LibCrypto: Split BigInteger operations into an Algorithms class

Since the operations are already complicated and will become even more
so soon, let's split them into their own files. We can also integrate
the NumberTheory operations that would better fit there into this class
as well.

This commit doesn't change behaviors, but moves the allocation of some
variables into caller classes.

Userland/Libraries/LibCrypto/BigInt/Algorithms/BitwiseOperations.cpp

@@ -0,0 +1,244 @@
+/*
+ * Copyright (c) 2020, Itamar S. <itamar8910@gmail.com>
+ * Copyright (c) 2020-2021, Dex♪ <dexes.ttp@gmail.com>
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ */
+
+#include "UnsignedBigIntegerAlgorithms.h"
+
+namespace Crypto {
+
+/**
+ * Complexity: O(N) where N is the number of words in the shorter value
+ * Method:
+ * Apply <op> word-wise until words in the shorter value are used up
+ * then copy the rest of the words verbatim from the longer value.
+ */
+FLATTEN void UnsignedBigIntegerAlgorithms::bitwise_or_without_allocation(
+    UnsignedBigInteger const& left,
+    UnsignedBigInteger const& right,
+    UnsignedBigInteger& output)
+{
+    // If either of the BigInts are invalid, the output is just the other one.
+    if (left.is_invalid()) {
+        output.set_to(right);
+        return;
+    }
+    if (right.is_invalid()) {
+        output.set_to(left);
+        return;
+    }
+
+    const UnsignedBigInteger *shorter, *longer;
+    if (left.length() < right.length()) {
+        shorter = &left;
+        longer = &right;
+    } else {
+        shorter = &right;
+        longer = &left;
+    }
+
+    output.m_words.resize_and_keep_capacity(longer->length());
+
+    size_t longer_offset = longer->length() - shorter->length();
+    for (size_t i = 0; i < shorter->length(); ++i)
+        output.m_words[i] = longer->words()[i] | shorter->words()[i];
+
+    __builtin_memcpy(output.m_words.data() + shorter->length(), longer->words().data() + shorter->length(), sizeof(u32) * longer_offset);
+}
+
+/**
+ * Complexity: O(N) where N is the number of words in the shorter value
+ * Method:
+ * Apply 'and' word-wise until words in the shorter value are used up
+ * and zero the rest.
+ */
+FLATTEN void UnsignedBigIntegerAlgorithms::bitwise_and_without_allocation(
+    UnsignedBigInteger const& left,
+    UnsignedBigInteger const& right,
+    UnsignedBigInteger& output)
+{
+    // If either of the BigInts are invalid, the output is just the other one.
+    if (left.is_invalid()) {
+        output.set_to(right);
+        return;
+    }
+    if (right.is_invalid()) {
+        output.set_to(left);
+        return;
+    }
+
+    const UnsignedBigInteger *shorter, *longer;
+    if (left.length() < right.length()) {
+        shorter = &left;
+        longer = &right;
+    } else {
+        shorter = &right;
+        longer = &left;
+    }
+
+    output.m_words.resize_and_keep_capacity(longer->length());
+
+    size_t longer_offset = longer->length() - shorter->length();
+    for (size_t i = 0; i < shorter->length(); ++i)
+        output.m_words[i] = longer->words()[i] & shorter->words()[i];
+
+    __builtin_memset(output.m_words.data() + shorter->length(), 0, sizeof(u32) * longer_offset);
+}
+
+/**
+ * Complexity: O(N) where N is the number of words in the shorter value
+ * Method:
+ * Apply 'xor' word-wise until words in the shorter value are used up
+ * and copy the rest.
+ */
+FLATTEN void UnsignedBigIntegerAlgorithms::bitwise_xor_without_allocation(
+    UnsignedBigInteger const& left,
+    UnsignedBigInteger const& right,
+    UnsignedBigInteger& output)
+{
+    // If either of the BigInts are invalid, the output is just the other one.
+    if (left.is_invalid()) {
+        output.set_to(right);
+        return;
+    }
+    if (right.is_invalid()) {
+        output.set_to(left);
+        return;
+    }
+
+    const UnsignedBigInteger *shorter, *longer;
+    if (left.length() < right.length()) {
+        shorter = &left;
+        longer = &right;
+    } else {
+        shorter = &right;
+        longer = &left;
+    }
+
+    output.m_words.resize_and_keep_capacity(longer->length());
+
+    size_t longer_offset = longer->length() - shorter->length();
+    for (size_t i = 0; i < shorter->length(); ++i)
+        output.m_words[i] = longer->words()[i] ^ shorter->words()[i];
+
+    __builtin_memcpy(output.m_words.data() + shorter->length(), longer->words().data() + shorter->length(), sizeof(u32) * longer_offset);
+}
+
+/**
+ * Complexity: O(N) where N is the number of words
+ */
+FLATTEN void UnsignedBigIntegerAlgorithms::bitwise_not_without_allocation(
+    UnsignedBigInteger const& right,
+    UnsignedBigInteger& output)
+{
+    // If the value is invalid, the output value is invalid as well.
+    if (right.is_invalid()) {
+        output.invalidate();
+        return;
+    }
+    if (right.length() == 0) {
+        output.set_to_0();
+        return;
+    }
+
+    output.m_words.resize_and_keep_capacity(right.length());
+
+    if (right.length() > 1) {
+        for (size_t i = 0; i < right.length() - 1; ++i)
+            output.m_words[i] = ~right.words()[i];
+    }
+
+    auto last_word_index = right.length() - 1;
+    auto last_word = right.words()[last_word_index];
+
+    output.m_words[last_word_index] = ((u32)0xffffffffffffffff >> __builtin_clz(last_word)) & ~last_word;
+}
+
+/**
+ * Complexity : O(N + num_bits % 8) where N is the number of words in the number
+ * Shift method :
+ * Start by shifting by whole words in num_bits (by putting missing words at the start),
+ * then shift the number's words two by two by the remaining amount of bits.
+ */
+FLATTEN void UnsignedBigIntegerAlgorithms::shift_left_without_allocation(
+    UnsignedBigInteger const& number,
+    size_t num_bits,
+    UnsignedBigInteger& temp_result,
+    UnsignedBigInteger& temp_plus,
+    UnsignedBigInteger& output)
+{
+    // We can only do shift operations on individual words
+    // where the shift amount is <= size of word (32).
+    // But we do know how to shift by a multiple of word size (e.g 64=32*2)
+    // So we first shift the result by how many whole words fit in 'num_bits'
+    shift_left_by_n_words(number, num_bits / UnsignedBigInteger::BITS_IN_WORD, temp_result);
+
+    output.set_to(temp_result);
+
+    // And now we shift by the leftover amount of bits
+    num_bits %= UnsignedBigInteger::BITS_IN_WORD;
+
+    if (num_bits == 0) {
+        return;
+    }
+
+    for (size_t i = 0; i < temp_result.length(); ++i) {
+        u32 current_word_of_temp_result = shift_left_get_one_word(temp_result, num_bits, i);
+        output.m_words[i] = current_word_of_temp_result;
+    }
+
+    // Shifting the last word can produce a carry
+    u32 carry_word = shift_left_get_one_word(temp_result, num_bits, temp_result.length());
+    if (carry_word != 0) {
+
+        // output += (carry_word << temp_result.length())
+        // FIXME : Using temp_plus this way to transform carry_word into a bigint is not
+        //         efficient nor pretty. Maybe we should have an "add_with_shift" method ?
+        temp_plus.set_to_0();
+        temp_plus.m_words.append(carry_word);
+        shift_left_by_n_words(temp_plus, temp_result.length(), temp_result);
+        add_without_allocation(output, temp_result, temp_plus);
+        output.set_to(temp_plus);
+    }
+}
+
+ALWAYS_INLINE void UnsignedBigIntegerAlgorithms::shift_left_by_n_words(
+    UnsignedBigInteger const& number,
+    size_t number_of_words,
+    UnsignedBigInteger& output)
+{
+    // shifting left by N words means just inserting N zeroes to the beginning of the words vector
+    output.set_to_0();
+    output.m_words.resize_and_keep_capacity(number_of_words + number.length());
+
+    __builtin_memset(output.m_words.data(), 0, number_of_words * sizeof(unsigned));
+    __builtin_memcpy(&output.m_words.data()[number_of_words], number.m_words.data(), number.m_words.size() * sizeof(unsigned));
+}
+
+/**
+ * Returns the word at a requested index in the result of a shift operation
+ */
+ALWAYS_INLINE u32 UnsignedBigIntegerAlgorithms::shift_left_get_one_word(
+    UnsignedBigInteger const& number,
+    size_t num_bits,
+    size_t result_word_index)
+{
+    // "<= length()" (rather than length() - 1) is intentional,
+    // The result inedx of length() is used when calculating the carry word
+    VERIFY(result_word_index <= number.length());
+    VERIFY(num_bits <= UnsignedBigInteger::BITS_IN_WORD);
+    u32 result = 0;
+
+    // we need to check for "num_bits != 0" since shifting right by 32 is apparently undefined behaviour!
+    if (result_word_index > 0 && num_bits != 0) {
+        result += number.m_words[result_word_index - 1] >> (UnsignedBigInteger::BITS_IN_WORD - num_bits);
+    }
+    if (result_word_index < number.length() && num_bits < 32) {
+        result += number.m_words[result_word_index] << num_bits;
+    }
+    return result;
+}
+
+}

Userland/Libraries/LibCrypto/BigInt/Algorithms/Division.cpp

@@ -0,0 +1,83 @@
+/*
+ * Copyright (c) 2020, Itamar S. <itamar8910@gmail.com>
+ * Copyright (c) 2020-2021, Dex♪ <dexes.ttp@gmail.com>
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ */
+
+#include "UnsignedBigIntegerAlgorithms.h"
+
+namespace Crypto {
+
+/**
+ * Complexity: O(N^2) where N is the number of words in the larger number
+ * Division method:
+ * We loop over the bits of the divisor, attempting to subtract divisor<<i from the dividend.
+ * If the result is non-negative, it means that divisor*2^i "fits" in the dividend,
+ * so we set the ith bit in the quotient and reduce divisor<<i from the dividend.
+ * When we're done, what's left from the dividend is the remainder.
+ */
+FLATTEN void UnsignedBigIntegerAlgorithms::divide_without_allocation(
+    UnsignedBigInteger const& numerator,
+    UnsignedBigInteger const& denominator,
+    UnsignedBigInteger& temp_shift_result,
+    UnsignedBigInteger& temp_shift_plus,
+    UnsignedBigInteger& temp_shift,
+    UnsignedBigInteger& temp_minus,
+    UnsignedBigInteger& quotient,
+    UnsignedBigInteger& remainder)
+{
+    quotient.set_to_0();
+    remainder.set_to(numerator);
+
+    // iterate all bits
+    for (int word_index = numerator.trimmed_length() - 1; word_index >= 0; --word_index) {
+        for (int bit_index = UnsignedBigInteger::BITS_IN_WORD - 1; bit_index >= 0; --bit_index) {
+            size_t shift_amount = word_index * UnsignedBigInteger::BITS_IN_WORD + bit_index;
+            shift_left_without_allocation(denominator, shift_amount, temp_shift_result, temp_shift_plus, temp_shift);
+
+            subtract_without_allocation(remainder, temp_shift, temp_minus);
+            if (!temp_minus.is_invalid()) {
+                remainder.set_to(temp_minus);
+                quotient.set_bit_inplace(shift_amount);
+            }
+        }
+    }
+}
+
+/**
+ * Complexity : O(N) where N is the number of digits in the numerator
+ * Division method :
+ * Starting from the most significant one, for each half-word of the numerator, combine it
+ * with the existing remainder if any, divide the combined number as a u32 operation and
+ * update the quotient / remainder as needed.
+ */
+FLATTEN void UnsignedBigIntegerAlgorithms::divide_u16_without_allocation(
+    UnsignedBigInteger const& numerator,
+    u32 denominator,
+    UnsignedBigInteger& quotient,
+    UnsignedBigInteger& remainder)
+{
+    VERIFY(denominator < (1 << 16));
+    u32 remainder_word = 0;
+    auto numerator_length = numerator.trimmed_length();
+    quotient.set_to_0();
+    quotient.m_words.resize(numerator_length);
+    for (int word_index = numerator_length - 1; word_index >= 0; --word_index) {
+        auto word_high = numerator.m_words[word_index] >> 16;
+        auto word_low = numerator.m_words[word_index] & ((1 << 16) - 1);
+
+        auto number_to_divide_high = (remainder_word << 16) | word_high;
+        auto quotient_high = number_to_divide_high / denominator;
+        remainder_word = number_to_divide_high % denominator;
+
+        auto number_to_divide_low = remainder_word << 16 | word_low;
+        auto quotient_low = number_to_divide_low / denominator;
+        remainder_word = number_to_divide_low % denominator;
+
+        quotient.m_words[word_index] = (quotient_high << 16) | quotient_low;
+    }
+    remainder.set_to(remainder_word);
+}
+
+}

Userland/Libraries/LibCrypto/BigInt/Algorithms/GCD.cpp

@@ -0,0 +1,43 @@
+/*
+ * Copyright (c) 2020, Ali Mohammad Pur <mpfard@serenityos.org>
+ * Copyright (c) 2020-2021, Dex♪ <dexes.ttp@gmail.com>
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ */
+
+#include "UnsignedBigIntegerAlgorithms.h"
+
+namespace Crypto {
+
+void UnsignedBigIntegerAlgorithms::destructive_GCD_without_allocation(
+    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)
+{
+    for (;;) {
+        if (temp_a == 0) {
+            output.set_to(temp_b);
+            return;
+        }
+
+        // temp_b %= temp_a
+        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
+        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);
+    }
+}
+
+}

Userland/Libraries/LibCrypto/BigInt/Algorithms/ModularInverse.cpp

@@ -0,0 +1,99 @@
+/*
+ * Copyright (c) 2020, Ali Mohammad Pur <mpfard@serenityos.org>
+ * Copyright (c) 2020-2021, Dex♪ <dexes.ttp@gmail.com>
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ */
+
+#include "UnsignedBigIntegerAlgorithms.h"
+
+namespace Crypto {
+
+void UnsignedBigIntegerAlgorithms::modular_inverse_without_allocation(
+    UnsignedBigInteger const& a,
+    UnsignedBigInteger const& b,
+    UnsignedBigInteger& temp_1,
+    UnsignedBigInteger& temp_2,
+    UnsignedBigInteger& temp_3,
+    UnsignedBigInteger& temp_4,
+    UnsignedBigInteger& temp_plus,
+    UnsignedBigInteger& temp_minus,
+    UnsignedBigInteger& temp_quotient,
+    UnsignedBigInteger& temp_d,
+    UnsignedBigInteger& temp_u,
+    UnsignedBigInteger& temp_v,
+    UnsignedBigInteger& temp_x,
+    UnsignedBigInteger& result)
+{
+    UnsignedBigInteger one { 1 };
+
+    temp_u.set_to(a);
+    if (a.words()[0] % 2 == 0) {
+        // u += b
+        add_without_allocation(temp_u, b, temp_plus);
+        temp_u.set_to(temp_plus);
+    }
+
+    temp_v.set_to(b);
+    temp_x.set_to(0);
+
+    // d = b - 1
+    subtract_without_allocation(b, one, temp_d);
+
+    while (!(temp_v == 1)) {
+        while (temp_v < temp_u) {
+            // u -= v
+            subtract_without_allocation(temp_u, temp_v, temp_minus);
+            temp_u.set_to(temp_minus);
+
+            // d += x
+            add_without_allocation(temp_d, temp_x, temp_plus);
+            temp_d.set_to(temp_plus);
+
+            while (temp_u.words()[0] % 2 == 0) {
+                if (temp_d.words()[0] % 2 == 1) {
+                    // d += b
+                    add_without_allocation(temp_d, b, temp_plus);
+                    temp_d.set_to(temp_plus);
+                }
+
+                // u /= 2
+                divide_u16_without_allocation(temp_u, 2, temp_quotient, temp_1);
+                temp_u.set_to(temp_quotient);
+
+                // d /= 2
+                divide_u16_without_allocation(temp_d, 2, temp_quotient, temp_1);
+                temp_d.set_to(temp_quotient);
+            }
+        }
+
+        // v -= u
+        subtract_without_allocation(temp_v, temp_u, temp_minus);
+        temp_v.set_to(temp_minus);
+
+        // x += d
+        add_without_allocation(temp_x, temp_d, temp_plus);
+        temp_x.set_to(temp_plus);
+
+        while (temp_v.words()[0] % 2 == 0) {
+            if (temp_x.words()[0] % 2 == 1) {
+                // x += b
+                add_without_allocation(temp_x, b, temp_plus);
+                temp_x.set_to(temp_plus);
+            }
+
+            // v /= 2
+            divide_u16_without_allocation(temp_v, 2, temp_quotient, temp_1);
+            temp_v.set_to(temp_quotient);
+
+            // x /= 2
+            divide_u16_without_allocation(temp_x, 2, temp_quotient, temp_1);
+            temp_x.set_to(temp_quotient);
+        }
+    }
+
+    // return x % b
+    divide_without_allocation(temp_x, b, temp_1, temp_2, temp_3, temp_4, temp_quotient, result);
+}
+
+}

Userland/Libraries/LibCrypto/BigInt/Algorithms/ModularPower.cpp

@@ -0,0 +1,52 @@
+/*
+ * Copyright (c) 2020, Ali Mohammad Pur <mpfard@serenityos.org>
+ * Copyright (c) 2020-2021, Dex♪ <dexes.ttp@gmail.com>
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ */
+
+#include "UnsignedBigIntegerAlgorithms.h"
+
+namespace Crypto {
+
+void UnsignedBigIntegerAlgorithms::destructive_modular_power_without_allocation(
+    UnsignedBigInteger& ep,
+    UnsignedBigInteger& base,
+    UnsignedBigInteger const& m,
+    UnsignedBigInteger& temp_1,
+    UnsignedBigInteger& temp_2,
+    UnsignedBigInteger& temp_3,
+    UnsignedBigInteger& temp_4,
+    UnsignedBigInteger& temp_multiply,
+    UnsignedBigInteger& temp_quotient,
+    UnsignedBigInteger& temp_remainder,
+    UnsignedBigInteger& exp)
+{
+    exp.set_to(1);
+    while (!(ep < 1)) {
+        if (ep.words()[0] % 2 == 1) {
+            // exp = (exp * base) % m;
+            multiply_without_allocation(exp, base, temp_1, temp_2, temp_3, temp_4, temp_multiply);
+            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;
+        divide_u16_without_allocation(ep, 2, temp_quotient, temp_remainder);
+        ep.set_to(temp_quotient);
+
+        // base = (base * base) % m;
+        multiply_without_allocation(base, base, temp_1, temp_2, temp_3, temp_4, temp_multiply);
+        divide_without_allocation(temp_multiply, m, temp_1, temp_2, temp_3, temp_4, temp_quotient, temp_remainder);
+        base.set_to(temp_remainder);
+
+        // Note that not clamping here would cause future calculations (multiply, specifically) to allocate even more unused space
+        // which would then persist through the temp bigints, and significantly slow down later loops.
+        // To avoid that, we can clamp to a specific max size, or just clamp to the min needed amount of space.
+        ep.clamp_to_trimmed_length();
+        exp.clamp_to_trimmed_length();
+        base.clamp_to_trimmed_length();
+    }
+}
+
+}

Userland/Libraries/LibCrypto/BigInt/Algorithms/Multiplication.cpp

@@ -0,0 +1,48 @@
+/*
+ * Copyright (c) 2020, Itamar S. <itamar8910@gmail.com>
+ * Copyright (c) 2020-2021, Dex♪ <dexes.ttp@gmail.com>
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ */
+
+#include "UnsignedBigIntegerAlgorithms.h"
+
+namespace Crypto {
+
+/**
+ * Complexity: O(N^2) where N is the number of words in the larger number
+ * Multiplication method:
+ * An integer is equal to the sum of the powers of two
+ * according to the indices of its 'on' bits.
+ * So to multiple x*y, we go over each '1' bit in x (say the i'th bit),
+ * and add y<<i to the result.
+ */
+FLATTEN void UnsignedBigIntegerAlgorithms::multiply_without_allocation(
+    UnsignedBigInteger const& left,
+    UnsignedBigInteger const& right,
+    UnsignedBigInteger& temp_shift_result,
+    UnsignedBigInteger& temp_shift_plus,
+    UnsignedBigInteger& temp_shift,
+    UnsignedBigInteger& temp_plus,
+    UnsignedBigInteger& output)
+{
+    output.set_to_0();
+
+    // iterate all bits
+    for (size_t word_index = 0; word_index < left.length(); ++word_index) {
+        for (size_t bit_index = 0; bit_index < UnsignedBigInteger::BITS_IN_WORD; ++bit_index) {
+            // If the bit is off - skip over it
+            if (!(left.m_words[word_index] & (1 << bit_index)))
+                continue;
+
+            size_t shift_amount = word_index * UnsignedBigInteger::BITS_IN_WORD + bit_index;
+
+            // output += (right << shift_amount);
+            shift_left_without_allocation(right, shift_amount, temp_shift_result, temp_shift_plus, temp_shift);
+            add_without_allocation(output, temp_shift, temp_plus);
+            output.set_to(temp_plus);
+        }
+    }
+}
+
+}

Userland/Libraries/LibCrypto/BigInt/Algorithms/SimpleOperations.cpp

@@ -0,0 +1,91 @@
+/*
+ * Copyright (c) 2020, Itamar S. <itamar8910@gmail.com>
+ * Copyright (c) 2020-2021, Dex♪ <dexes.ttp@gmail.com>
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ */
+
+#include "UnsignedBigIntegerAlgorithms.h"
+
+namespace Crypto {
+
+/**
+ * Complexity: O(N) where N is the number of words in the larger number
+ */
+void UnsignedBigIntegerAlgorithms::add_without_allocation(
+    UnsignedBigInteger const& left,
+    UnsignedBigInteger const& right,
+    UnsignedBigInteger& output)
+{
+    const UnsignedBigInteger* const longer = (left.length() > right.length()) ? &left : &right;
+    const UnsignedBigInteger* const shorter = (longer == &right) ? &left : &right;
+
+    u8 carry = 0;
+
+    output.set_to_0();
+    output.m_words.resize_and_keep_capacity(longer->length());
+
+    for (size_t i = 0; i < shorter->length(); ++i) {
+        u32 word_addition_result = shorter->m_words[i] + longer->m_words[i];
+        u8 carry_out = 0;
+        // if there was a carry, the result will be smaller than any of the operands
+        if (word_addition_result + carry < shorter->m_words[i]) {
+            carry_out = 1;
+        }
+        if (carry) {
+            word_addition_result++;
+        }
+        carry = carry_out;
+        output.m_words[i] = word_addition_result;
+    }
+
+    for (size_t i = shorter->length(); i < longer->length(); ++i) {
+        u32 word_addition_result = longer->m_words[i] + carry;
+
+        carry = 0;
+        if (word_addition_result < longer->m_words[i]) {
+            carry = 1;
+        }
+        output.m_words[i] = word_addition_result;
+    }
+    if (carry) {
+        output.m_words.append(carry);
+    }
+}
+
+/**
+ * Complexity: O(N) where N is the number of words in the larger number
+ */
+void UnsignedBigIntegerAlgorithms::subtract_without_allocation(
+    UnsignedBigInteger const& left,
+    UnsignedBigInteger const& right,
+    UnsignedBigInteger& output)
+{
+    if (left < right) {
+        output.invalidate();
+        return;
+    }
+
+    u8 borrow = 0;
+    auto own_length = left.length();
+    auto other_length = right.length();
+
+    output.set_to_0();
+    output.m_words.resize_and_keep_capacity(own_length);
+
+    for (size_t i = 0; i < own_length; ++i) {
+        u32 other_word = (i < other_length) ? right.m_words[i] : 0;
+        i64 temp = static_cast<i64>(left.m_words[i]) - static_cast<i64>(other_word) - static_cast<i64>(borrow);
+        // If temp < 0, we had an underflow
+        borrow = (temp >= 0) ? 0 : 1;
+        if (temp < 0) {
+            temp += (UINT32_MAX + 1);
+        }
+        output.m_words[i] = temp;
+    }
+
+    // This assertion should not fail, because we verified that *this>=other at the beginning of the function
+    VERIFY(borrow == 0);
+}
+
+}

Userland/Libraries/LibCrypto/BigInt/Algorithms/UnsignedBigIntegerAlgorithms.h

@@ -0,0 +1,35 @@
+/*
+ * Copyright (c) 2021, Dex♪ <dexes.ttp@gmail.com>
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ */
+
+#pragma once
+
+#include <LibCrypto/BigInt/UnsignedBigInteger.h>
+
+namespace Crypto {
+
+class UnsignedBigIntegerAlgorithms {
+public:
+    static void add_without_allocation(UnsignedBigInteger const& left, UnsignedBigInteger const& right, UnsignedBigInteger& output);
+    static void subtract_without_allocation(UnsignedBigInteger const& left, UnsignedBigInteger const& right, UnsignedBigInteger& output);
+    static void bitwise_or_without_allocation(UnsignedBigInteger const& left, UnsignedBigInteger const& right, UnsignedBigInteger& output);
+    static void bitwise_and_without_allocation(UnsignedBigInteger const& left, UnsignedBigInteger const& right, UnsignedBigInteger& output);
+    static void bitwise_xor_without_allocation(UnsignedBigInteger const& left, UnsignedBigInteger const& right, UnsignedBigInteger& output);
+    static void bitwise_not_without_allocation(UnsignedBigInteger const& left, UnsignedBigInteger& output);
+    static void shift_left_without_allocation(UnsignedBigInteger const& number, size_t bits_to_shift_by, UnsignedBigInteger& temp_result, UnsignedBigInteger& temp_plus, UnsignedBigInteger& output);
+    static void multiply_without_allocation(UnsignedBigInteger const& left, UnsignedBigInteger const& right, UnsignedBigInteger& temp_shift_result, UnsignedBigInteger& temp_shift_plus, UnsignedBigInteger& temp_shift, UnsignedBigInteger& temp_plus, UnsignedBigInteger& output);
+    static void divide_without_allocation(UnsignedBigInteger const& numerator, UnsignedBigInteger const& denominator, UnsignedBigInteger& temp_shift_result, UnsignedBigInteger& temp_shift_plus, UnsignedBigInteger& temp_shift, UnsignedBigInteger& temp_minus, UnsignedBigInteger& quotient, UnsignedBigInteger& remainder);
+    static void divide_u16_without_allocation(UnsignedBigInteger const& numerator, u32 denominator, UnsignedBigInteger& quotient, UnsignedBigInteger& remainder);
+
+    static void destructive_GCD_without_allocation(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);
+    static void modular_inverse_without_allocation(UnsignedBigInteger const& a_, UnsignedBigInteger const& b, UnsignedBigInteger& temp_1, UnsignedBigInteger& temp_2, UnsignedBigInteger& temp_3, UnsignedBigInteger& temp_4, UnsignedBigInteger& temp_plus, UnsignedBigInteger& temp_minus, UnsignedBigInteger& temp_quotient, UnsignedBigInteger& temp_d, UnsignedBigInteger& temp_u, UnsignedBigInteger& temp_v, UnsignedBigInteger& temp_x, UnsignedBigInteger& result);
+    static void destructive_modular_power_without_allocation(UnsignedBigInteger& ep, UnsignedBigInteger& base, UnsignedBigInteger const& m, UnsignedBigInteger& temp_1, UnsignedBigInteger& temp_2, UnsignedBigInteger& temp_3, UnsignedBigInteger& temp_4, UnsignedBigInteger& temp_multiply, UnsignedBigInteger& temp_quotient, UnsignedBigInteger& temp_remainder, UnsignedBigInteger& result);
+
+private:
+    ALWAYS_INLINE static void shift_left_by_n_words(UnsignedBigInteger const& number, size_t number_of_words, UnsignedBigInteger& output);
+    ALWAYS_INLINE static u32 shift_left_get_one_word(UnsignedBigInteger const& number, size_t num_bits, size_t result_word_index);
+};
+
+}