Changes to get rid of dependency on openssl in the dart VM.

vm/bigint_operations.cc

-// Copyright (c) 2012, the Dart project authors.  Please see the AUTHORS file
-// for details. All rights reserved. Use of this source code is governed by a
-// BSD-style license that can be found in the LICENSE file.
+// Copyright 2012 Google Inc. All Rights Reserved.
 
 #include "vm/bigint_operations.h"
 
-#include <openssl/crypto.h>
+#include "platform/utils.h"
 
-#include "platform/utils.h"
-#include "vm/bigint_store.h"
 #include "vm/double_internals.h"
 #include "vm/exceptions.h"
 #include "vm/zone.h"
 
 namespace dart {
 
-bool Bigint::IsZero() const { return BN_is_zero(BNAddr()); }
-bool Bigint::IsNegative() const { return !!BNAddr()->neg; }
-
-
-void Bigint::SetSign(bool is_negative) const {
-  BIGNUM* bn = MutableBNAddr();
-  // Danger Will Robinson! Use of OpenSSL internals!
-  // FIXME(benl): can be changed to use BN_set_negative() on more
-  // recent OpenSSL releases (> 1.0.0).
-  if (!is_negative || BN_is_zero(bn)) {
-    bn->neg = 0;
-  } else {
-    bn->neg = 1;
-  }
-}
-
-
-BIGNUM* BigintOperations::TmpBN() {
-  BigintStore* store = BigintStore::Get();
-  if (store->bn_ == NULL) {
-    store->bn_ = BN_new();
-  }
-  return store->bn_;
-}
-
-
-BN_CTX* BigintOperations::TmpBNCtx() {
-  BigintStore* store = BigintStore::Get();
-  if (store->bn_ctx_ == NULL) {
-    store->bn_ctx_ = BN_CTX_new();
-  }
-  return store->bn_ctx_;
-}
-
-
 RawBigint* BigintOperations::NewFromSmi(const Smi& smi, Heap::Space space) {
   intptr_t value = smi.Value();
-  bool is_negative = value < 0;
+  if (value == 0) {
+    return Zero();
+  }
 
+  bool is_negative = (value < 0);
   if (is_negative) {
     value = -value;
   }
+  // Assert that there are no overflows. Smis reserve a bit for themselves, but
+  // protect against future changes.
+  ASSERT(-Smi::kMinValue > 0);
 
-  BN_set_word(TmpBN(), value);
+  // A single digit of a Bigint might not be sufficient to store a Smi.
+  // Count number of needed Digits.
+  intptr_t digit_count = 0;
+  intptr_t count_value = value;
+  while (count_value > 0) {
+    digit_count++;
+    count_value >>= kDigitBitSize;
+  }
 
-  const Bigint& result = Bigint::Handle(Bigint::New(TmpBN(), space));
+  // Allocate a bigint of the correct size and copy the bits.
+  const Bigint& result = Bigint::Handle(Bigint::Allocate(digit_count, space));
+  for (int i = 0; i < digit_count; i++) {
+    result.SetChunkAt(i, static_cast<Chunk>(value & kDigitMask));
+    value >>= kDigitBitSize;
+  }
   result.SetSign(is_negative);
-
+  ASSERT(IsClamped(result));
   return result.raw();
 }
 
 
 RawBigint* BigintOperations::NewFromInt64(int64_t value, Heap::Space space) {
   bool is_negative = value < 0;
 
   if (is_negative) {
     value = -value;
   }
 
   const Bigint& result = Bigint::Handle(NewFromUint64(value, space));
   result.SetSign(is_negative);
 
   return result.raw();
 }
 
 
 RawBigint* BigintOperations::NewFromUint64(uint64_t value, Heap::Space space) {
-  const int kNumBytes = sizeof(value);
-  unsigned char pch[kNumBytes];
-  for (int i = kNumBytes - 1; i >= 0; i--) {
-    unsigned char c = value & 0xFF;
-    value >>=8;
-    pch[i] = c;
+  if (value == 0) {
+    return Zero();
   }
-  BN_bin2bn(pch, kNumBytes, TmpBN());
-  return Bigint::New(TmpBN(), space);
+  // A single digit of a Bigint might not be sufficient to store the value.
+  // Count number of needed Digits.
+  intptr_t digit_count = 0;
+  uint64_t count_value = value;
+  while (count_value > 0) {
+    digit_count++;
+    count_value >>= kDigitBitSize;
+  }
+
+  // Allocate a bigint of the correct size and copy the bits.
+  const Bigint& result = Bigint::Handle(Bigint::Allocate(digit_count, space));
+  for (int i = 0; i < digit_count; i++) {
+    result.SetChunkAt(i, static_cast<Chunk>(value & kDigitMask));
+    value >>= kDigitBitSize;
+  }
+  result.SetSign(false);
+  ASSERT(IsClamped(result));
+  return result.raw();
 }
 
 
 RawBigint* BigintOperations::NewFromCString(const char* str,
                                             Heap::Space space) {
   ASSERT(str != NULL);
   if (str[0] == '\0') {
-    return NewFromInt64(0, space);
+    return Zero();
   }
 
   // If the string starts with '-' recursively restart the whole operation
   // without the character and then toggle the sign.
   // This allows multiple leading '-' (which will cancel each other out), but
   // we have added an assert, to make sure that the returned result of the
   // recursive call is not negative.
   // We don't catch leading '-'s for zero. Ex: "--0", or "---".
   if (str[0] == '-') {
     const Bigint& result = Bigint::Handle(NewFromCString(&str[1], space));
-    if (!result.IsNull()) {
-      result.ToggleSign();
-      // FIXME(benl): this will fail if there is more than one leading '-'.
-      ASSERT(result.IsZero() || result.IsNegative());
-    }
+    result.ToggleSign();
+    ASSERT(result.IsZero() || result.IsNegative());
+    ASSERT(IsClamped(result));
     return result.raw();
   }
 
   intptr_t str_length = strlen(str);
   if ((str_length > 2) &&
       (str[0] == '0') &&
       ((str[1] == 'x') || (str[1] == 'X'))) {
     const Bigint& result = Bigint::Handle(FromHexCString(&str[2], space));
+    ASSERT(IsClamped(result));
     return result.raw();
   } else {
-    return FromDecimalCString(str, space);
+    return FromDecimalCString(str);
   }
 }
 
 
+RawBigint* BigintOperations::FromHexCString(const char* hex_string,
+                                            Heap::Space space) {
+  // If the string starts with '-' recursively restart the whole operation
+  // without the character and then toggle the sign.
+  // This allows multiple leading '-' (which will cancel each other out), but
+  // we have added an assert, to make sure that the returned result of the
+  // recursive call is not negative.
+  // We don't catch leading '-'s for zero. Ex: "--0", or "---".
+  if (hex_string[0] == '-') {
+    const Bigint& value = Bigint::Handle(FromHexCString(&hex_string[1], space));
+    value.ToggleSign();
+    ASSERT(value.IsZero() || value.IsNegative());
+    ASSERT(IsClamped(value));
+    return value.raw();
+  }
+
+  ASSERT(kDigitBitSize % 4 == 0);
+  const int kHexCharsPerDigit = kDigitBitSize / 4;
+
+  intptr_t hex_length = strlen(hex_string);
+  // Round up.
+  intptr_t bigint_length = ((hex_length - 1) / kHexCharsPerDigit) + 1;
+  const Bigint& result =
+      Bigint::Handle(Bigint::Allocate(bigint_length, space));
+  // The bigint's least significant digit (lsd) is at position 0, whereas the
+  // given string has it's lsd at the last position.
+  // The hex_i index, pointing into the string, starts therefore at the end,
+  // whereas the bigint-index (i) starts at 0.
+  intptr_t hex_i = hex_length - 1;
+  for (intptr_t i = 0; i < bigint_length; i++) {
+    Chunk digit = 0;
+    int shift = 0;
+    for (int j = 0; j < kHexCharsPerDigit; j++) {
+      // Reads a block of hexadecimal digits and stores it in 'digit'.
+      // Ex: "0123456" with kHexCharsPerDigit == 3, hex_i == 6, reads "456".
+      if (hex_i < 0) {
+        break;
+      }
+      ASSERT(hex_i >= 0);
+      char c = hex_string[hex_i--];
+      ASSERT(Utils::IsHexDigit(c));
+      digit += static_cast<Chunk>(Utils::HexDigitToInt(c)) << shift;
+      shift += 4;
+    }
+    result.SetChunkAt(i, digit);
+  }
+  ASSERT(hex_i == -1);
+  Clamp(result);
+  return result.raw();
+}
+
+
+RawBigint* BigintOperations::FromDecimalCString(const char* str,
+                                                Heap::Space space) {
+  // Read 8 digits a time. 10^8 < 2^27.
+  const int kDigitsPerIteration = 8;
+  const Chunk kTenMultiplier = 100000000;
+  ASSERT(kDigitBitSize >= 27);
+
+  intptr_t str_length = strlen(str);
+  intptr_t str_pos = 0;
+
+  // Read first digit separately. This avoids a multiplication and addition.
+  // The first digit might also not have kDigitsPerIteration decimal digits.
+  int first_digit_decimal_digits = str_length % kDigitsPerIteration;
+  Chunk digit = 0;
+  for (intptr_t i = 0; i < first_digit_decimal_digits; i++) {
+    char c = str[str_pos++];
+    ASSERT(('0' <= c) && (c <= '9'));
+    digit = digit * 10 + c - '0';
+  }
+  Bigint& result = Bigint::Handle(Bigint::Allocate(1, space));
+  result.SetChunkAt(0, digit);
+  Clamp(result);  // Multiplication requires the inputs to be clamped.
+
+  // Read kDigitsPerIteration at a time, and store it in 'increment'.
+  // Then multiply the temporary result by 10^kDigitsPerIteration and add
+  // 'increment' to the new result.
+  const Bigint& increment = Bigint::Handle(Bigint::Allocate(1, space));
+  while (str_pos < str_length - 1) {
+    Chunk digit = 0;
+    for (intptr_t i = 0; i < kDigitsPerIteration; i++) {
+      char c = str[str_pos++];
+      ASSERT(('0' <= c) && (c <= '9'));
+      digit = digit * 10 + c - '0';
+    }
+    result ^= MultiplyWithDigit(result, kTenMultiplier);
+    if (digit != 0) {
+      increment.SetChunkAt(0, digit);
+      result ^= Add(result, increment);
+    }
+  }
+  Clamp(result);
+  return result.raw();
+}
+
+
 RawBigint* BigintOperations::NewFromDouble(double d, Heap::Space space) {
   if ((-1.0 < d) && (d < 1.0)) {
     // Shortcut for small numbers. Also makes the right-shift below
     // well specified.
     Smi& zero = Smi::Handle(Smi::New(0));
     return NewFromSmi(zero, space);
   }
   DoubleInternals internals = DoubleInternals(d);
   if (internals.IsSpecial()) {
     GrowableArray<const Object*> exception_arguments;
@@ skipping 19 matching lines @@
       Bigint::Handle(NewFromInt64(static_cast<int64_t>(significand), space));
   result.SetSign(sign < 0);
   if (exponent > 0) {
     return ShiftLeft(result, exponent);
   } else {
     return result.raw();
   }
 }
 
 
-RawBigint* BigintOperations::FromHexCString(const char* hex_string,
-                                            Heap::Space space) {
-  BIGNUM *bn = TmpBN();
-  BN_hex2bn(&bn, hex_string);
-  ASSERT(bn == TmpBN());
-  const Bigint& result = Bigint::Handle(Bigint::New(TmpBN(), space));
-  return result.raw();
-}
+const char* BigintOperations::ToHexCString(intptr_t length,
+                                           bool is_negative,
+                                           void* data,
+                                           uword (*allocator)(intptr_t size)) {
+  NoGCScope no_gc;
 
+  ASSERT(kDigitBitSize % 4 == 0);
+  const int kHexCharsPerDigit = kDigitBitSize / 4;
 
-RawBigint* BigintOperations::FromDecimalCString(const char* str,
-                                                Heap::Space space) {
-  BIGNUM *bn = TmpBN();
-  int len = BN_dec2bn(&bn, str);
-  if (len == 0) {
-    return Bigint::null();
+  intptr_t chunk_length = length;
+  Chunk* chunk_data = reinterpret_cast<Chunk*>(data);
+  if (length == 0) {
+    const char* zero = "0x0";
+    const int kLength = strlen(zero);
+    char* result = reinterpret_cast<char*>(allocator(kLength + 1));
+    ASSERT(result != NULL);
+    memmove(result, zero, kLength);
+    result[kLength] = '\0';
+    return result;
   }
-  ASSERT(bn == TmpBN());
-  const Bigint& result = Bigint::Handle(Bigint::New(TmpBN(), space));
-  return result.raw();
-}
+  ASSERT(chunk_data != NULL);
 
-
-const char* BigintOperations::ToHexCString(const BIGNUM *bn,
-                                           uword (*allocator)(intptr_t size)) {
-  char* str = BN_bn2hex(bn);
-  char* to_free = str;
-  intptr_t neg = 0;
-  if (str[0] == '-') {
-    ++str;
-    neg = 1;
+  // Compute the number of hex-digits that are needed to represent the
+  // leading bigint-digit. All other digits need exactly kHexCharsPerDigit
+  // characters.
+  int leading_hex_digits = 0;
+  Chunk leading_digit = chunk_data[chunk_length - 1];
+  while (leading_digit != 0) {
+    leading_hex_digits++;
+    leading_digit >>= 4;
   }
-  if (str[0] == '0' && str[1] != '\0') {
-    ++str;
+  // Sum up the space that is needed for the string-representation.
+  intptr_t required_size = 0;
+  if (is_negative) {
+    required_size++;  // For the leading "-".
   }
-  intptr_t length = strlen(str) + 3 + neg;
-  char *result = reinterpret_cast<char*>(allocator(length));
-  if (neg) {
-    result[0] = '-';
-    result[1] = '0';
-    result[2] = 'x';
-    // memcpy would suffice
-    memmove(result + 3, str, length - 3);
-  } else {
-    result[0] = '0';
-    result[1] = 'x';
-    // memcpy would suffice
-    memmove(result + 2, str, length - 2);
+  required_size += 2;  // For the "0x".
+  required_size += leading_hex_digits;
+  required_size += (chunk_length - 1) * kHexCharsPerDigit;
+  required_size++;  // For the trailing '\0'.
+  char* result = reinterpret_cast<char*>(allocator(required_size));
+  // Print the number into the string.
+  // Start from the last position.
+  intptr_t pos = required_size - 1;
+  result[pos--] = '\0';
+  for (intptr_t i = 0; i < (chunk_length - 1); i++) {
+    // Print all non-leading characters (which are printed with
+    // kHexCharsPerDigit characters.
+    Chunk digit = chunk_data[i];
+    for (int j = 0; j < kHexCharsPerDigit; j++) {
+      result[pos--] = Utils::IntToHexDigit(static_cast<int>(digit & 0xF));
+      digit >>= 4;
+    }
   }
-  OPENSSL_free(to_free);
-  return result;
-}
-
-
-const char* BigintOperations::ToDecCString(const Bigint& bigint,
-                                           uword (*allocator)(intptr_t size)) {
-  return ToDecCString(bigint.BNAddr(), allocator);
-}
-
-
-const char* BigintOperations::ToDecCString(const BIGNUM *bn,
-                                           uword (*allocator)(intptr_t size)) {
-  char* str = BN_bn2dec(bn);
-  intptr_t length = strlen(str) + 1;  // '\0'-terminated.
-  char* result = reinterpret_cast<char*>(allocator(length));
-  memmove(result, str, length);
-  OPENSSL_free(str);
+  // Print the leading digit.
+  leading_digit = chunk_data[chunk_length - 1];
+  while (leading_digit != 0) {
+    result[pos--] = Utils::IntToHexDigit(static_cast<int>(leading_digit & 0xF));
+    leading_digit >>= 4;
+  }
+  result[pos--] = 'x';
+  result[pos--] = '0';
+  if (is_negative) {
+    result[pos--] = '-';
+  }
+  ASSERT(pos == -1);
   return result;
 }
 
 
 const char* BigintOperations::ToHexCString(const Bigint& bigint,
                                            uword (*allocator)(intptr_t size)) {
-  return ToHexCString(bigint.BNAddr(), allocator);
+  NoGCScope no_gc;
+
+  intptr_t length = bigint.Length();
+  return ToHexCString(length,
+                      bigint.IsNegative(),
+                      length ? bigint.ChunkAddr(0) : NULL,
+                      allocator);
 }
 
 
 bool BigintOperations::FitsIntoSmi(const Bigint& bigint) {
-  ASSERT(!bigint.IsNull());
-  const BIGNUM *bn = bigint.BNAddr();
-  int bits = BN_num_bits(bn);
-  // Special case for kMinValue as the absolute value is 1 bit longer
-  // than anything else
-  if (bits == Smi::kBits + 1 && BN_abs_is_word(bn, -Smi::kMinValue)
-      && bigint.IsNegative()) {
+  intptr_t bigint_length = bigint.Length();
+  if (bigint_length == 0) {
     return true;
   }
-  // All other cases must have no more bits than the size of an Smi
-  if (bits > Smi::kBits) {
+  if ((bigint_length == 1) &&
+      (static_cast<size_t>(kDigitBitSize) < sizeof(intptr_t) * 8)) {

[Ivan Posva, 2012/02/28 01:38:34]

8 -> kBitsPerByte

[siva, 2012/02/28 18:22:18]

Done.

+    return true;
+  }
+
+  uintptr_t limit;
+  if (bigint.IsNegative()) {
+    limit = static_cast<uintptr_t>(-Smi::kMinValue);
+  } else {
+    limit = static_cast<uintptr_t>(Smi::kMaxValue);
+  }
+  bool bigint_is_greater = false;
+  // Consume the least-significant digits of the bigint.
+  // If bigint_is_greater is set, then the processed sub-part of the bigint is
+  // greater than the corresponding part of the limit.
+  for (int i = 0; i < bigint_length - 1; i++) {
+    Chunk limit_digit = static_cast<Chunk>(limit & kDigitMask);
+    Chunk bigint_digit = bigint.GetChunkAt(i);
+    if (limit_digit < bigint_digit) {
+      bigint_is_greater = true;
+    } else if (limit_digit > bigint_digit) {
+      bigint_is_greater = false;
+    }  // else don't change the boolean.
+    limit >>= kDigitBitSize;
+
+    // Bail out if the bigint is definitely too big.
+    if (limit == 0) {
+      return false;
+    }
+  }
+  Chunk most_significant_digit = bigint.GetChunkAt(bigint_length - 1);
+  if (limit > most_significant_digit) {
+    return true;
+  }
+  if (limit < most_significant_digit) {
     return false;
   }
-  return true;
+  return !bigint_is_greater;
 }
 
 
 RawSmi* BigintOperations::ToSmi(const Bigint& bigint) {

[Ivan Posva, 2012/02/28 01:38:34]

Will we need a FitsIntoMint and ToMint as well?

[siva, 2012/02/28 18:22:18]

They are FitsIntoInt64 and ToInt64 respectively.

Renamed these methods to FitsIntoMint and ToMint

On 2012/02/28 01:38:34, Ivan Posva wrote:

   ASSERT(FitsIntoSmi(bigint));
-  unsigned char bytes[kBitsPerWord / kBitsPerByte];
-  ASSERT(BN_num_bytes(bigint.BNAddr()) <= static_cast<int>(sizeof bytes));
-  int n = BN_bn2bin(bigint.BNAddr(), bytes);
-  ASSERT(n >= 0);
   intptr_t value = 0;
-  ASSERT(n <= static_cast<int>(sizeof value));
-  for (int i = 0; i < n; ++i) {
-    value <<= 8;
-    value |= bytes[i];
+  for (int i = bigint.Length() - 1; i >= 0; i--) {
+    value <<= kDigitBitSize;
+    value += static_cast<intptr_t>(bigint.GetChunkAt(i));
   }
   if (bigint.IsNegative()) {
     value = -value;
   }
   return Smi::New(value);
 }
 
 
 RawDouble* BigintOperations::ToDouble(const Bigint& bigint) {
   // TODO(floitsch/benl): This is a quick and dirty implementation to unblock
   // other areas of the code. It does not handle all bit-twiddling correctly.
+  const double shift_value = (1 << kDigitBitSize);
   double value = 0.0;
-  for (int i = bigint.NumberOfBits() - 1; i >= 0; --i) {
-    value *= 2;
-    value += static_cast<double>(bigint.Bit(i));
+  for (int i = bigint.Length() - 1; i >= 0; i--) {
+    value *= shift_value;
+    value += static_cast<double>(bigint.GetChunkAt(i));
   }
   if (bigint.IsNegative()) {
     value = -value;
   }
   return Double::New(value);
 }
 
 
 bool BigintOperations::FitsIntoInt64(const Bigint& bigint) {
-  const BIGNUM *bn = bigint.BNAddr();
-  int bits = BN_num_bits(bn);
-  if (bits <= 63) return true;
-  if (bits > 64) return false;
-  if (!bigint.IsNegative()) return false;
-  // Special case for negative values, since Int64 representation may lose
-  // one bit.
-  ASSERT(bigint.Bit(63) != 0);
-  for (int i = 0; i < 63; i++) {
-    // Verify that all 63 least significant bits are 0.
-    if (bigint.Bit(i) != 0) return false;
+  intptr_t bigint_length = bigint.Length();
+  if (bigint_length == 0) {
+    return true;
   }
-  return true;
+  if ((bigint_length < 3) &&
+      (static_cast<size_t>(kDigitBitSize) < sizeof(intptr_t) * 8)) {
+    return true;
+  }
+
+  uint64_t limit;
+  if (bigint.IsNegative()) {
+    limit = static_cast<uint64_t>(Mint::kMinValue);
+  } else {
+    limit = static_cast<uint64_t>(Mint::kMaxValue);
+  }
+  bool bigint_is_greater = false;
+  // Consume the least-significant digits of the bigint.
+  // If bigint_is_greater is set, then the processed sub-part of the bigint is
+  // greater than the corresponding part of the limit.
+  for (int i = 0; i < bigint_length - 1; i++) {
+    Chunk limit_digit = static_cast<Chunk>(limit & kDigitMask);
+    Chunk bigint_digit = bigint.GetChunkAt(i);
+    if (limit_digit < bigint_digit) {
+      bigint_is_greater = true;
+    } else if (limit_digit > bigint_digit) {
+      bigint_is_greater = false;
+    }  // else don't change the boolean.
+    limit >>= kDigitBitSize;
+
+    // Bail out if the bigint is definitely too big.
+    if (limit == 0) {
+      return false;
+    }
+  }
+  Chunk most_significant_digit = bigint.GetChunkAt(bigint_length - 1);
+  if (limit > most_significant_digit) {
+    return true;
+  }
+  if (limit < most_significant_digit) {
+    return false;
+  }
+  return !bigint_is_greater;
 }
 
 
 uint64_t BigintOperations::AbsToUint64(const Bigint& bigint) {
-  unsigned char bytes[8];
-  ASSERT(BN_num_bytes(bigint.BNAddr()) <= static_cast<int>(sizeof bytes));
-  int n = BN_bn2bin(bigint.BNAddr(), bytes);
-  ASSERT(n >= 0);
   uint64_t value = 0;
-  ASSERT(n <= static_cast<int>(sizeof value));
-  for (int i = 0; i < n; ++i) {
-    value <<= 8;
-    value |= bytes[i];
+  for (int i = bigint.Length() - 1; i >= 0; i--) {
+    value <<= kDigitBitSize;
+    value += static_cast<intptr_t>(bigint.GetChunkAt(i));
   }
   return value;
 }
 
 
 int64_t BigintOperations::ToInt64(const Bigint& bigint) {
   ASSERT(FitsIntoInt64(bigint));
   int64_t value = AbsToUint64(bigint);
   if (bigint.IsNegative()) {
     value = -value;
   }
   return value;
 }
 
 
 bool BigintOperations::FitsIntoUint64(const Bigint& bigint) {
-  const BIGNUM *bn = bigint.BNAddr();
   if (bigint.IsNegative()) return false;
-  int bits = BN_num_bits(bn);
-  if (bits > 64) return false;
+  intptr_t b_length = bigint.Length();
+  int num_bits = CountBits(bigint.GetChunkAt(b_length - 1));
+  num_bits += (kDigitBitSize * (b_length - 1));
+  if (num_bits > 64) return false;
   return true;
 }
 
 
 uint64_t BigintOperations::ToUint64(const Bigint& bigint) {
   ASSERT(FitsIntoUint64(bigint));
   return AbsToUint64(bigint);
 }
 
 
-RawBigint* BigintOperations::Add(const Bigint& a, const Bigint& b) {
-  int status = BN_add(TmpBN(), a.BNAddr(), b.BNAddr());
-  if (status == 0) {
-    GrowableArray<const Object*> exception_arguments;
-    exception_arguments.Add(
-        &Object::ZoneHandle(String::New("BigintOperations::Add")));
-    Exceptions::ThrowByType(Exceptions::kInternalError, exception_arguments);
-  }
-  const Bigint& result = Bigint::Handle(Bigint::New(TmpBN()));
+RawBigint* BigintOperations::Multiply(const Bigint& a, const Bigint& b) {
+  ASSERT(IsClamped(a));
+  ASSERT(IsClamped(b));
+
+  intptr_t a_length = a.Length();
+  intptr_t b_length = b.Length();
+  intptr_t result_length = a_length + b_length;
+  const Bigint& result = Bigint::Handle(Bigint::Allocate(result_length));
+
+  if (a.IsNegative() != b.IsNegative()) {
+    result.ToggleSign();
+  }
+
+  // Comba multiplication: compute each column separately.
+  // Example: r = a2a1a0 * b2b1b0.
+  //    r =  1    * a0b0 +
+  //        10    * (a1b0 + a0b1) +
+  //        100   * (a2b0 + a1b1 + a0b2) +
+  //        1000  * (a2b1 + a1b2) +
+  //        10000 * a2b2
+  //
+  // Each column will be accumulated in an integer of type DoubleChunk. We
+  // must guarantee that the column-sum will not overflow.
+  //
+  // In the worst case we have to accumulate k = Min(a.length, b.length)
+  // products plus the carry from the previous round.
+  // Each bigint-digit is smaller than beta = 2^kDigitBitSize.
+  // Each product is at most (beta - 1)^2.
+  // If we want to use Comba multiplication the following condition must hold:
+  // k * (beta - 1)^2 + (2^(kDoubleChunkBitSize - kDigitBitSize) - 1) <
+  //        2^kDoubleChunkBitSize.
+  const DoubleChunk square =
+      static_cast<DoubleChunk>(kDigitMaxValue) * kDigitMaxValue;
+  const DoubleChunk kDoubleChunkMaxValue = static_cast<DoubleChunk>(-1);
+  const DoubleChunk left_over_carry = kDoubleChunkMaxValue >> kDigitBitSize;
+  const intptr_t kMaxDigits = (kDoubleChunkMaxValue - left_over_carry) / square;
+  if (Utils::Minimum(a_length, b_length) > kMaxDigits) {
+    UNIMPLEMENTED();
+  }
+
+  DoubleChunk accumulator = 0;  // Accumulates the result of one column.
+  for (intptr_t i = 0; i < result_length; i++) {
+    // Example: r = a2a1a0 * b2b1b0.
+    //   For i == 0, compute a0b0.
+    //       i == 1,         a1b0 + a0b1 + overflow from i == 0.
+    //       i == 2,         a2b0 + a1b1 + a0b2 + overflow from i == 1.
+    //       ...
+    // The indices into a and b are such that their sum equals i.
+    intptr_t a_index = Utils::Minimum(a_length - 1, i);
+    intptr_t b_index = i - a_index;
+    ASSERT(a_index + b_index == i);
+
+    // Instead of testing for a_index >= 0 && b_index < b_length we compute the
+    // number of iterations first.
+    intptr_t iterations = Utils::Minimum(b_length - b_index, a_index + 1);
+    for (intptr_t j = 0; j < iterations; j++) {
+      DoubleChunk chunk_a = a.GetChunkAt(a_index);
+      DoubleChunk chunk_b = b.GetChunkAt(b_index);
+      accumulator += chunk_a * chunk_b;
+      a_index--;
+      b_index++;
+    }
+    result.SetChunkAt(i, static_cast<Chunk>(accumulator & kDigitMask));
+    accumulator >>= kDigitBitSize;
+  }
+  ASSERT(accumulator == 0);
+
+  Clamp(result);
   return result.raw();
 }
 
 
-RawBigint* BigintOperations::Subtract(const Bigint& a, const Bigint& b) {
-  int status = BN_sub(TmpBN(), a.BNAddr(), b.BNAddr());
-  if (status == 0) {
-    GrowableArray<const Object*> exception_arguments;
-    exception_arguments.Add(
-        &Object::ZoneHandle(String::New("BigintOperations::Subtract")));
-    Exceptions::ThrowByType(Exceptions::kInternalError, exception_arguments);
-  }
-  const Bigint& result = Bigint::Handle(Bigint::New(TmpBN()));
-  return result.raw();
-}
-
-
-RawBigint* BigintOperations::Multiply(const Bigint& a, const Bigint& b) {
-  int status = BN_mul(TmpBN(), a.BNAddr(), b.BNAddr(), TmpBNCtx());
-  if (status == 0) {
-    GrowableArray<const Object*> exception_arguments;
-    exception_arguments.Add(
-        &Object::ZoneHandle(String::New("BigintOperations::Multiply")));
-    Exceptions::ThrowByType(Exceptions::kInternalError, exception_arguments);
-  }
-  const Bigint& result = Bigint::Handle(Bigint::New(TmpBN()));
-  return result.raw();
-}
-
-
 RawBigint* BigintOperations::Divide(const Bigint& a, const Bigint& b) {
-  int status = BN_div(TmpBN(), NULL, a.BNAddr(), b.BNAddr(), TmpBNCtx());
-  if (status == 0) {
-    GrowableArray<const Object*> exception_arguments;
-    exception_arguments.Add(
-        &Object::ZoneHandle(String::New("BigintOperations::Divide")));
-    Exceptions::ThrowByType(Exceptions::kInternalError, exception_arguments);
-  }
-  const Bigint& quotient = Bigint::Handle(Bigint::New(TmpBN()));
+  Bigint& quotient = Bigint::Handle();
+  Bigint& remainder = Bigint::Handle();
+  DivideRemainder(a, b, &quotient, &remainder);
   return quotient.raw();
 }
 
 
 RawBigint* BigintOperations::Modulo(const Bigint& a, const Bigint& b) {
-  int status = BN_nnmod(TmpBN(), a.BNAddr(), b.BNAddr(), TmpBNCtx());
-  if (status == 0) {
-    GrowableArray<const Object*> exception_arguments;
-    exception_arguments.Add(
-        &Object::ZoneHandle(String::New("BigintOperations::Modulo")));
-    Exceptions::ThrowByType(Exceptions::kInternalError, exception_arguments);
-  }
-  const Bigint& modulo = Bigint::Handle(Bigint::New(TmpBN()));
+  Bigint& quotient = Bigint::Handle();
+  Bigint& modulo = Bigint::Handle();
+  DivideRemainder(a, b, &quotient, &modulo);
   return modulo.raw();
 }
 
 
 RawBigint* BigintOperations::Remainder(const Bigint& a, const Bigint& b) {
-  int status = BN_div(NULL, TmpBN(), a.BNAddr(), b.BNAddr(), TmpBNCtx());
-  if (status == 0) {
-    GrowableArray<const Object*> exception_arguments;
-    exception_arguments.Add(
-        &Object::ZoneHandle(String::New("BigintOperations::Remainder")));
-    Exceptions::ThrowByType(Exceptions::kInternalError, exception_arguments);
-  }
-  const Bigint& remainder = Bigint::Handle(Bigint::New(TmpBN()));
+  Bigint& quotient = Bigint::Handle();
+  Bigint& remainder = Bigint::Handle();
+  DivideRemainder(a, b, &quotient, &remainder);
   return remainder.raw();
 }
 
 
 RawBigint* BigintOperations::ShiftLeft(const Bigint& bigint, intptr_t amount) {
-  int status = BN_lshift(TmpBN(), bigint.BNAddr(), amount);
-  if (status == 0) {
-    GrowableArray<const Object*> exception_arguments;
-    exception_arguments.Add(
-        &Object::ZoneHandle(String::New("BigintOperations::ShiftLeft")));
-    Exceptions::ThrowByType(Exceptions::kInternalError, exception_arguments);
-  }
-  const Bigint& result = Bigint::Handle(Bigint::New(TmpBN()));
-  return result.raw();
+  ASSERT(IsClamped(bigint));
+  ASSERT(amount >= 0);
+  intptr_t bigint_length = bigint.Length();
+  if (bigint.IsZero()) {
+    return Zero();
+  }
+  // TODO(floitsch): can we reuse the input?
+  if (amount == 0) {
+    return Copy(bigint);
+  }
+  intptr_t digit_shift = amount / kDigitBitSize;
+  intptr_t bit_shift = amount % kDigitBitSize;
+  if (bit_shift == 0) {
+    const Bigint& result =
+        Bigint::Handle(Bigint::Allocate(bigint_length + digit_shift));
+    for (intptr_t i = 0; i < digit_shift; i++) {
+      result.SetChunkAt(i, 0);
+    }
+    for (intptr_t i = 0; i < bigint_length; i++) {
+      result.SetChunkAt(i + digit_shift, bigint.GetChunkAt(i));
+    }
+    if (bigint.IsNegative()) {
+      result.ToggleSign();
+    }
+    return result.raw();
+  } else {
+    const Bigint& result =
+        Bigint::Handle(Bigint::Allocate(bigint_length + digit_shift + 1));
+    for (intptr_t i = 0; i < digit_shift; i++) {
+      result.SetChunkAt(i, 0);
+    }
+    Chunk carry = 0;
+    for (intptr_t i = 0; i < bigint_length; i++) {
+      Chunk digit = bigint.GetChunkAt(i);
+      Chunk shifted_digit = ((digit << bit_shift) & kDigitMask) + carry;
+      result.SetChunkAt(i + digit_shift, shifted_digit);
+      carry = digit >> (kDigitBitSize - bit_shift);
+    }
+    result.SetChunkAt(bigint_length + digit_shift, carry);
+    if (bigint.IsNegative()) {
+      result.ToggleSign();
+    }
+    Clamp(result);
+    return result.raw();
+  }
 }
 
 
 RawBigint* BigintOperations::ShiftRight(const Bigint& bigint, intptr_t amount) {
+  ASSERT(IsClamped(bigint));
   ASSERT(amount >= 0);
-  int status = BN_rshift(TmpBN(), bigint.BNAddr(), amount);
-  if (status == 0) {
-    GrowableArray<const Object*> exception_arguments;
-    exception_arguments.Add(
-        &Object::ZoneHandle(String::New("BigintOperations::ShiftRight")));
-    Exceptions::ThrowByType(Exceptions::kInternalError, exception_arguments);
-  }
-  // OpenSSL doesn't take account of sign when shifting - this fixes it.
+  intptr_t bigint_length = bigint.Length();
+  if (bigint.IsZero()) {
+    return Zero();
+  }
+  // TODO(floitsch): can we reuse the input?
+  if (amount == 0) {
+    return Copy(bigint);
+  }
+  intptr_t digit_shift = amount / kDigitBitSize;
+  intptr_t bit_shift = amount % kDigitBitSize;
+  if (digit_shift >= bigint_length) {
+    return bigint.IsNegative() ? MinusOne() : Zero();
+  }
+
+  const Bigint& result =
+      Bigint::Handle(Bigint::Allocate(bigint_length - digit_shift));
+  if (bit_shift == 0) {
+    for (intptr_t i = 0; i < bigint_length - digit_shift; i++) {
+      result.SetChunkAt(i, bigint.GetChunkAt(i + digit_shift));
+    }
+  } else {
+    Chunk carry = 0;
+    for (intptr_t i = bigint_length - 1; i >= digit_shift; i--) {
+      Chunk digit = bigint.GetChunkAt(i);
+      Chunk shifted_digit = (digit >> bit_shift) + carry;
+      result.SetChunkAt(i - digit_shift, shifted_digit);
+      carry = (digit << (kDigitBitSize - bit_shift)) & kDigitMask;
+    }
+    Clamp(result);
+  }
+
   if (bigint.IsNegative()) {
-    for (intptr_t i = 0; i < amount; ++i) {
-      if (bigint.IsBitSet(i)) {
-        int status = BN_sub_word(TmpBN(), 1);
-        ASSERT(status == 1);
+    result.ToggleSign();
+    // If the input is negative then the result needs to be rounded down.
+    // Example: -5 >> 2 => -2
+    bool needs_rounding = false;
+    for (intptr_t i = 0; i < digit_shift; i++) {
+      if (bigint.GetChunkAt(i) != 0) {
+        needs_rounding = true;
         break;
       }
     }
-  }
-  const Bigint& result = Bigint::Handle(Bigint::New(TmpBN()));
+    if (!needs_rounding && (bit_shift > 0)) {
+      Chunk digit = bigint.GetChunkAt(digit_shift);
+      needs_rounding = (digit << (kChunkBitSize - bit_shift)) != 0;
+    }
+    if (needs_rounding) {
+      Bigint& one = Bigint::Handle(One());
+      return Subtract(result, one);
+    }
+  }
+
   return result.raw();
 }
 
-/* Bit operations are complicated by negatives: BIGNUMs don't use 2's
- * complement, but instead store the absolute value and a sign
- * indicator. However bit operations are defined on 2's complement
- * representations. This function handles the necessary
- * invert-and-carry operations to deal with the fact that -x = ~x + 1
- * both on the operands and result. The actual operation performed is
- * specified by the truth table |tt|, which is in the order of input
- * bits (00, 01, 10, 11).
- */
-RawBigint* BigintOperations::BitTT(const Bigint& a, const Bigint& b,
-                                   bool tt[4]) {
-  BN_zero(TmpBN());
-  int n;
-  int rflip = 0;
-  int rborrow = 0;
-  // There's probably a clever way to figure out why these are what
-  // they are, but I confess I worked them out pragmatically.
-  if (a.IsNegative() && b.IsNegative()) {
-    if (tt[3]) {
-      rflip = -1;
-      rborrow = -1;
-    }
-    if (tt[2] != tt[3]) {
-      n = Utils::Maximum(a.NumberOfBits(), b.NumberOfBits());
+
+RawBigint* BigintOperations::BitAnd(const Bigint& a, const Bigint& b) {
+  ASSERT(IsClamped(a));
+  ASSERT(IsClamped(b));
+
+  if (a.IsZero() || b.IsZero()) {
+    return Zero();
+  }
+  if (a.IsNegative() && !b.IsNegative()) {
+    return BitAnd(b, a);
+  }
+  if ((a.IsNegative() == b.IsNegative()) && (a.Length() < b.Length())) {
+    return BitAnd(b, a);
+  }
+
+  intptr_t a_length = a.Length();
+  intptr_t b_length = b.Length();
+  intptr_t min_length = Utils::Minimum(a_length, b_length);
+  intptr_t max_length = Utils::Maximum(a_length, b_length);
+  if (!b.IsNegative()) {
+    ASSERT(!a.IsNegative());
+    intptr_t result_length = min_length;
+    const Bigint& result = Bigint::Handle(Bigint::Allocate(result_length));
+
+    for (intptr_t i = 0; i < min_length; i++) {
+      result.SetChunkAt(i, a.GetChunkAt(i) & b.GetChunkAt(i));
+    }
+    Clamp(result);
+    return result.raw();
+  }
+
+  // Bigints encode negative values by storing the absolute value and the sign
+  // separately. To do bit operations we need to simulate numbers that are
+  // implemented as two's complement.
+  // The negation of a positive number x would be encoded as follows in
+  // two's complement: n = ~(x - 1).
+  // The inverse transformation is hence (~n) + 1.
+
+  if (!a.IsNegative()) {
+    ASSERT(b.IsNegative());
+    // The result will be positive.
+    intptr_t result_length = a_length;
+    const Bigint& result = Bigint::Handle(Bigint::Allocate(result_length));
+    Chunk borrow = 1;
+    for (intptr_t i = 0; i < min_length; i++) {
+      Chunk b_digit = b.GetChunkAt(i) - borrow;
+      result.SetChunkAt(i, a.GetChunkAt(i) & (~b_digit) & kDigitMask);
+      borrow = b_digit >> (kChunkBitSize - 1);
+    }
+    for (intptr_t i = min_length; i < a_length; i++) {
+      result.SetChunkAt(i, a.GetChunkAt(i) & (kDigitMaxValue - borrow));
+      borrow = 0;
+    }
+    Clamp(result);
+    return result.raw();
+  }
+
+  ASSERT(a.IsNegative());
+  ASSERT(b.IsNegative());
+  // The result will be negative.
+  // We need to convert a and b to two's complement. Do the bit-operation there,
+  // and transform the resulting bits from two's complement back to separated
+  // magnitude and sign.
+  // a & b is therefore computed as ~((~(a - 1)) & (~(b - 1))) + 1 which is
+  //   equal to ((a-1) | (b-1)) + 1.
+  intptr_t result_length = max_length + 1;
+  const Bigint& result = Bigint::Handle(Bigint::Allocate(result_length));
+  result.ToggleSign();
+  Chunk a_borrow = 1;
+  Chunk b_borrow = 1;
+  Chunk result_carry = 1;
+  ASSERT(a_length >= b_length);
+  for (intptr_t i = 0; i < b_length; i++) {
+    Chunk a_digit = a.GetChunkAt(i) - a_borrow;
+    Chunk b_digit = b.GetChunkAt(i) - b_borrow;
+    Chunk result_chunk = ((a_digit | b_digit) & kDigitMask) + result_carry;
+    result.SetChunkAt(i, result_chunk & kDigitMask);
+    a_borrow = a_digit >> (kChunkBitSize - 1);
+    b_borrow = b_digit >> (kChunkBitSize - 1);
+    result_carry = result_chunk >> kDigitBitSize;
+  }
+  for (intptr_t i = b_length; i < a_length; i++) {
+    Chunk a_digit = a.GetChunkAt(i) - a_borrow;
+    Chunk b_digit = -b_borrow;
+    Chunk result_chunk = ((a_digit | b_digit) & kDigitMask) + result_carry;
+    result.SetChunkAt(i, result_chunk & kDigitMask);
+    a_borrow = a_digit >> (kChunkBitSize - 1);
+    b_borrow = 0;
+    result_carry = result_chunk >> kDigitBitSize;
+  }
+  Chunk a_digit = -a_borrow;
+  Chunk b_digit = -b_borrow;
+  Chunk result_chunk = ((a_digit | b_digit) & kDigitMask) + result_carry;
+  result.SetChunkAt(a_length, result_chunk & kDigitMask);
+  Clamp(result);
+  return result.raw();
+}
+
+
+RawBigint* BigintOperations::BitOr(const Bigint& a, const Bigint& b) {
+  ASSERT(IsClamped(a));
+  ASSERT(IsClamped(b));
+
+  if (a.IsNegative() && !b.IsNegative()) {
+    return BitOr(b, a);
+  }
+  if ((a.IsNegative() == b.IsNegative()) && (a.Length() < b.Length())) {
+    return BitOr(b, a);
+  }
+
+  intptr_t a_length = a.Length();
+  intptr_t b_length = b.Length();
+  intptr_t min_length = Utils::Minimum(a_length, b_length);
+  intptr_t max_length = Utils::Maximum(a_length, b_length);
+  if (!b.IsNegative()) {
+    ASSERT(!a.IsNegative());
+    intptr_t result_length = max_length;
+    const Bigint& result = Bigint::Handle(Bigint::Allocate(result_length));
+
+    ASSERT(a_length >= b_length);
+    for (intptr_t i = 0; i < b_length; i++) {
+      result.SetChunkAt(i, a.GetChunkAt(i) | b.GetChunkAt(i));
+    }
+    for (intptr_t i = b_length; i < a_length; i++) {
+      result.SetChunkAt(i, a.GetChunkAt(i));
+    }
+    return result.raw();
+  }
+
+  // Bigints encode negative values by storing the absolute value and the sign
+  // separately. To do bit operations we need to simulate numbers that are
+  // implemented as two's complement.
+  // The negation of a positive number x would be encoded as follows in
+  // two's complement: n = ~(x - 1).
+  // The inverse transformation is hence (~n) + 1.
+
+  if (!a.IsNegative()) {
+    ASSERT(b.IsNegative());
+    if (a.IsZero()) {
+      return Copy(b);
+    }
+    // The result will be negative.
+    // We need to convert  b to two's complement. Do the bit-operation there,
+    // and transform the resulting bits from two's complement back to separated
+    // magnitude and sign.
+    // a | b is therefore computed as ~((a & (~(b - 1))) + 1 which is
+    //   equal to ((~a) & (b-1)) + 1.
+    intptr_t result_length = b_length;
+    const Bigint& result = Bigint::Handle(Bigint::Allocate(result_length));
+    result.ToggleSign();
+    Chunk borrow = 1;
+    Chunk result_carry = 1;
+    for (intptr_t i = 0; i < min_length; i++) {
+      Chunk a_digit = a.GetChunkAt(i);
+      Chunk b_digit = b.GetChunkAt(i) - borrow;
+      Chunk result_digit = ((~a_digit) & b_digit & kDigitMask) + result_carry;
+      result.SetChunkAt(i, result_digit & kDigitMask);
+      borrow = b_digit >> (kChunkBitSize - 1);
+      result_carry = result_digit >> kDigitBitSize;
+    }
+    ASSERT(result_carry == 0);
+    for (intptr_t i = min_length; i < b_length; i++) {
+      Chunk b_digit = b.GetChunkAt(i) - borrow;
+      Chunk result_digit = (b_digit & kDigitMask) + result_carry;
+      result.SetChunkAt(i, result_digit & kDigitMask);
+      borrow = b_digit >> (kChunkBitSize - 1);
+      result_carry = result_digit >> kDigitBitSize;
+    }
+    ASSERT(result_carry == 0);
+    Clamp(result);
+    return result.raw();
+  }
+
+  ASSERT(a.IsNegative());
+  ASSERT(b.IsNegative());
+  // The result will be negative.
+  // We need to convert a and b to two's complement. Do the bit-operation there,
+  // and transform the resulting bits from two's complement back to separated
+  // magnitude and sign.
+  // a & b is therefore computed as ~((~(a - 1)) | (~(b - 1))) + 1 which is
+  //   equal to ((a-1) & (b-1)) + 1.
+  intptr_t result_length = min_length + 1;
+  const Bigint& result = Bigint::Handle(Bigint::Allocate(result_length));
+  result.ToggleSign();
+  Chunk a_borrow = 1;
+  Chunk b_borrow = 1;
+  Chunk result_carry = 1;
+  ASSERT(a_length >= b_length);
+  for (intptr_t i = 0; i < b_length; i++) {
+    Chunk a_digit = a.GetChunkAt(i) - a_borrow;
+    Chunk b_digit = b.GetChunkAt(i) - b_borrow;
+    Chunk result_chunk = ((a_digit & b_digit) & kDigitMask) + result_carry;
+    result.SetChunkAt(i, result_chunk & kDigitMask);
+    a_borrow = a_digit >> (kChunkBitSize - 1);
+    b_borrow = b_digit >> (kChunkBitSize - 1);
+    result_carry = result_chunk >> kDigitBitSize;
+  }
+  result.SetChunkAt(a_length, result_carry);
+  Clamp(result);
+  return result.raw();
+}
+
+
+RawBigint* BigintOperations::BitXor(const Bigint& a, const Bigint& b) {
+  ASSERT(IsClamped(a));
+  ASSERT(IsClamped(b));
+
+  if (a.IsZero()) {
+    return Copy(b);
+  }
+  if (b.IsZero()) {
+    return Copy(a);
+  }
+  if (a.IsNegative() && !b.IsNegative()) {
+    return BitXor(b, a);
+  }
+  if ((a.IsNegative() == b.IsNegative()) && (a.Length() < b.Length())) {
+    return BitXor(b, a);
+  }
+
+  intptr_t a_length = a.Length();
+  intptr_t b_length = b.Length();
+  intptr_t min_length = Utils::Minimum(a_length, b_length);
+  intptr_t max_length = Utils::Maximum(a_length, b_length);
+  if (!b.IsNegative()) {
+    ASSERT(!a.IsNegative());
+    intptr_t result_length = max_length;
+    const Bigint& result = Bigint::Handle(Bigint::Allocate(result_length));
+
+    ASSERT(a_length >= b_length);
+    for (intptr_t i = 0; i < b_length; i++) {
+      result.SetChunkAt(i, a.GetChunkAt(i) ^ b.GetChunkAt(i));
+    }
+    for (intptr_t i = b_length; i < a_length; i++) {
+      result.SetChunkAt(i, a.GetChunkAt(i));
+    }
+    Clamp(result);
+    return result.raw();
+  }
+
+  // Bigints encode negative values by storing the absolute value and the sign
+  // separately. To do bit operations we need to simulate numbers that are
+  // implemented as two's complement.
+  // The negation of a positive number x would be encoded as follows in
+  // two's complement: n = ~(x - 1).
+  // The inverse transformation is hence (~n) + 1.
+
+  if (!a.IsNegative()) {
+    ASSERT(b.IsNegative());
+    // The result will be negative.
+    // We need to convert  b to two's complement. Do the bit-operation there,
+    // and transform the resulting bits from two's complement back to separated
+    // magnitude and sign.
+    // a ^ b is therefore computed as ~((a ^ (~(b - 1))) + 1.
+    intptr_t result_length = max_length + 1;
+    const Bigint& result = Bigint::Handle(Bigint::Allocate(result_length));
+    result.ToggleSign();
+    Chunk borrow = 1;
+    Chunk result_carry = 1;
+    for (intptr_t i = 0; i < min_length; i++) {
+      Chunk a_digit = a.GetChunkAt(i);
+      Chunk b_digit = b.GetChunkAt(i) - borrow;
+      Chunk result_digit =
+          ((~(a_digit ^ ~b_digit)) & kDigitMask) + result_carry;
+      result.SetChunkAt(i, result_digit & kDigitMask);
+      borrow = b_digit >> (kChunkBitSize - 1);
+      result_carry = result_digit >> kDigitBitSize;
+    }
+    for (intptr_t i = min_length; i < a_length; i++) {
+      Chunk a_digit = a.GetChunkAt(i);
+      Chunk b_digit = -borrow;
+      Chunk result_digit =
+          ((~(a_digit ^ ~b_digit)) & kDigitMask) + result_carry;
+      result.SetChunkAt(i, result_digit & kDigitMask);
+      borrow = b_digit >> (kChunkBitSize - 1);
+      result_carry = result_digit >> kDigitBitSize;
+    }
+    for (intptr_t i = min_length; i < b_length; i++) {
+      // a_digit = 0.
+      Chunk b_digit = b.GetChunkAt(i) - borrow;
+      Chunk result_digit = (b_digit & kDigitMask) + result_carry;
+      result.SetChunkAt(i, result_digit & kDigitMask);
+      borrow = b_digit >> (kChunkBitSize - 1);
+      result_carry = result_digit >> kDigitBitSize;
+    }
+    result.SetChunkAt(max_length, result_carry);
+    Clamp(result);
+    return result.raw();
+  }
+
+  ASSERT(a.IsNegative());
+  ASSERT(b.IsNegative());
+  // The result will be positive.
+  // We need to convert a and b to two's complement, do the bit-operation there,
+  // and simply store the result.
+  // a ^ b is therefore computed as (~(a - 1)) ^ (~(b - 1)).
+  intptr_t result_length = max_length;
+  const Bigint& result = Bigint::Handle(Bigint::Allocate(result_length));
+  Chunk a_borrow = 1;
+  Chunk b_borrow = 1;
+  ASSERT(a_length >= b_length);
+  for (intptr_t i = 0; i < b_length; i++) {
+    Chunk a_digit = a.GetChunkAt(i) - a_borrow;
+    Chunk b_digit = b.GetChunkAt(i) - b_borrow;
+    Chunk result_chunk = (~a_digit) ^ (~b_digit);
+    result.SetChunkAt(i, result_chunk & kDigitMask);
+    a_borrow = a_digit >> (kChunkBitSize - 1);
+    b_borrow = b_digit >> (kChunkBitSize - 1);
+  }
+  ASSERT(b_borrow == 0);
+  for (intptr_t i = b_length; i < a_length; i++) {
+    Chunk a_digit = a.GetChunkAt(i) - a_borrow;
+    result.SetChunkAt(i, (~a_digit) & kDigitMask);
+    a_borrow = a_digit >> (kChunkBitSize - 1);
+  }
+  ASSERT(a_borrow == 0);
+  Clamp(result);
+  return result.raw();
+}
+
+
+RawBigint* BigintOperations::BitNot(const Bigint& bigint) {
+  if (bigint.IsZero()) {
+    return MinusOne();
+  }
+  const Bigint& one_bigint = Bigint::Handle(One());
+  if (bigint.IsNegative()) {
+    return UnsignedSubtract(bigint, one_bigint);
+  } else {
+    const Bigint& result = Bigint::Handle(UnsignedAdd(bigint, one_bigint));
+    result.ToggleSign();
+    return result.raw();
+  }
+}
+
+
+int BigintOperations::Compare(const Bigint& a, const Bigint& b) {
+  bool a_is_negative = a.IsNegative();
+  bool b_is_negative = b.IsNegative();
+  if (a_is_negative != b_is_negative) {
+    return a_is_negative ? -1 : 1;
+  }
+
+  if (a_is_negative) {
+    return -UnsignedCompare(a, b);
+  }
+  return UnsignedCompare(a, b);
+}
+
+
+RawBigint* BigintOperations::AddSubtract(const Bigint& a,
+                                         const Bigint& b,
+                                         bool negate_b) {
+  ASSERT(IsClamped(a));
+  ASSERT(IsClamped(b));
+  Bigint& result = Bigint::Handle();
+  // We perform the subtraction by simulating a negation of the b-argument.
+  bool b_is_negative = negate_b ? !b.IsNegative() : b.IsNegative();
+
+  // If both are of the same sign, then we can compute the unsigned addition
+  // and then simply adjust the sign (if necessary).
+  // Ex: -3 + -5 -> -(3 + 5)
+  if (a.IsNegative() == b_is_negative) {
+    result = UnsignedAdd(a, b);
+    result.SetSign(b_is_negative);
+    ASSERT(IsClamped(result));
+    return result.raw();
+  }
+
+  // The signs differ.
+  // Take the number with small magnitude and subtract its absolute value from
+  // the absolute value of the other number. Then adjust the sign, if necessary.
+  // The sign is the same as for the number with the greater magnitude.
+  // Ex:  -8 + 3  -> -(8 - 3)
+  //       8 + -3 ->  (8 - 3)
+  //      -3 + 8  ->  (8 - 3)
+  //       3 + -8 -> -(8 - 3)
+  int comp = UnsignedCompare(a, b);
+  if (comp < 0) {
+    result = UnsignedSubtract(b, a);
+    result.SetSign(b_is_negative);
+  } else if (comp > 0) {
+    result = UnsignedSubtract(a, b);
+    result.SetSign(a.IsNegative());
+  } else {
+    return Zero();
+  }
+  ASSERT(IsClamped(result));
+  return result.raw();
+}
+
+
+int BigintOperations::UnsignedCompare(const Bigint& a, const Bigint& b) {
+  ASSERT(IsClamped(a));
+  ASSERT(IsClamped(b));
+  intptr_t a_length = a.Length();
+  intptr_t b_length = b.Length();
+  if (a_length < b_length) return -1;
+  if (a_length > b_length) return 1;
+  for (intptr_t i = a_length - 1; i >= 0; i--) {
+    Chunk digit_a = a.GetChunkAt(i);
+    Chunk digit_b = b.GetChunkAt(i);
+    if (digit_a < digit_b) return -1;
+    if (digit_a > digit_b) return 1;
+    // Else look at the next digit.
+  }
+  return 0;  // They are equal.
+}
+
+
+int BigintOperations::UnsignedCompareNonClamped(
+    const Bigint& a, const Bigint& b) {
+  intptr_t a_length = a.Length();
+  intptr_t b_length = b.Length();
+  while (a_length > b_length) {
+    if (a.GetChunkAt(a_length - 1) != 0) return 1;
+    a_length--;
+  }
+  while (b_length > a_length) {
+    if (b.GetChunkAt(b_length - 1) != 0) return -1;
+    b_length--;
+  }
+  for (intptr_t i = a_length - 1; i >= 0; i--) {
+    Chunk digit_a = a.GetChunkAt(i);
+    Chunk digit_b = b.GetChunkAt(i);
+    if (digit_a < digit_b) return -1;
+    if (digit_a > digit_b) return 1;
+    // Else look at the next digit.
+  }
+  return 0;  // They are equal.
+}
+
+
+RawBigint* BigintOperations::UnsignedAdd(const Bigint& a, const Bigint& b) {
+  ASSERT(IsClamped(a));
+  ASSERT(IsClamped(b));
+
+  intptr_t a_length = a.Length();
+  intptr_t b_length = b.Length();
+  if (a_length < b_length) {
+    return UnsignedAdd(b, a);
+  }
+
+  // We might request too much space, in which case we will adjust the length
+  // afterwards.
+  intptr_t result_length = a_length + 1;
+  const Bigint& result = Bigint::Handle(Bigint::Allocate(result_length));
+
+  Chunk carry = 0;
+  // b has fewer digits than a.
+  ASSERT(b_length <= a_length);
+  for (intptr_t i = 0; i < b_length; i++) {
+    Chunk sum = a.GetChunkAt(i) + b.GetChunkAt(i) + carry;
+    result.SetChunkAt(i, sum & kDigitMask);
+    carry = sum >> kDigitBitSize;
+  }
+  // Copy over the remaining digits of a, but don't forget the carry.
+  for (intptr_t i = b_length; i < a_length; i++) {
+    Chunk sum = a.GetChunkAt(i) + carry;
+    result.SetChunkAt(i, sum & kDigitMask);
+    carry = sum >> kDigitBitSize;
+  }
+  // Shrink the result if there was no overflow. Otherwise apply the carry.
+  if (carry == 0) {
+    // TODO(floitsch): We change the size of bigint-objects here.
+    result.SetLength(a_length);
+  } else {
+    result.SetChunkAt(a_length, carry);
+  }
+  ASSERT(IsClamped(result));
+  return result.raw();
+}
+
+
+RawBigint* BigintOperations::UnsignedSubtract(const Bigint& a,
+                                              const Bigint& b) {
+  ASSERT(IsClamped(a));
+  ASSERT(IsClamped(b));
+  ASSERT(UnsignedCompare(a, b) >= 0);
+
+  const int kSignBitPos = Bigint::kChunkSize * kBitsPerByte - 1;
+
+  intptr_t a_length = a.Length();
+  intptr_t b_length = b.Length();
+
+  // We might request too much space, in which case we will adjust the length
+  // afterwards.
+  intptr_t result_length = a_length;
+  const Bigint& result = Bigint::Handle(Bigint::Allocate(result_length));
+
+  Chunk borrow = 0;
+  ASSERT(b_length <= a_length);
+  for (intptr_t i = 0; i < b_length; i++) {
+    Chunk difference = a.GetChunkAt(i) - b.GetChunkAt(i) - borrow;
+    result.SetChunkAt(i, difference & kDigitMask);
+    borrow = difference >> kSignBitPos;
+    ASSERT((borrow == 0) || (borrow == 1));
+  }
+  // Copy over the remaining digits of a, but don't forget the borrow.
+  for (intptr_t i = b_length; i < a_length; i++) {
+    Chunk difference = a.GetChunkAt(i) - borrow;
+    result.SetChunkAt(i, difference & kDigitMask);
+    borrow = (difference >> kSignBitPos);
+    ASSERT((borrow == 0) || (borrow == 1));
+  }
+  ASSERT(borrow == 0);
+  Clamp(result);
+  return result.raw();
+}
+
+
+RawBigint* BigintOperations::MultiplyWithDigit(
+    const Bigint& bigint, Chunk digit) {
+  // TODO(floitsch): implement MultiplyWithDigit.
+  ASSERT(digit <= kDigitMaxValue);
+  if (digit == 0) return Zero();
+
+  Bigint& tmp = Bigint::Handle(Bigint::Allocate(1));
+  tmp.SetChunkAt(0, digit);
+  return Multiply(bigint, tmp);
+}
+
+
+void BigintOperations::DivideRemainder(
+    const Bigint& a, const Bigint& b, Bigint* quotient, Bigint* remainder) {
+  // TODO(floitsch): This function is very memory-intensive since all
+  // intermediate bigint results are allocated in new memory. It would be
+  // much more efficient to reuse the space of temporary intermediate variables.
+  ASSERT(IsClamped(a));
+  ASSERT(IsClamped(b));
+  ASSERT(!b.IsZero());
+
+  int comp = UnsignedCompare(a, b);
+  if (comp < 0) {
+    (*quotient) ^= Zero();
+    (*remainder) ^= Copy(a);  // TODO(floitsch): can we reuse the input?
+    return;
+  } else if (comp == 0) {
+    (*quotient) ^= One();
+    quotient->SetSign(a.IsNegative() != b.IsNegative());
+    (*remainder) ^= Zero();
+    return;
+  }
+
+  // High level description:
+  // The algorithm is basically the algorithm that is taught in school:
+  // Let a the dividend and b the divisor. We are looking for
+  // the quotient q = truncate(a / b), and
+  // the remainder r = a - q * b.
+  // School algorithm:
+  // q = 0
+  // n = number_of_digits(a) - number_of_digits(b)
+  // for (i = n; i >= 0; i--) {
+  //   Maximize k such that k*y*10^i is less than or equal to a and
+  //                  (k + 1)*y*10^i is greater.
+  //   q = q + k * 10^i   // Add new digit to result.
+  //   a = a - k * b * 10^i
+  // }
+  // r = a
+  //
+  // Instead of working in base 10 we work in base kDigitBitSize.
+
+  intptr_t b_length = b.Length();
+  int normalization_shift =
+      kDigitBitSize - CountBits(b.GetChunkAt(b_length - 1));
+  Bigint& dividend = Bigint::Handle(ShiftLeft(a, normalization_shift));
+  const Bigint& divisor = Bigint::Handle(ShiftLeft(b, normalization_shift));
+  dividend.SetSign(false);
+  divisor.SetSign(false);
+
+  intptr_t dividend_length = dividend.Length();
+  intptr_t divisor_length = b_length;
+  ASSERT(divisor_length == divisor.Length());
+
+  intptr_t quotient_length = dividend_length - divisor_length + 1;
+  *quotient ^= Bigint::Allocate(quotient_length);
+  quotient->SetSign(a.IsNegative() != b.IsNegative());
+
+  intptr_t quotient_pos = dividend_length - divisor_length;
+  // Find the first quotient-digit.
+  // The first digit must be computed separately from the other digits because
+  // the preconditions for the loop are not yet satisfied.
+  // For simplicity use a shifted divisor, so that the comparison and
+  // subtraction are easier.
+  int divisor_shift_amount = dividend_length - divisor_length;
+  Bigint& shifted_divisor =
+      Bigint::Handle(DigitsShiftLeft(divisor, divisor_shift_amount));
+  Chunk first_quotient_digit = 0;
+  while (UnsignedCompare(dividend, shifted_divisor) >= 0) {
+    first_quotient_digit++;
+    dividend ^= Subtract(dividend, shifted_divisor);
+  }
+  quotient->SetChunkAt(quotient_pos--, first_quotient_digit);
+
+  // Find the remainder of the digits.
+
+  Chunk first_divisor_digit = divisor.GetChunkAt(divisor_length - 1);
+  // The short divisor only represents the first two digits of the divisor.
+  // If the divisor has only one digit, then the second part is zeroed out.
+  Bigint& short_divisor = Bigint::Handle(Bigint::Allocate(2));
+  if (divisor_length > 1) {
+    short_divisor.SetChunkAt(0, divisor.GetChunkAt(divisor_length - 2));
+  } else {
+    short_divisor.SetChunkAt(0, 0);
+  }
+  short_divisor.SetChunkAt(1, first_divisor_digit);
+  // The following bigint will be used inside the loop. It is allocated outside
+  // the loop to avoid repeated allocations.
+  Bigint& target = Bigint::Handle(Bigint::Allocate(3));
+  // The dividend_length here must be from the initial dividend.
+  for (intptr_t i = dividend_length - 1; i >= divisor_length; i--) {
+    // Invariant: let t = i - divisor_length
+    //   then dividend / (divisor << (t * kDigitBitSize)) <= kDigitMaxValue.
+    // Ex: dividend: 53451232, and divisor: 535  (with t == 5) is ok.
+    //     dividend: 56822123, and divisor: 563  (with t == 5) is bad.
+    //     dividend:  6822123, and divisor: 563  (with t == 5) is ok.
+
+    // The dividend has changed. So recompute its length.
+    dividend_length = dividend.Length();
+    Chunk dividend_digit;
+    if (i > dividend_length) {
+      quotient->SetChunkAt(quotient_pos--, 0);
+      continue;
+    } else if (i == dividend_length) {
+      dividend_digit = 0;
     } else {
-      n = Utils::Minimum(a.NumberOfBits(), b.NumberOfBits());
-    }
-  } else if (a.IsNegative() || b.IsNegative()) {
-    if (tt[2]) {
-      rflip = rborrow = -1;
-    }
-    n = Utils::Maximum(a.NumberOfBits(), b.NumberOfBits());
-  } else {
-    if (tt[2]) {
-      n = Utils::Maximum(a.NumberOfBits(), b.NumberOfBits());
+      ASSERT(i + 1 == dividend_length);
+      dividend_digit = dividend.GetChunkAt(i);
+    }
+    Chunk quotient_digit;
+    // Compute an estimate of the quotient_digit. The estimate will never
+    // be too small.
+    if (dividend_digit == first_divisor_digit) {
+      // Small shortcut: the else-branch would compute a value > kDigitMaxValue.
+      // However, by hypothesis, we know that the quotient_digit must fit into
+      // a digit. Avoid going through repeated iterations of the adjustment
+      // loop by directly assigning kDigitMaxValue to the quotient_digit.
+      // Ex:  51235 / 523.
+      //    51 / 5 would yield 10 (if computed in the else branch).
+      // However we know that 9 is the maximal value.
+      quotient_digit = kDigitMaxValue;
     } else {
-      n = Utils::Minimum(a.NumberOfBits(), b.NumberOfBits());
-    }
-  }
-  bool aflip = false;
-  int acarry = 0;
-  if (a.IsNegative()) {
-    aflip = true;
-    acarry = 1;
-  }
-  bool bflip = false;
-  int bcarry = 0;
-  if (b.IsNegative()) {
-    bflip = true;
-    bcarry = 1;
-  }
-  for (int i = 0 ; i < n ; ++i) {
-    int ab = (a.Bit(i) ^ (aflip ? 1 : 0)) + acarry;
-    ASSERT(ab <= 2 && ab >= 0);
-    int bb = (b.Bit(i) ^ (bflip ? 1 : 0)) + bcarry;
-    ASSERT(bb <= 2 && bb >= 0);
-    int r = tt[(ab & 1) + ((bb & 1) << 1)];
-    r = r + rborrow;
-    ASSERT(r >= -1 && r <= 1);
-    if ((r ^ rflip) & 1) {
-      int status = BN_set_bit(TmpBN(), i);
-      ASSERT(status == 1);
-    }
-    acarry = ab >> 1;
-    bcarry = bb >> 1;
-    rborrow = r >> 1;
-    }
-  if (rborrow) {
-    int status = BN_set_bit(TmpBN(), n);
-    ASSERT(status == 1);
-  }
-  if (rflip) {
-    // FIXME(benl): can be changed to use BN_set_negative() on more
-    // recent OpenSSL releases (> 1.0.0).
-    ASSERT(!BN_is_zero(TmpBN()));
-    TmpBN()->neg = 1;
-  }
-  const Bigint& result = Bigint::Handle(Bigint::New(TmpBN()));
-  return result.raw();
-}
-
-
-RawSmi* BigintOperations::BitOpWithSmi(Token::Kind kind,
-                                       const Bigint& bigint,
-                                       const Smi& smi) {
-  ASSERT((kind == Token::kBIT_OR) || (kind == Token::kBIT_AND));
-  intptr_t smi_value = smi.Value();
-  intptr_t big_value = 0;
-  // We take Smi::kBits + 1 (one more bit), in case bigint is negative.
-  ASSERT((Smi::kBits + 1) <= (sizeof(big_value) * kBitsPerByte));
-  intptr_t num_bits = bigint.NumberOfBits();
-  int n = Utils::Minimum(num_bits, Smi::kBits + 1);
-  for (int i = n - 1; i >= 0; i--) {
-    big_value <<= 1;
-    big_value += bigint.Bit(i);
-  }
-  if (bigint.IsNegative()) {
-    big_value = -big_value;
-  }
-  intptr_t result;
-  if (kind == Token::kBIT_OR) {
-    result = smi_value | big_value;
-  } else {
-    result = smi_value & big_value;
-  }
-  ASSERT(Smi::IsValid(result));
-  return Smi::New(result);
-}
-
-
-RawBigint* BigintOperations::BitAnd(const Bigint& a, const Bigint& b) {
-  static bool tt_and[] = { false, false, false, true };
-  return BitTT(a, b, tt_and);
-}
-
-
-RawInteger* BigintOperations::BitAndWithSmi(const Bigint& bigint,
-                                            const Smi& smi) {
-  if (smi.IsNegative()) {
-    Bigint& other = Bigint::Handle(NewFromSmi(smi));
-    return BitAnd(bigint, other);
-  }
-  return BitOpWithSmi(Token::kBIT_AND, bigint, smi);
-}
-
-
-RawBigint* BigintOperations::BitOr(const Bigint& a, const Bigint& b) {
-  static bool tt_or[] = { false, true, true, true };
-  return BitTT(a, b, tt_or);
-}
-
-
-RawInteger* BigintOperations::BitOrWithSmi(const Bigint& bigint,
-                                           const Smi& smi) {
-  if (!smi.IsNegative()) {
-    Bigint& other = Bigint::Handle(NewFromSmi(smi));
-    return BitOr(bigint, other);
-  }
-  return BitOpWithSmi(Token::kBIT_OR, bigint, smi);
-}
-
-
-RawBigint* BigintOperations::BitXor(const Bigint& a, const Bigint& b) {
-  static bool tt_xor[] = { false, true, true, false };
-  return BitTT(a, b, tt_xor);
-}
-
-
-RawInteger* BigintOperations::BitXorWithSmi(const Bigint& bigint,
-                                            const Smi& smi) {
-  Bigint& other = Bigint::Handle(NewFromSmi(smi));
-  return BitXor(bigint, other);
-}
-
-
-RawBigint* BigintOperations::BitNot(const Bigint& bigint) {
-  const Bigint& one_bigint = Bigint::Handle(One());
-  const Bigint& result = Bigint::Handle(Add(bigint, one_bigint));
-  result.ToggleSign();
-  return result.raw();
-}
-
-
-int BigintOperations::Compare(const Bigint& a, const Bigint& b) {
-  return BN_cmp(a.BNAddr(), b.BNAddr());
+      // Compute the estimate by using two digits of the dividend and one of
+      // the divisor.
+      // Ex: 32421 / 535
+      //    32 / 5 -> 6
+      // The estimate would hence be 6.
+      DoubleChunk two_dividend_digits = dividend_digit;
+      two_dividend_digits <<= kDigitBitSize;
+      two_dividend_digits += dividend.GetChunkAt(i - 1);
+      DoubleChunk q = two_dividend_digits / first_divisor_digit;
+      if (q > kDigitMaxValue) q = kDigitMaxValue;
+      quotient_digit = static_cast<Chunk>(q);
+    }
+
+    // Refine estimation.
+    quotient_digit++;  // The following loop will start by decrementing.
+    Bigint& estimation_product = Bigint::Handle();
+    target.SetChunkAt(0, ((i - 2) < 0) ? 0 : dividend.GetChunkAt(i - 2));
+    target.SetChunkAt(1, ((i - 1) < 0) ? 0 : dividend.GetChunkAt(i - 1));
+    target.SetChunkAt(2, dividend_digit);
+    do {
+      quotient_digit = (quotient_digit - 1) & kDigitMask;
+      estimation_product ^= MultiplyWithDigit(short_divisor, quotient_digit);
+    } while (UnsignedCompareNonClamped(estimation_product, target) > 0);
+    // At this point the quotient_digit is fairly accurate.
+    // At the worst it is off by one.
+    // Remove a multiple of the divisor. If the estimate is incorrect we will
+    // subtract the divisor another time.
+    // Let t = i - divisor_length.
+    // dividend -= (quotient_digit * divisor) << (t * kDigitBitSize);
+    shifted_divisor ^= MultiplyWithDigit(divisor, quotient_digit);
+    shifted_divisor ^= DigitsShiftLeft(shifted_divisor, i - divisor_length);
+    dividend = Subtract(dividend, shifted_divisor);
+    if (dividend.IsNegative()) {
+      // The estimation was still too big.
+      quotient_digit--;
+      // TODO(floitsch): allocate space for the shifted_divisor once and reuse
+      // it at every iteration.
+      shifted_divisor ^= DigitsShiftLeft(divisor, i - divisor_length);
+      // TODO(floitsch): reuse the space of the previous dividend.
+      dividend = Add(dividend, shifted_divisor);
+    }
+    quotient->SetChunkAt(quotient_pos--, quotient_digit);
+  }
+  ASSERT(quotient_pos == -1);
+  Clamp(*quotient);
+  *remainder ^= ShiftRight(dividend, normalization_shift);
+  remainder->SetSign(a.IsNegative());
+}
+
+
+void BigintOperations::Clamp(const Bigint& bigint) {
+  intptr_t length = bigint.Length();
+  while (length > 0 && (bigint.GetChunkAt(length - 1) == 0)) {
+    length--;
+  }
+  // TODO(floitsch): We change the size of bigint-objects here.
+  bigint.SetLength(length);
+}
+
+
+RawBigint* BigintOperations::Copy(const Bigint& bigint) {
+  intptr_t bigint_length = bigint.Length();
+  Bigint& copy = Bigint::Handle(Bigint::Allocate(bigint_length));
+  for (intptr_t i = 0; i < bigint_length; i++) {
+    copy.SetChunkAt(i, bigint.GetChunkAt(i));
+  }
+  copy.SetSign(bigint.IsNegative());
+  return copy.raw();
+}
+
+
+int BigintOperations::CountBits(Chunk digit) {
+  int result = 0;
+  while (digit != 0) {
+    digit >>= 1;
+    result++;
+  }
+  return result;
 }
 
 }  // namespace dart