LibCrypto: Add UnsignedBigInteger multiplication

Also added documentation for the runtime complexity of some operations.

Libraries/LibCrypto/BigInt/UnsignedBigInteger.cpp

@@ -28,7 +28,10 @@
 
 namespace Crypto {
 
-UnsignedBigInteger UnsignedBigInteger::add(const UnsignedBigInteger& other)
+/**
+ * Complexity: O(N) where N is the number of words in the larger number
+ */
+UnsignedBigInteger UnsignedBigInteger::add(const UnsignedBigInteger& other) const
 {
     const UnsignedBigInteger* const longer = (length() > other.length()) ? this : &other;
     const UnsignedBigInteger* const shorter = (longer == &other) ? this : &other;
@@ -64,7 +67,10 @@ UnsignedBigInteger UnsignedBigInteger::add(const UnsignedBigInteger& other)
     return result;
 }
 
-UnsignedBigInteger UnsignedBigInteger::sub(const UnsignedBigInteger& other)
+/**
+ * Complexity: O(N) where N is the number of words in the larger number
+ */
+UnsignedBigInteger UnsignedBigInteger::sub(const UnsignedBigInteger& other) const
 {
     UnsignedBigInteger result;
 
@@ -96,6 +102,92 @@ UnsignedBigInteger UnsignedBigInteger::sub(const UnsignedBigInteger& other)
     return result;
 }
 
+/**
+ * Complexity: O(N^2) where N is the number of words in the larger number
+ * Multiplcation method:
+ * An integer is equal to the sum of the powers of two
+ * according to the indexes 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.
+ */
+UnsignedBigInteger UnsignedBigInteger::multiply(const UnsignedBigInteger& other) const
+{
+    UnsignedBigInteger result;
+    // iterate all bits
+    for (size_t word_index = 0; word_index < 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 (!(m_words[word_index] & (1 << bit_index)))
+                continue;
+
+            const size_t shift_amount = word_index * UnsignedBigInteger::BITS_IN_WORD + bit_index;
+            auto shift_result = other.shift_left(shift_amount);
+            result = result.add(shift_result);
+        }
+    }
+    return result;
+}
+
+UnsignedBigInteger UnsignedBigInteger::shift_left(size_t num_bits) const
+{
+    // 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'
+    UnsignedBigInteger temp_result = shift_left_by_n_words(num_bits / UnsignedBigInteger::BITS_IN_WORD);
+
+    // And now we shift by the leftover amount of bits
+    num_bits %= UnsignedBigInteger::BITS_IN_WORD;
+
+    UnsignedBigInteger result(temp_result);
+
+    for (size_t i = 0; i < temp_result.length(); ++i) {
+        u32 current_word_of_temp_result = temp_result.shift_left_get_one_word(num_bits, i);
+        result.m_words[i] = current_word_of_temp_result;
+    }
+
+    // Shifting the last word can produce a carry
+    u32 carry_word = temp_result.shift_left_get_one_word(num_bits, temp_result.length());
+    if (carry_word != 0) {
+        result = result.add(UnsignedBigInteger(carry_word).shift_left_by_n_words(temp_result.length()));
+    }
+    return result;
+}
+
+UnsignedBigInteger UnsignedBigInteger::shift_left_by_n_words(const size_t number_of_words) const
+{
+    // shifting left by N words means just inserting N zeroes to the beginning of the words vector
+    UnsignedBigInteger result;
+    for (size_t i = 0; i < number_of_words; ++i) {
+        result.m_words.append(0);
+    }
+    for (size_t i = 0; i < length(); ++i) {
+        result.m_words.append(m_words[i]);
+    }
+    return result;
+}
+
+/**
+ * Returns the word at a requested index in the result of a shift operation
+ */
+u32 UnsignedBigInteger::shift_left_get_one_word(const size_t num_bits, const size_t result_word_index) const
+{
+    // "<= length()" (rather than length() - 1) is intentional,
+    // The result inedx of length() is used when calculating the carry word
+    ASSERT(result_word_index <= length());
+    ASSERT(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 += m_words[result_word_index - 1] >> (UnsignedBigInteger::BITS_IN_WORD - num_bits);
+    }
+    if (result_word_index < length() && num_bits < 32) {
+        result += m_words[result_word_index] << num_bits;
+    }
+    return result;
+}
+
 bool UnsignedBigInteger::operator==(const UnsignedBigInteger& other) const
 {
     if (trimmed_length() != other.trimmed_length()) {