInteger::divide_u16_without_allocation(d, 2, temp_quotient, temp_remainder);
d.set_to(temp_quotient);
}
}
// v -= u
UnsignedBigInteger::subtract_without_allocation(v, u, temp_minus);
v.set_to(temp_minus);
// x += d
UnsignedBigInteger::add_without_allocation(x, d, temp_plus);
x.set_to(temp_plus);
while (v.words()[0] % 2 == 0) {
if (x.words()[0] % 2 == 1) {
// x += b
UnsignedBigInteger::add_without_allocation(x, b, temp_plus);
x.set_to(temp_plus);
}
// v /= 2
UnsignedBigInteger::divide_u16_without_allocation(v, 2, temp_quotient, temp_remainder);
v.set_to(temp_quotient);
// x /= 2
UnsignedBigInteger::divide_u16_without_allocation(x, 2, temp_quotient, temp_remainder);
x.set_to(temp_quotient);
}
}
// x % b
UnsignedBigInteger::divide_without_allocation(x, b, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder);
return temp_remainder;
}
static auto ModularPower(const UnsignedBigInteger& b, const UnsignedBigInteger& e, const UnsignedBigInteger& m) -> UnsignedBigInteger
{
if (m == 1)
return 0;
UnsignedBigInteger ep { e };
UnsignedBigInteger base { b };
UnsignedBigInteger exp { 1 };
UnsignedBigInteger temp_1;
UnsignedBigInteger temp_2;
UnsignedBigInteger temp_3;
UnsignedBigInteger temp_4;
UnsignedBigInteger temp_multiply;
UnsignedBigInteger temp_quotient;
UnsignedBigInteger temp_remainder;
while (!(ep < 1)) {
#ifdef NT_DEBUG
dbg() << ep.to_base10();
#endif
if (ep.words()[0] % 2 == 1) {
// exp = (exp * base) % m;
UnsignedBigInteger::multiply_without_allocation(exp, base, temp_1, temp_2, temp_3, temp_4, temp_multiply);
UnsignedBigInteger::divide_without_allocation(temp_multiply, m, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder);
exp.set_to(temp_remainder);
}
// ep = ep / 2;
UnsignedBigInteger::divide_u16_without_allocation(ep, 2, temp_quotient, temp_remainder);
ep.set_to(temp_quotient);
// base = (base * base) % m;
UnsignedBigInteger::multiply_without_allocation(base, base, temp_1, temp_2, temp_3, temp_4, temp_multiply);
UnsignedBigInteger::divide_without_allocation(temp_multiply, m, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder);
base.set_to(temp_remainder);
}
return exp;
}
static void GCD_without_allocation(
const UnsignedBigInteger& a,
const UnsignedBigInteger& b,
UnsignedBigInteger& temp_a,
UnsignedBigInteger& temp_b,
UnsignedBigInteger& temp_1,
UnsignedBigInteger& temp_2,
UnsignedBigInteger& temp_3,
UnsignedBigInteger& temp_4,
UnsignedBigInteger& temp_quotient,
UnsignedBigInteger& temp_remainder,
UnsignedBigInteger& output)
{
temp_a.set_to(a);
temp_b.set_to(b);
for (;;) {
if (temp_a == 0) {
output.set_to(temp_b);
return;
}
// temp_b %= temp_a
UnsignedBigInteger::divide_without_allocation(temp_b, temp_a, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder);
temp_b.set_to(temp_remainder);
if (temp_b == 0) {
output.set_to(temp_a);
return;
}
// temp_a %= temp_b
UnsignedBigInteger::divide_without_allocation(temp_a, temp_b, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder);
temp_a.set_to(temp_remainder);
}
}
static UnsignedBigInteger GCD(const UnsignedBigInteger& a, const UnsignedBigInteger& b)
{
UnsignedBigInteger temp_a;
UnsignedBigInteger temp_b;
UnsignedBigInteger temp_1;
UnsignedBigInteger temp_2;
UnsignedBigInteger temp_3;
UnsignedBigInteger temp_4;
UnsignedBigInteger temp_quotient;
UnsignedBigInteger temp_remainder;
UnsignedBigInteger output;
GCD_without_allocation(a, b, temp_a, temp_b, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder, output);
return output;
}
static auto LCM(const UnsignedBigInteger& a, const UnsignedBigInteger& b) -> UnsignedBigInteger
{
UnsignedBigInteger temp_a;
UnsignedBigInteger temp_b;
UnsignedBigInteger temp_1;
UnsignedBigInteger temp_2;
UnsignedBigInteger temp_3;
UnsignedBigInteger temp_4;
UnsignedBigInteger temp_quotient;
UnsignedBigInteger temp_remainder;
UnsignedBigInteger gcd_output;
UnsignedBigInteger output { 0 };
GCD_without_allocation(a, b, temp_a, temp_b, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder, gcd_output);
if (gcd_output == 0) {
#ifdef NT_DEBUG
dbg() << "GCD is zero";
#endif
return output;
}
// output = (a / gcd_output) * b
UnsignedBigInteger::divide_without_allocation(a, gcd_output, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder);
UnsignedBigInteger::multiply_without_allocation(temp_quotient, b, temp_1, temp_2, temp_3, temp_4, output);
#ifdef NT_DEBUG
dbg() << "quot: " << temp_quotient << " rem: " << temp_remainder << " out: " << output;
#endif
return output;
}
template<size_t test_count>
static bool MR_primality_test(UnsignedBigInteger n, const Vector<UnsignedBigInteger, test_count>& tests)
{
auto prev = n.minus({ 1 });
auto b = prev;
auto r = 0;
auto div_result = b.divided_by(2);
while (div_result.quotient == 0) {
div_result = b.divided_by(2);
b = div_result.quotient;
++r;
}
for (size_t i = 0; i < tests.size(); ++i) {
auto return_ = true;
if (n < tests[i])
continue;
auto x = ModularPower(tests[i], b, n);
if (x == 1 || x == prev)
continue;
for (auto d = r - 1; d != 0; --d) {
x = ModularPower(x, 2, n);
if (x == 1)
return false;
if (x == prev) {
return_ = false;
break;
}
}
if (return_)
return false;
}
return true;
}
static UnsignedBigInteger random_number(const UnsignedBigInteger& min, const UnsignedBigInteger& max)
{
ASSERT(min < max);
auto range = max.minus(min);
UnsignedBigInteger base;
// FIXME: Need a cryptographically secure rng
auto size = range.trimmed_length() * sizeof(u32);
u8 buf[size];
arc4random_buf(buf, size);
Vector<u32> vec;
for (size_t i = 0; i < size / sizeof(u32); ++i) {
vec.append(*(u32*)buf + i);
}
UnsignedBigInteger offset { move(vec) };
return offset.plus(min);
}
static bool is_probably_prime(const UnsignedBigInteger& p)
{
if (p == 2 || p == 3 || p == 5)
return true;
if (p < 49)
return true;
Vector<UnsignedBigInteger, 256> tests;
UnsignedBigInteger seven { 7 };
for (size_t i = 0; i < tests.size(); ++i)
tests.append(random_number(seven, p.minus(2)));
return MR_primality_test(p, tests);
}
static UnsignedBigInteger random_big_prime(size_t bits)
{
ASSERT(bits >= 33);
UnsignedBigInteger min = UnsignedBigInteger::from_base10("6074001000").shift_left(bits - 33);
UnsignedBigInteger max = UnsignedBigInteger { 1 }.shift_left(bits).minus(1);
for (;;) {
auto p = random_number(min, max);
if (is_probably_prime(p))
return p;
}
}
}
}
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